Schuijer - Analyzing Atonal Music - Pitch-Class Set Theory and Its Contexts-University of Rochester Press (2008) [PDF]

Zitiervorschau

Analyzing Atonal Music

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Eastman Studies in Music Ralph P. Locke, Senior Editor Eastman School of Music

Additional Titles in Music Analysis and on Music since 1920 Analyzing Wagner’s Operas: Alfred Lorenz and German Nationalist Ideology Stephen McClatchie Aspects of Unity in J. S. Bach’s Partitas and Suites: An Analytical Study David W. Beach CageTalk: Dialogues with and about John Cage Edited by Peter Dickinson Concert Music, Rock, and Jazz since 1945: Essays and Analytical Studies Edited by Elizabeth West Marvin and Richard Hermann Elliott Carter: Collected Essays and Lectures, 1937–1995 Edited by Jonathan W. Bernard Explaining Tonality: Schenkerian Theory and Beyond Matthew Brown The Music of Luigi Dallapiccola Raymond Fearn Music Theory and Mathematics Chords, Collections, and Transformations Edited by Jack Douthett, Martha M. Hyde, and Charles J. Smith Music Theory in Concept and Practice Edited by James M. Baker, David W. Beach, and Jonathan W. Bernard

Opera and Ideology in Prague: Polemics and Practice at the National Theater, 1900–1938 Brian S. Locke Pentatonicism from the Eighteenth Century to Debussy Jeremy Day-O’Connell The Pleasure of Modernist Music: Listening, Meaning, Intention, Ideology Edited by Arved Ashby Ruth Crawford Seeger’s Worlds: Innovation and Tradition in Twentieth-Century American Music Edited by Ray Allen and Ellie M. Hisama The Sea on Fire: Jean Barraqué Paul Griffiths The Substance of Things Heard: Writings about Music Paul Griffiths Theories of Fugue from the Age of Josquin to the Age of Bach Paul Mark Walker Variations on the Canon: Essays on Music from Bach to Boulez in Honor of Charles Rosen on His Eightieth Birthday Edited by Robert Curry, David Gable, and Robert L. Marshall

A complete list of titles in the Eastman Studies in Music Series, in order of publication, may be found at the end of this book.

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Analyzing Atonal Music Pitch-Class Set Theory and Its Contexts michiel schuijer

University of Rochester Press

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Copyright © 2008 Michiel Schuijer All rights reserved. Except as permitted under current legislation, no part of this work may be photocopied, stored in a retrieval system, published, performed in public, adapted, broadcast, transmitted, recorded, or reproduced in any form or by any means, without the prior permission of the copyright owner. First published 2008 University of Rochester Press 668 Mt. Hope Avenue, Rochester, NY 14620, USA www.urpress.com and Boydell & Brewer Limited PO Box 9, Woodbridge, Suffolk IP12 3DF, UK www.boydellandbrewer.com ISBN-13: 978–1–58046–270-9 ISBN-10: 1–58046–270-7 ISSN: 1071–9989 Library of Congress Cataloging-in-Publication Data Schuijer, Michiel. Analyzing atonal music : pitch-class set theory and its contexts / Michiel Schuijer. p. cm.—(Eastman studies in music, ISSN 1071–9989 ; v. 60) Includes bibliographical references and index. ISBN-13: 978–1-58046–270–9 (hardcover : alk. paper) ISBN-10: 1–58046–270–7 (alk. paper) 1. Musical analysis—Data processing. 2. Atonality. 3. Computer composition. 4. Musical pitch. 5. Set theory. I. Title. MT6.S35294A63 2008 781.2'67—dc22 2008030245 Portions of chapters 2 and 3 first appeared as “Was ist ein pitch-class set?” in Wahrnehmung und Begriff: Dokumentation des Internationalen Symposions, Freiburg 2.–3. Juni 2000 (Kassel: Gustav Bosse Verlag, 2000). A large part of chapter 3 was published as “T & I: A History of Abstraction” in Tijdschrift voor Muziektheorie 6/1 (February 2001). A catalogue record for this title is available from the British Library. This publication is printed on acid-free paper. Printed in the United States of America.

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in liefdevolle herinnering

Korien de Boer

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Contents List of Illustrations

ix

Preface

xv

Acknowledgments

xvii

1

Pitch-Class Set Theory: An Overture

2

Objects and Entities

29

3

Operations

49

4

Equivalence

84

5

Similarity

130

6

Inclusion

179

7

“Blurring the Boundaries”: Analysis, Performance, and History

218

8

Mise-en-Scène

236

Reference List

279

Index

293

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Illustrations Examples 1.1 1.2

Obvious and non-obvious combinations of tones in the opening measures of Arnold Schoenberg’s Piano Piece, Op. 23, no. 3

5

Igor Stravinsky, Pas de Deux from Agon. An analysis of the opening sonic field, mm. 411–13

8

1.3

A device of progression. The Webern-group

10

1.4

The octatonic scale as the common ground between the materials of Schoenberg’s piano piece and Stravinsky’s Pas de Deux

13

Heinrich Schenker’s analysis of a song by Robert Schumann: “Aus meinen Tränen sprießen” from Dichterliebe

14

2.1

Pitch numbers

30

2.2

Conversion of pitch numbers to pitch-class (PC) numbers

30

2.3

Pitch intervals

37

2.4

Conversion of pitch intervals to pitch-interval classes (PICs)

37

2.5

PICs are order-dependent

38

2.6

PICs associated with PC pairs

39

2.7

PICs and interval classes according to Allen Forte

39

2.8

Schoenberg, Piano Piece, Op. 11, no. 1, mm. 49–50

42

2.9

Instances of a pitch-class set. An example from Joseph Straus’s Introduction to Post-Tonal Theory

42

The transposition of a melody. A musical example from Heinrich Christoph Koch’s Musikalisches Lexikon

53

Twelve-tone transposition. The beginning of the prelude from Schoenberg’s Suite for Piano, Op. 25

54

PC-set transposition in Anton Webern’s Movement for String Quartet, Op. 5, no. 5

56

Instances of T9 in Webern’s Opus 5, no. 5

58

1.5

3.1 3.2 3.3 3.4

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x



3.5

illustrations Excerpts from Webern’s Opus 5, no. 5, selected by Allen Forte in The Atonal Music of Anton Webern

61

The inversion of a fugue subject. An example from Nicola Vicentino’s L’Antica Musica ridotta alla moderna prattica

63

The inversion of a fugue subject around B. An example from Gioseffo Zarlino’s Le Istitutioni harmoniche

63

The inversion of a fugue subject according to Giovanni Maria Bononcini

65

Friedrich Wilhelm Marpurg’s derivation of inverted forms of the major and minor scales

66

Tonally modified inverted forms of subjects in fugues by Johann Sebastian Bach

66

3.11

Axial symmetry in Webern’s Variations for Piano, Op. 27

70

3.12

The symmetrical arrangement of PCs in the closing section of Schoenberg’s Piano Piece, Op. 23, no. 3

71

Inversionally related PC sets and symmetrical voice leading in Webern’s Piece for Orchestra, Op. 6, no. 3

72

3.14

Inversion applied to the division of an interval (major sixth)

72

3.15

Inversionally related pitch intervals with respect to a referential major sixth

73

3.16

The major sixth as a referential interval in Luciano Berio’s O King

76

3.17

Multiplication in PITCH

77

3.18

Interval expansion in Bela Bartók’s Music for Strings, Percussion and Celesta

79

3.19

Evolution of the cycle-of-fourths and cycle-of-fifths transforms

80

4.1

PC sets extracted by Allen Forte from Schoenberg’s Little Piano Pieces, Op. 19, no. 2

90

4.2

A part of Forte’s 1964 table of set-classes

93

4.3

Three instances of twelve-tone combinatoriality

95

4.4

The source set of Schoenberg’s Piano Piece, Op. 33a

96

4.5

“Z-related” pitch-class sets

99

4.6

Successive-interval arrays

101

3.6 3.7 3.8 3.9 3.10

3.13

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illustrations 4.7



xi

The opening of the third movement of Schoenberg’s Fourth String Quartet, Op. 37

103

4.8

A part of Forte’s 1973 table of set-classes

108

4.9

The first two bars of Schoenberg’s Piano Piece, Op. 33a

112

4.10

John Rahn’s analysis of the opening measures of Wagner’s Tristan und Isolde

112

4.11

Class types of pitch sets

125

4.12

PC set equivalence and sonic difference (Berg and Stravinsky)

127

5.1

Similarity relations in the Seventh Piano Sonata, Op. 64. by Alexander Scriabin. An analysis by Allen Forte

135

5.2

Ernst Krenek’s chord classification

138

5.3

Paul Hindemith’s chord classification

140

5.4

Harmonisches Gefälle (Hindemith)

143

5.5

Maximally similar PC sets according to Forte (Webern, Movement for String Quartet, Opus 5, no. 4)

145

5.6

Richard Teitelbaum’s computation of the “similarity index” (s.i.)

152

5.7

Vector subtraction

154

5.8

Maximal similarity in terms of REL2. Six pairs of PC sets singled out by David Rogers

171

A segmentation of the main melody of Gabriel Fauré’s Élégie for Cello and Piano, Op. 24

182

6.2

Members of a contour class

183

6.3

Segmentation and inclusion in Arnold Schoenberg’s Piano Piece, Op. 11, no.1, mm. 1–11

184

The representation of set-class “6–16” in Schoenberg’s Opus 11, no. 1

186

6.5

The representation of set-class “6-Z10” in the same piece

188

6.6

A family of subsets

192

6.7

A family of supersets

192

6.8

The inclusion of families of subsets and supersets

192

6.9

An analysis of the march section of Claude Debussy’s Fêtes

196

6.1

6.4

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6.10

illustrations

A tonal progression in Messiaen’s Mode 2, the octatonic. An example from Le Banquet Céleste (1928)

201

6.11

The “Petrushka chord”

202

6.12

Johann Sebastian Bach, Nun lob, mein Seel, den Herren, BWV 17/7

203

6.13

Gottfried Weber’s chart of key relationships

204

6.14

Set-complex relations

208

6.15

The set complex about “5–33”/“7–33”

210

6.16

Analysis of Webern’s Bagatelle for String Quartet, Op. 9, no. 5

211

6.17

Families of chords in C major and C minor, projected on Weber’s chart of key relationships

214

Harmonic dualism: the symmetrical alignment of major and minor keys

216

Allen Forte’s and Richard Taruskin’s interpretations of a passage from Stravinsky’s Rite of Spring

226

Nicholas Cook’s analysis of Schoenberg’s Little Piano Piece, Op. 19, no. 6

231

Rudolph Réti’s reading of the opening measures of Mozart’s Symphony No. 40, K. 550

233

Anton Webern, Piece for Violin and Piano, Op. 7, no. 1, represented in the Ford-Columbia language (DARMS)

243

Webern’s Short Piece for Cello and Piano, Op. 11, no. 1, mm. 2–3. A segmentation following the rules of Forte’s score-reading program

245

Mm. 1–3 of Webern’s Opus 11, no. 1 as segmented by Forte in 1965

247

Dyadic invariance in Schoenberg’s Third String Quartet, Op. 30

253

6.18 7.1 7.2 7.3 8.1 8.2

8.3 8.4 Tables 1.1

Pitch-class set theory in the curricula of twenty teaching institutions in the United States and Canada (1996)

19

2.1

The sum total of PC sets of successive cardinal numbers

47

2.2

PIC content

48

2.3

Absolute-PIC content

48

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illustrations



xiii

3.1

An operation as defined by PC set theory

51

3.2

Milton Babbitt’s “group multiplication table”

51

3.3

The multiplication (mod 12) of PC values by factors from 1 through 11

77

4.1

Determination of the prime form of a PC set

106

4.2

The numbers of set-classes generated by three canonical transformation groups

115

4.3

Early comprehensive chord counts in the twelve-tone universe

117

4.4

Fritz Heinrich Klein’s algorithm applied to dyads and triads

118

5.1

Distributive patterns in the relation 5:1 (Forte: second-order maximal similarity)

146

Distributive patterns in the relation 5:1 (Forte: first-order maximal similarity)

147

Distributive patterns in the relation 4:2 (Forte: first-order maximal similarity)

147

5.4

Distributive patterns in the relation 6:4 (Forte: minimal similarity)

149

5.5

Distributive patterns in the relation 4:6 (Forte: minimal similarity)

149

5.6

Distributive patterns of PC sets with different numbers of elements

151

5.7

Distributive patterns in the relation 3:3. The difference is spread over six interval classes

156

Distributive patterns in the relation 3:3. The difference only involves two interval classes

156

5.9

The partition function of two PC sets

159

5.10

The partition function of two PC sets

159

5.11

The constant ratios between the subset classes of all cardinal numbers in David Rogers’s six maximally similar PC set pairs

170

5.2 5.3

5.8

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Preface This book contains a musicological reflection on a body of music theory covering the second half of the twentieth century, early beginnings and late developments included. Few musicologists have concerned themselves with pitch-class set theory. This is not surprising. Pitch-class set theory is a product of music theory’s remarkable rise (or renaissance) as an independent academic discipline in the United States—independent, that is, from musicology. A theory of pitch relations in post-tonal music, it has chimed in with a deep interest in matters of musical structure shown by composers and music theorists. And as far as the study of musical works from the past is concerned, it has viewed the score as its primary source of information. This goes against the grain of the musicologist who, instead of isolating music, seeks to place it in a larger context—social, cultural, historical, or otherwise. However, I see no reason why this musicologist should avoid dealing with pitch-class set theory altogether. Although it has developed into a highly distinct area of competence, it is part of the cultural environment in which music is made, heard, and studied. It has provided a vocabulary for the description of western art music from beyond the common practice period. It has established an image of this music as being tightly, though not always conspicuously, structured, opening new avenues for analytical practice. Today, it is taught at colleges and universities, and it is amply represented in the music literature. Seen thus, pitch-class theory definitely falls within the scope of a contextualist approach to music. By addressing its history, taking into account the variety of factors that contributed to its development, this study aims to establish a new rapport between musicology and music theory. As a consequence, the reader will find historical narrative side by side—and sometimes inextricably entwined—with theoretical and analytical discourse. It is my hope that this combination of perspectives will be to the enrichment of them both.

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Acknowledgments The research for this book was conducted at the University of Utrecht as part of a project funded by The Netherlands Organization for Scientific Research (NWO). Paul op de Coul was a kind and stimulating supervisor. I retain good memories of our spirited discussions. Rudolf Rasch provided expert guidance. Over the course of twelve years, he has shown unbroken confidence in the project, leaving me the freedom to shape it as I saw fit, and offering me good advice when this freedom became unsettling. I thank him wholeheartedly for his enduring support. Several other people were involved in the review and improvement of the manuscript: Henk Borgdorff, as a specialist in art theory and a close-enough friend not to spare me the least bit of his criticism; Tijn Borghuis, because of his expertise in mathematics and formal logic; and Paul van Emmerik, because of his scrupulous scholarship, his knowledge of twentieth-century music, and his interest in American culture. I also would like to acknowledge the perceptive remarks of Patrick van Deurzen, Erkki Huovinen, and Willem Wander van Nieuwkerk. For answers to specific questions I am indebted to Jaap van Benthem, David Fowler, Clemens Kemme, and Paul Scheepers. A special note of thanks is due to Barbara Bleij, who provided critical comments and editorial advice during the preparation of the manuscript, and to Gerard Goossens and Ger Vaessen of Ascolta Music Publishing, who set the music examples with great skill. This book was written directly in English. In matters of language, Johan Herrenberg was an invaluable and always available mentor. Furthermore, I was happy to receive English corrections from Ian Borthwick, Andrew Meyer, and Daniele Sahr, who read parts of the manuscript in their capacity as native speakers. Outside the circle of people who offered direct help in the process of writing, there are some whose names deserve mention as well. The person with whom I took the first steps to become knowledgeable about pitch-class set theory, in the early 1990s, was Diderik Wagenaar, my analysis teacher at the Royal Conservatory of The Hague. I recall with much pleasure the hours we spent on mastering an analytical vocabulary that was new to both of us. I also enjoyed my conversations and discussions with Allen Forte and Joseph Straus, which helped me in developing a balanced perspective on the theory. Allen kindly granted me permission for the use of the photo on the cover. The road to book publication can be long and hard, but I was fortunate enough to have such an outstanding colleague and friend in Jonathan Cross, whose enthusiasm for the project brought it to the attention of the University of

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xviii



acknowledgments

Rochester Press. And it was a privilege to work with URP’s friendly and professional staff. Special thanks to Eastman series editor Ralph Locke for his interest and advice, and to managing editor Katie Hurley for her patient guidance. I owe another debt to David McCarthy for his meticulous copyediting, and to Tracey Engel, for helping me through the last stage of the production phase. This project could only be finished with the support of two great families, who did everything in their capacity to mend the loss that hit us all very hard in 1998. In particular, I want to thank my parents, Jan Schuijer and Elly SchuijerBrandenburg, and my parents-in-law, Otto de Boer and Thoma de Boer-Campagne, for all their help and encouragement. I also want to mention my daughters, Jantien and Marleen: tiny tots when the project started, strong allies when it had to be finished under considerable pressure. Finally, my thoughts are with the person who, although she has died, has never ceased reading over my shoulder. This work is dedicated to her memory.

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Chapter One

Pitch-Class Set Theory: An Overture A Tale of Two Continents In the late afternoon of October 24, 1999, about one hundred people were gathered in a large rehearsal room of the Rotterdam Conservatory. They were listening to a discussion between representatives of nine European countries about the teaching of music theory and music analysis. It was the third day of the Fourth European Music Analysis Conference.1 Most participants in the conference (which included a number of music theorists from Canada and the United States) had been looking forward to this session: meetings about the various analytical traditions and pedagogical practices in Europe were rare, and a broad survey of teaching methods was lacking. Most felt a need for information from beyond their country’s borders. This need was reinforced by the mobility of music students and the resulting hodgepodge of nationalities at renowned conservatories and music schools. Moreover, the European systems of higher education were on the threshold of a harmonization operation. Earlier that year, on June 19, the governments of 29 countries had ratified the “Bologna Declaration,” a document that envisaged a unified European area for higher education. Its enforcement added to the urgency of the meeting in Rotterdam. However, this meeting would not be remembered for the unusually broad representation of nationalities or for its political timeliness. What would be remembered was an incident which took place shortly after the audience had been invited to join in the discussion. Somebody had raised a question about classroom analysis of twentieth-century music, a recurring topic among music theory teachers: whereas the music of the eighteenth and nineteenth centuries lent itself to general analytical methodologies, the extremely diverse repertoire of the twentieth century seemed only to invite ad hoc approaches; how could the analysis of

1. I have checked my account of the events described here with Patrick van Deurzen, then coordinator of the Fourth European Music Analysis Conference. I am grateful for his willingness to share his memories with me. For another account, see Cross 2000, 35. For general reports of the conference mentioned, see Maas 2000 and Bernnat 2000.

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pitch-class set theory: an overture

works from this repertoire be tailored to the purpose of systematical training without placing too much emphasis on particular styles of composition? A late visitor entered the room, and seated himself on a chair in the middle of the front row. He listened for a while to the discussion, his face expressing growing astonishment. Then he raised his hand and said: “You guys are discussing methods of analyzing twentieth-century music. Why don’t you talk about pitch-class sets?”

He was American. The chairman, a professor from the Sorbonne, was quick to respond: “We do not talk about pitch-class sets, because we do not hear them!”

This dismissive response was not effective. The visitor said it was not his intention to discuss whether pitch-class sets could be heard, or had been used by composers. He wanted to stress their value as an analytical tool. Pitch-class set theory, he argued, enabled students to come to grips with complex music of the posttonal era. It was successfully applied in the United States, and he could hardly believe that it was not taught anywhere in Europe! Indeed it wasn’t, as he could infer from the reactions of some of the Europeans present who appeared to be knowledgeable about the theory. They were willing to admit that it worked well in some cases—for example, it was helpful in clarifying the pitch structure of an early atonal composition by Anton Webern— but they made it clear they would never encourage its general use. In their view, it would force most music onto a Procrustean bed of preconceived relations. The meeting had turned into a confrontation between European and American approaches to music analysis. And in the absence of a unified European methodology, the participants ended up debating American practice only. Although this turn took many by surprise—most of all the unhappy chairman— it was to be expected at some point. For decades, the professional discourse on music theory and analysis2 had been dominated by Americans. Whereas in most European countries music theory offered training in basic musical skills as part 2. A note is necessary on my use of the terms “music theory” and “music analysis.” These do not signify two separate disciplines. “Music theory” often functions as an umbrella term comprising music analysis and other subjects, such as harmony and counterpoint. However, it can also refer to a framework of concepts and/or protocols underlying the analysis of musical works. It is in this sense that I will use it. The corollary that music analysis is “applied music theory” is not commonly embraced, as will be clear from what follows below. European music theorists, including the British, often distinguish themselves from their American counterparts by stressing the priority of analysis over theory. Characteristically, one British study is entitled Music Analysis in Theory and Practice (Dunsby and Whittall 1988), whereas the title of an American study is Music Theory in Concept and Practice (ed. James M. Baker, David W. Beach, and Jonathan W. Bernard, 1997). For a historic discussion of the relation between theory and analysis, see Cone 1967 and Lewin 1969.

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pitch-class set theory: an overture



3

of the conservatory tradition, in the United States it had grown into an academic discipline as well. Traditional pedagogy had been supplemented by an intensive research program, which had made the Americans pre-eminent in the production of music-theoretical knowledge. This pre-eminence was strongly apparent in the field of music analysis, mainly because of the American adherence to two distinctive bodies of analytical theory: the theory of Heinrich Schenker (1869–1935) for tonal music, and pitch-class set theory3—commonly identified with the name of Allen Forte (b. 1926)—for atonal music. On these twin pillars Forte himself based the first American graduate program in music theory (at Yale University) in 1965.4 A division of analytical practice along the lines of these two theoretical bodies is crude, of course. And today, as other perspectives on music analysis have gained prominence in the United States, it is also outdated. However, it has been of crucial importance to the image of American music theory. The Canadian composer and music theorist William Benjamin has—perhaps unwillingly— helped spread this image by addressing, in an article, the “curious marriage of convenience” between what he saw as two contradictory streams of thought (Benjamin 1981, 171). This metaphor came to the notice of the musicologist Joseph Kerman, who cited it in Contemplating Music, his widely read, critical account of postwar musical studies.5 It confirmed Kerman’s own critical assessment of American music theory: “a small field built around one or two intense, dogmatic personalities and their partisans” (Kerman 1985, 63). Meanwhile, Kerman saw more sense in the “marriage” than Benjamin, as did Patrick McCreless, a commentator of later years: Schenkerian theory and the current theories of atonal and twelve-tone music, however mutually exclusive in terms of the repertoires that constitute their objects, both share a value system that explicitly privileges rigor, system, and theory-based analysis and implicitly share an aesthetic ideology whereby analysis validates masterworks that exhibit an unquestioned structural autonomy. (McCreless 1997, 32)

In Rotterdam, the dual image of American analytical practice was prevalent in the minds of most conference-goers, and—much more importantly—so was its privileging of “rigor” and “system.” When the chairman said: “We do not hear these pitch-class sets,” he appealed to a common concern about “theory-based 3. Some people call it “set theory” (without the prefix “pitch-class”), reducing its name to that of its mathematical model: Georg Cantor’s Mannigfaltigkeitslehre or Mengenlehre. Others speak of “set-class theory.” Although these alternative names are sometimes used for good reasons—which will become obvious as we proceed—I have decided to stick to the designation “pitch-class set theory” throughout this study. 4. Significantly, this program was not founded within Yale University’s professional music school, but within the Department of Music, where it was associated with graduate studies in musicology. Forte 1998b, 10. 5. Published in England under the title Musicology.

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pitch-class set theory: an overture

analysis” as practiced in the United States—a concern shared not only by those European music theorists involved in the teaching of musical performers, but also by American scholars like Kerman. He made himself the spokesman for those who suspected such analysis of seeking to demonstrate the workings of the theory rather than to reveal what is special about the music. There may well have been grounds for this suspicion, although in Europe it cannot always be dissociated from an anxiety about American hegemony over the discipline. After all, Europe has its own breeds of analytical systems, and Heinrich Schenker lived in Vienna! The European Music Analysis Conferences, for all the work toward an entente cordiale between the national schools of musical analysis, have also been motivated by a desire to “remind the theoretical community of its European heritage” (Street 1990, 357). Pitch-class set theory—which is not part of that heritage—already divided Europeans and Americans at the very first of these conferences, held in Colmar in 1989. Then, Allen Forte crossed swords with the Belgian music theorist Célestin Deliège, who, in a preparatory paper, had criticized the theory for lacking explanatory power (Deliège 1989). More than a decade later, the controversy was still alive. What was it that aroused such persistent antagonism?

Paradigmatic Pieces Pitch-class set theory addresses the notion of musical coherence. There are many ways in which music can be said to cohere. For example, it may be in one key, or be in a familiar form, it may obey a model of rhetoric, or it may have been set to a single text. Pitch-class set theory seeks coherence in the relations between various combinations of tones. A work susceptible to this approach is Arnold Schoenberg’s Piano Piece Op. 23, no. 3 (1923). Example 1.1a shows its beginning.6 The piece first presents a single melodic line, involving the pitches B♭4, D4, E4, B3, and C♯4 respectively.7 The same melody, now starting on F3, is formed by the successive bass tones in measures 2–3. And in measure 4 there is an inverted form of this melody starting on D5 in the top voice. These three statements are isolated and bracketed in the lower half of example 1.1a. 6. This piece is a favorite among analysts, especially those concerned with Schoenberg’s path toward serial composition. Analyses of these opening measures have been provided by Stein (1925), Perle (1962), De Leeuw (1977), and Simms (2000), among others. I do not pretend to add anything new to the work of these authors. It is my sole intention to give the reader a basic understanding of what pitch-class set theory is about, in anticipation of the more systematic presentation in subsequent chapters. 7. With regard to the letter names of pitches, I follow the rule proclaimed by the Acoustical Society of America. According to that rule, a pitch is identified by a capital letter, a sharp or a flat if so required, and by the number of its octave range. Octave ranges cover the pitches from any C through the next higher B, and are numbered from low to high. Middle C is C4.

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pitch-class set theory: an overture 1

Langsam ( = ca 54)

2

dolce



5

3

poco rit. 5

4

6

3

2

3

5 4

Example 1.1a. The beginning of Schoenberg’s Piano Piece, Op. 23, no. 3. Obvious combinations of tones: three statements of a five-tone melody. FÜNF KLAVIERSTÜCKE, OP. 23, by Arnold Schoenberg. Copyright © Edition Wilhelm Hansen AS. International Copyright Secured. All Rights Reserved. Used by Permission.

So far, an analysis of this piece does not seem to require a language other than that used for any work featuring imitative counterpoint. The melody actually functions like the subject of a fugue, particularly since it is first answered “at the fifth.” The rhythm of the melody is slightly different at the second entry, and then is transformed beyond recognition at the third. Still, imitation—another device to achieve musical coherence—sanctions the combinations shown in example 1.1a. It is obvious that these tones belong together. On closer scrutiny, however, less obvious groups of tones appear to relate to the opening melody as well. These are shown in example 1.1b. Each of them can be transformed into another statement of this melody, in recto or in inverso

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6



pitch-class set theory: an overture α 2

3

δ

β

ε

γ 2

3

Example 1.1b. Non-obvious combinations related to the opening melody of Schoenberg’s Piano Piece, Op. 23, no. 3. β:

γ:

8va

δ:

ε:

8va

Example 1.1c. New statements of the melody derived from the combinations in ex. 1.1b.

(ex. 1.1c). We only need to reorder the pitches and replace some of them by a higher or lower octave. Although these combinations are not musically articulated, we can define them on the basis of their hidden “substance.” Thus, a dense web of relations emerges, in which the opening melody imposes its structure on the harmonies and accompanying figures. We can continue this exploration, and add to the first statement of the melody the first tone of the second statement (F3), which sounds between the former’s B3

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pitch-class set theory: an overture

2



7

3

Example 1.1d. Non-obvious six-tone combinations.

and C♯4. Now we have six tones instead of five (see ex. 1.1d, first 4 beats). These six tones form a shape that similarly recurs a couple of times in the first measures. Let us look at the second five-tone combination in example 1.1b, which is marked β. If we enlarge it to include E♭3—a tone originally assigned to the next combination (γ)—we have another instance of our six-tone combination (as shown in ex. 1.1d, in the center). It appears that β and γ are connected in the same way as the first two statements of the melody: γ answers β “at the fifth.” In both cases, the sixtone combinations result from the progression from one five-tone combination to the next. In measure 3, the progression from δ to ε is not a “fifth progression”; however, it yields a similar six-tone combination (see ex. 1.1d, to the right). The brief analysis above reveals a tightly woven pattern of recurrence. This tightness is not characteristic for each measure of this piece. For example, measure 4 is already less “close-knit” than the previous measures.8 But it is the coherence of measures 1–3 that concerns us here. What is it that recurs so consistently throughout these measures? We have noted several related entities, but what do these have in common? They can all be traced to the opening melody, but some of them are not in the slightest way a representation of that melody. What they actually represent is a basic property of the melody’s pitch material. Pitch-class set theory is concerned with such properties. A pitch-class set is an abstract concept of a combination of musical tones. It does not include the durations and octave ranges of these tones; nor does it include the order in which they appear. It reduces the combination to an unordered set of collective pitch designations (“pitch classes”). By using this concept, analysts can trace relations that are independent of shape and actual pitch content.

8. I agree with Bryan Simms, who writes: “As the piece progresses, dividing the entire texture into variants of the initial shape becomes ever more difficult and requires analytic strategies that fully bypass the musical context and, in all likelihood, the composer’s intentions” (Simms 2000, 198). It is by no means foreign to the idea of contrapuntal writing to vary the pace of thematic development. In a fugue, thematic passages often alternate with freely constructed episodes. This holds true for Opus 23, no. 3 as well.

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8



pitch-class set theory: an overture Adagio

= 112

3

Violino Solo

marc. espress. 411

412

413 3

I marc. espress.

Violini 3

3

3

II

arco

3

Viola ma marc.

tutti

arco

Violoncelli

pizz. Contrabassi poco

Example 1.2a. Stravinsky, Pas de deux from Agon. The opening sonic field, mm. 411–13. © Copyright 1957 by Hawkes & Son (London) LTD. Reprinted by kind permission of Boosey & Hawkes Music Publishers LTD. 1

2

1

3

Example 1.2b. Abstract forms of the motifs in the first-violin part, mm. 412–13.

Another composition in which such relations have been found is Igor Stravinsky’s ballet Agon (completed 1957). Example 1.2a shows the first three measures of the “Pas de deux” (mm. 411–13). These have a more open texture than the first three measures of Schoenberg’s Opus 23, no. 3. Rather than a contrapuntal fabric, they present a sonic field. This field is built from several tone combinations. Most conspicuous are the motifs played by the first violins in measures 412–13, which mark the end of these introductory measures. We can project the tones of each motif in a single octave range, and then place them in ascending or descending order (ex. 1.2b). It appears that the motifs reduce to different forms: a succession of a minor and major second (in chromatic steps: 1 and 2), and a succession of a minor second and a minor third (in chromatic steps: 1 and 3). One motif cannot be transformed into a statement of the other by the rearrangement and octave displacement of pitches.

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pitch-class set theory: an overture 411

3

412

immobile segment

3

3



9

413

3

411

arco

412

413

pizz.

mobile segment

Example 1.2c. The mobile and immobile segments of the sonic field.

However, each of these motifs reflects the structure of a segment of the sonic field from which they emerge. Example 1.2c shows that this field consists of a mobile and an immobile segment. The former is a progression involving the pitches C3 (mm. 411–12, violas), D♭3 (mm. 412–13, violas and cellos), and E♭2 (m. 413, double basses); the latter consists of the repeated D4s, B♭3s and C♭5s played by second violins and violas. The first motif reflects the structure of the mobile segment, and the second that of the immobile segment. The motifs and segments, it should be noted, partition the total chromatic; the sonic field grows into a twelve-tone field. Significantly, each of the two pieces presented above was written just before or at the time of its composer’s turn to twelve-tone serialism. Neither of them is strictly twelve-tone, but they both display a remarkable consistency in the use of pitch intervals. Henri Pousseur (1972) has noted that Stravinsky’s “Pas de deux” employs a group of intervals taken from the twelve-tone series of Anton Webern’s Variations for Orchestra, Op. 30—a work Stravinsky is known to have admired (ex. 1.3a). This group of intervals—scalewise in chromatic steps: 1 2 1—combines the two intervallic patterns that emerged from our analysis of the first three measures (see ex. 1.3b). Indeed, from measure 414 onwards we can see that it is repeatedly generated by adjacent or overlapping occurrences of these smaller patterns (ex. 1.3c and 1.3d). It thus seems to function as a device of progression, similarly to the six-tone combination in Schoenberg’s Piano Piece, Op. 23, no. 3 (ex. 1.1b and 1.1d above). In any case, it is the product of an intervallic consistency reminiscent of—if not inspired by—twelve-tone serialism. It is not by accident that two musical works approximating dodecaphonic practice reflect so well the focus of interest of pitch-class set theory, for this theory is itself an outgrowth of the theory of twelve-tone serialism. Its pioneer was the composer Milton Babbitt (b. 1916), who had dedicated himself to extending, and strengthening the theoretical underpinning of, Schoenberg’s twelve-tone technique. Babbitt set the example for the confluence of music theory and contemporary composition under the aegis of the university, America’s principal employer of composers. He taught at the music department of Princeton University,

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pitch-class set theory: an overture

Example 1.3a. A source of inspiration: a recurring group of intervals in the twelvetone series of Webern’s Variations for Orchestra, Op. 30. 1

2

1

3

1

2

1

3

Example 1.3b. The derivation of this group from the abstract forms of the motifs in ex. 1.2b.

which owes to him its reputation as an American center of musical innovation. His theory proceeded from a mathematical description of the familiar twelvetone operations (transposition, inversion and retrogradation). A good deal of its conceptual apparatus was adopted by Allen Forte, who tailored it to the analysis of non-serial music. For example, whereas Babbitt used the word “set” for “twelve-tone series,” Forte also applied it to combinations involving less than twelve tones and showing no definite ordering—combinations such as found in Schoenberg’s piano piece and Stravinsky’s “Pas de deux.” The relations shown in example 1.1 through 1.3 are instances of what Schoenberg saw as a continuous process of variation, the “endless reshaping of a basic shape” (Schoenberg 1984, 290). The term “basic shape” is not very specific. It might signify a twelve-tone series. If, alternatively, we replace it by “pitch-class set,” we interpret it more generally—in other words, we impose fewer constraints on the process of its “reshaping.” However, pitch-class sets do not necessarily function as “basic shapes” in the Schoenbergian sense. The recurrence of a basic shape is not the only measure of musical coherence posited by pitch-class set theory. There are other categories of relations between groups of tones: relations between their interval contents, relations based on common tones, and relations delineating hierarchical levels of structure, to mention but a few. Pitchclass set theory imposes no limit on the number and nature of relations between tone combinations.

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Vln. 3

solo 414

415

Vla.

pizz. 3 arco

Vc.

poco

arco

DB

poco

1: 3

2:

F G A A

3:

A A B C

F G A A

3

3

4:

B C D E

Example 1.3c. Stravinsky, Pas de deux from Agon. Four occurrences of the Webern-group in mm. 414–15. The circled pitches form the same intervallic patterns as those extracted from the motifs in mm. 412–13. © Copyright 1957 by Hawkes & Son (London) LTD. Reprinted by kind permission of Boosey & Hawkes Music Publishers LTD.

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12



pitch-class set theory: an overture pizz.

Vln. I/II 459

460

Va. /Vc.

arco

461

462

arco

pizz.

Example 1.3d. Stravinsky, Pas de deux from Agon, mm. 458–62. The Webern-group as a device of progression. Each three-tone chord (circled) is complemented by a member of the next, or the previous chord. © Copyright 1957 by Hawkes & Son (London) LTD. Reprinted by kind permission of Boosey & Hawkes Music Publishers LTD.

For example, the theory also deals with the question of how to conceive a relation between the two motifs in measures 412–13 of Stravinsky’s Agon (ex. 1.2). It has provided ways to measure the degree to which such motifs are similar or different. (As shown in example 1.2b, the abstract forms of these motifs both contain the intervals of a minor second and a minor third. The remaining intervals are different: in one we find a major second, in the other a major third.) Even if there is not a single basic shape underlying the different melodies, motifs, and simultaneities, a composition may be shown to make a consistent use of particular tone combinations. These combinations may feature the same intervals, or follow one another according to a rationale of progression. We can take yet another view of the two motifs in Agon and stress that they are both abstractly contained in the four-tone combination derived from Webern’s

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pitch-class set theory: an overture



13

Schoenberg (‘melody’)

1

2

1 Stravinsky

3

Example 1.4. The octatonic scale as the common ground between the materials of Schoenberg’s piano piece and Stavinsky’s Pas de Deux.

Opus 30 (as can be seen from ex. 1.3b). Each of them “descends” from the latter. In turn, this four-tone combination “descends” from combinations of yet higher magnitudes. In this way, we can plot entire family trees of pitch-class sets. It is interesting that the five- and six-tone combinations found in Schoenberg’s Piano Piece, Op. 23, no. 3, and the three- and four-tone combinations found in the “Pas de deux” from Agon can be assigned to one family, since they all “descend” from the octatonic scale (ex. 1.4). The octatonic background is most conspicuous in measures 413–15 of the “Pas de deux,” where the overlapping three- and four-tone combinations actually complement this scale (ex. 1.3c). These family-like relations have enabled pitch-class set theory to account for higher levels of organization in a musical work, and to hypothesize principles of structure governing an entire repertoire of music. This aspect of the theory—its working toward a hierarchy of structural levels—is reminiscent of its tonal-music counterpart in the United States: the analytical theory of Heinrich Schenker.9 This is not surprising when we consider that Allen Forte was a devoted Schenkerian. In 1982, he and his former student Steven Gilbert published a much-consulted introduction to Schenker’s method (Introduction to Schenkerian Analysis); and throughout his career he has used it for the analysis of a wide range of music, including works from the late nineteenth and early twentieth centuries (Contemporary Tone-Structures, 1955; “Schenker’s Conception of Musical Structure,” 1959), and American popular songs from the 1930s and 40s (The American Popular Ballad of the Golden Era, 1995). Therefore, a brief digression on Schenker’s approach to tonal music is now appropriate.

A Short Detour to Schenker Schenkerian theory claims that harmonic progressions and voice-leading patterns governing single phrases of tonal music are also at the basis of entire 9. For a general reflection on the relationship between Schenkerian theory and pitch-class set theory, stressing their common roots in nineteenth-century organicism, see Hinton 1988. For an analysis combining pitch-class sets with a Schenkerian approach, see Forte 1988b.

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14



pitch-class set theory: an overture ∧ 3

∧ 2

∧ 1

I

V

I

Example 1.5a. A Schenkerian fundamental structure (Ursatz). Schumann, Dichterliebe, Op. 48II ∧ 3 b)

∧ 3

∧ 2

(= 3

2

∧ 2

∧ 1

1)

(Nbn)

( ) (8

7) 12

T

4

8

(Nbn)

(!)

(Wdh)

Vdg

I (A1

IV

Kons Dg.

V

I

V B

(

)

I A 2)

Example 1.5b. Heinrich Schenker’s analysis of a song by Robert Schumann: “Aus meinen Tränen sprießen” from Dichterliebe. This analysis was published in Schenker’s treatise Der Freie Satz (1935). © 1935 by Universal Edition A.G., Wien/UE 6869A.

compositions. Small-scale patterns thus mark stages in the unfolding of larger ones; they “prolong” these stages. Through a series of reductions a Schenkerian analysis retraces the process of the work’s unfolding to the “fundamental structure” (Ursatz), which is itself the prolongation of the tonic triad. This fundamental structure consists of two simultaneous parts: a fundamental line—a stepwise descent from the tonic third (or fifth, or octave) to the tonic—and a bass arpeggiation I V I that supports it (ex. 1.5a). It is important to note that the purpose of the analysis is the discovery, not of the fundamental structure, but of the way that leads from this fundament to the actual work. Example 1.5b has been taken from Schenker’s treatise Der Freie Satz (1935).10 It shows three successive reductions of Schumann’s song “Aus meinen Tränen sprießen” from the cycle Dichterliebe, Op. 48. Schenker himself did not waste many words on this analytical graph. However, it has provoked much comment from others. It has often been used as an example showing both the pros and cons of Schenker’s analytical method: the insights it can offer, and the problems it may raise. The lowest staff looks like a rough draft, with analytical markings indicating important tones and connections. A particular emphasis appears to be placed on 10.

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Translated in English as Free Composition (1979). The example is numbered 22a.

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pitch-class set theory: an overture



15

the first tone of the opening phrase (C♯5; beam, open note-head) and the first tone of the contrasting middle section (B4 in m. 8; beam). These tones form a large-scale progression: a descent from the third (^3) to the second (^2)of the A-major scale. The same progression starts in measure 12, which makes sense, as an altered recapitulation of the opening phrase begins here. But now the progression is shown to reach the tonic A4. This tone and the B4 that precedes it are represented by open note-heads, which suggests that they are more important than the corresponding tones in measure 4. The middle staff gives much less detail. Its function is to highlight the progression from C♯5 to B4, and later that from C♯5 to A4. The brief middle section (“Und wenn du mich lieb hast”) is shown here to prolong the B4 in measure 8, and to connect it to the C♯5 of the recapitulation. The middle staff thus reinforces the emphasis placed on the outer phrases. And we can now see why these phrases are emphasized: they contain the elements of the fundamental structure shown in ex. 1.5a. The upper staff represents a basic elaboration of this fundamental structure. It shows us that in Schumann’s song the descending third progression is first interrupted and then stated completely; the resolution of the treble and bass into the octave is suspended until just before the end.11 In Schenkerian theory, “interruption” is an over-arching concept. It covers a span of music that is composed of different parts. This can be a small span (like an eight-bar period) or a large one (like a movement in sonata form). In other words, the term “interruption” can be applied to different levels of structure. It epitomizes Schenker’s hierarchical approach to music analysis. There are two reasons to dwell on example 1.5b. First, Forte presented Schenker’s analytical graph in an article for Journal of Music Theory in 1959, and went into great detail to explain it. This article—entitled “Schenker’s Conception of Musical Structure”—was also important for other reasons, which will be clarified in chapter 8. Second, it may provoke the objection that it imposes an a priori structural model on the song. It thus reinforces the image of a “theory-based” or “system-oriented” analytical practice, an image that Schenkerian analysis shares with pitch-class set analysis. Joseph Kerman, in his commentary on both Schenker’s interpretation and Forte’s rendering of it, raised this objection (Kerman 1980).12 It is questionable whether the graph of example 1.5b provides a good example of Schenkerian analysis. In any case, Schumann’s song is a questionable example of a Schenkerian interruption. This concept—a premature halt, followed by a 11. It is unclear to me why Schenker chose not to beam the progression B–A in the last two measures, but did beam the corresponding progression in measure 15. It is obvious that the latter does not end the final phrase. 12. Kerman involved yet another analysis, which I take the liberty to exclude from consideration: an analysis published by Arthur Komar in 1971.

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16



pitch-class set theory: an overture

new start—is perfectly illustrated by classical themes in period form;13 in “Aus meinen Tränen sprießen,” however, the descending third progression starting on C♯5 is not interrupted at the end of the first phrase—as it would be in a period—but is stretched to the beginning of the next phrase. Furthermore, one remarkable thing about this song is the stealthy beginning of its recapitulation. The tonic third C♯5 is now the leading tone of a dominant seventh chord applied to D. As a consequence, it does not demand the same kind of attention that it received at the opening of the song. Neither the break in the continuity of the descending third progression in measure 8, nor the new start of that progression in measure 12 is musically articulated as such. Therefore, the term “interruption” does not seem to apply naturally to Schumann’s song.14 Schenker’s approach to Schumann’s song seems to be geared too much toward the validation of a general concept of tonal structure, ignoring what is inconsistent with that concept. But what is it that we value in an analysis? Forte selected this one as typical for Schenker’s achievement, which he summarized as follows: Schenker opened the way for a deeper understanding of musical structure with his discovery that the manifold of surface events in a given composition is related in specific ways to a fundamental organization. (Forte 1959, 4)

Forte liked the systematic, generalizing tendency of Schenker’s late output, and saw his own role in the light of this tendency: Although Schenker came very close to constructing a complete system, further refinement and amplification are required if it is to fulfill its promise. (ibid., 16)

In spite of the dangers of overgeneralization, we should consider the reason for imposing such concepts as the Schenkerian interruption: it strengthens the image of music theory as a domain of competence distinct from musicology, musical performance, and composition. Within these disciplines music theory had traditionally played a subservient role. It provides historical musicologists with the knowledge and tools to access musical sources from the past; it enables performers to maximize their awareness and control of musical processes; and

13. As defined by William Caplin: “Essential to the concept of the period is the idea that a musical unit of partial cadential closure is repeated so as to produce a stronger cadential closure” (Caplin 1998, 49). An interruption in the Schenkerian sense demands that the first unit end with a half cadence. 14. Forte did not make this point; nor did Kerman. This, however, does not detract from the value of their observations. Forte saw the G in measures 12–13—which turns the initial A-major chord into a dominant seventh chord—as resulting from a chromatically descending inner voice (Forte 1959, 24). Kerman stressed this tone’s expressive quality in relation to the word “klingen” (Kerman 1980, 326).

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pitch-class set theory: an overture



17

it offers composers a foundation for critical self-reflection. However, it does not stand on its own two feet. It was for analytical theories like those of Schenker and Forte to achieve that self-consistency (Forte’s “promise”), sanctioning interpretations that, from other disciplinary perspectives, may strike one as subversive rather than subservient.

Institutionalization and Criticism Judging from the reactions it elicited at the European Music Analysis Conferences, pitch-class set theory has not found an easy entry into Europe. Indeed, only in Britain has it become part of the standard repertoire of analytical resources. In North America, however, it has enjoyed a wide dissemination. As noted earlier, it has played an important role in the emancipation of music theory as an academic discipline in its own right. This emancipation was marked, first, by the appearance of specialized journals such as Journal of Music Theory (Yale University, 1957), Perspectives of New Music (Princeton University, 1961), and The Music Forum (Columbia University, 1967–87); second, by the increasing number of degree programs in music theory at American universities, especially the PhD programs that were established after the example of Yale’s (1965); and, third, by the foundation of the Society for Music Theory (SMT) in 1977. The SMT was instrumental in, among other things, the organization of a large number of conferences, and provided its members with another important journal from 1979 onwards (Music Theory Spectrum). By 1980, then, an infrastructure was available that encouraged research, facilitated an ongoing professional debate, and raised the profile of music theory as a “body of knowledge and a set of shared practices.”15 Pitch-class set theory has not itself spurred this development from the beginning. However, it has contributed to it by putting on the agenda the theoretical underpinnings of the twentieth-century post-tonal repertoire. Apart from providing a technical vocabulary, it has helped to formulate the premises and questions from which to proceed in the analysis of works from this repertoire: what was to 15. This is another quote from Patrick McCreless (1997, 17). He has described the birth and growth of academic music theory from a perspective developed by the French philosopher Michel Foucault, stressing the dependence of knowledge on power: “Music theory is in fact, like all academic disciplines, a ‘docile body’—an object of control—with respect to the university, just as, in another sense, most music theorists, as individuals and employees of the university, are ‘docile bodies.’” (ibid., 35) What is characteristic for this approach is the emphasis on a self-regulating academic discourse dictating the contributions of its individual participants. Although the present study will pay a good deal of attention to the intrinsic dynamics of the evolving discourse on pitch-class set theory, it will not refrain from also addressing the decisive role of personal involvement.

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18



pitch-class set theory: an overture

be searched for, and how this was to be done.16 The influence it thus exerted on the study (and the composition) of post-tonal music has been profound and long lasting. The rise of pitch-class set theory has earned American music theory a reputation for engagement with musical modernism. However, the flurry of scholarly activity has left the modernist tradition of composition with mixed fortunes. On the one hand, this activity is a genuine response to that tradition, which has yielded textbooks and manuals of analysis giving students access to the music of such “difficult” composers as Schoenberg, Webern, Boulez, Carter, Babbitt, and (late) Stravinsky. On the other hand, this music’s exposure in the literature—combined with its minor role in public concert life—has added to its reputation of being “cerebral” and “academic.” Pitch-class set theory employs a mathematical vocabulary and mathematical models of presentation; and to designate pitches, it has substituted integers for the traditional letter names. Analyses proceeding along set-theoretical lines, then, convey an image of rationality that may confirm people in their rejection of the music concerned. The integration of pitch-class set theory into the curricula of American colleges and universities is probably the most significant measure of its institutionalization. Table 1.1 may give us a tentative impression of the extent to which it has influenced music theory teaching at the college and graduate levels. It shows its place in the curricula of twenty institutions spread over the United States and Canada. The table reflects the situation in 1996. At that time, all of these institutions offered degree programs in music. Apart from their geographical distribution, they differed by type. They included private research universities with music schools (University of Rochester) or music departments (Harvard University, Columbia University), a private university with an integrated college (Bradley University), state universities with music schools (Florida State University, Universities of Iowa and Michigan) or music departments (Universities of Virginia and New Mexico), liberal arts colleges (Simpson College and Davidson College), a college conservatory (Oberlin), and a community college (Pima, the music program of which may have been exceptional for this type of institution). In view of this diversity, it is significant that only one institution from this group did not teach pitch-class set theory in 1996. This was Davidson College. All the others had included it in their programs, although it received little coverage at the University of California in San Diego, at Memorial University of 16. Seen thus, pitch-class set theory has played a role comparable to that which Thomas Kuhn (1962) called a “paradigm.” There are different interpretations of this term, but in one sense a paradigm is an intellectual achievement that sets the course for subsequent research in a particular field. More specifically, it confronts researchers with “puzzles,” while at the same time providing them with tools for solving these “puzzles.” For another comparison, Larry Laudan’s concept of a “research tradition” is worth considering: “a set of general assumptions about the entities and processes in a domain of study, and about the appropriate methods to be used for investigating the problems and constructing the theories in that domain.” (Laudan 1977, 81)

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Schuijer.indd Sec1:19

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Representation Separate course

Integrated (Music Theory and Practice II ) Separate Course

None Integrated (20th-Century Styles, Atonal Analysis) Integrated (Mathematical Models for Music Theory) Integrated (20th-Century Techniques) Separate courses Integrated (Basic Musicianship: Writing Skills; Analysis 20th-Century Music Separate course

Institute

Bradley University, Peoria, IL.

University of California, San Diego

Columbia University, New York

Davidson College, Davidson, NC

Florida State University, Tallahassee

Harvard University

University of Iowa

University of Michigan, Ann Arbor

Undergraduate, graduate

Undergraduate and graduate

Graduate

Undergraduate and graduate

Graduate

Undergaduate

Undergraduate

Academic level

Own materials (Morris 1991)

Rahn 1980 (Morris 1991, Lewin 1993)

Own materials (Lewin 1987, Lewin 1993)

Straus 1990 (Forte 1973, Rahn 1980, et al.)

Own materials (Perle 1962, Forte 1973, Rahn 1980, Straus 1990, Morris 1991)

(Rahn 1980 and Straus 1990)

Rahn 1980 (Forte 1973 and Straus 1990, articles, dissertations)

Textbooks/manuals used

Table 1.1. Pitch-class set theory in the curricula of twenty teaching institutions in the United States and Canada (1996)

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Representation Integrated (Composition I), Separate courses Integrated (Music Theory IV ) Integrated (Theory and Structure of Modern Music) Integrated (Various Courses) Separate Courses Integrated (Basic Theory) Integrated Separate course Integrated (Materials of Contemporary Music) Integrated (Post-Tonal Theory and Analysis, Analytical Techniques 1900-1950) Integrated (Music Theory IV, Literature Survey of 20th-Century Music) Separate course Integrated (Music Analysis)

Institute

University of New Mexico

Oberlin Conservatory

Pima Community College, Tucson, AZ

University of Rochester Eastman School of Music

Simpson College, Indianola, IA

University of Texas at Austin

University of Virginia, Charlottesville

University of Washington, Seattle

University of Wisconsin, Madison

University of Alberta, Edmonton, Canada

Undergraduate and graduate

Undergraduate and graduate

Undergraduate and graduate

Undergraduate

Undergradute and graduate

Undergraduate

Undergraduate and graduate

Undergraduate

Undergraduate

Undergraduate and graduate

Academic level

Forte 1973, Rahn 1980, Lewin 1987, own materials

Forte 1973, Morris 1987

Forte 1973, Rahn 1980, Straus 1990

Straus 1990

Rahn 1980, Straus 1990

Kostka and Payne 1984

Straus 1990, Morris 1991 (Forte 1973, Rahn 1980)

Own materials, Straus 1990

Lester 1989

Straus 1990, Morris 1991

Textbooks/manuals used

Table 1.1. Pitch-class set theory in the curricula of twenty teaching institutions in the United States and Canada (1996)—(cont’d)

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Integrated (20th-Century Form and Analysis) Integrated (Materials and Techniques of Music IV, 20th-Century Harmony, Theory/Composition Seminar)

Dalhousie University, Halifax, Nova Scotia, Canada

Memorial University of Newfoundland, St. John’s, Canada

Undergraduate

Undergraduate

Academic level

Kostka and Payne 1984 (Forte 1973, Rahn 1980, Straus 1990)

Straus 1990

Textbooks/manuals used

Note: In the second column, one can see whether pitch-class set theory is the topic of a separate course, or is addressed in a more general context. In the latter case, I have included the title or subject of the course in brackets. The third column specifies the academic level(s) on which the theory is taught. The fourth column shows which texts students read. Here, the brackets indicate that a text is used for reference only, or is kept in reserve.

Representation

Institute

Table 1.1. Pitch-class set theory in the curricula of twenty teaching institutions in the United States and Canada (1996)—(cont’d)

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pitch-class set theory: an overture

Newfoundland, at Oberlin Conservatory, and at Simpson College.17 In some cases, it was a recent addition to the program (Bradley University and Simpson College, 1993; University of Virginia, 1992; University of New Mexico, 1991). From table 1.1 it appears that several institutions offered separate courses on pitch-class set theory. Apart from the course at Bradley University—an undergraduate course entitled “Theories of Atonal Music” but entirely devoted to the theory of pitch-class sets—it concerned specialized graduate courses. A more important indicator of the theory’s acceptance, however, was its appearance in general courses, especially those intended for students at the undergraduate level. This signified its inclusion in what was considered “basic knowledge” of music. In this connection, we should take special note of its place in the music theory core curriculum at universities like Michigan and Wisconsin. And the two weeks spent on pitch-class set theory at Oberlin Conservatory may be a better illustration of its success than its strong presence in the curriculum of the Eastman School of Music, an institution with a sixty-year tradition in the field of advanced music theory. Although it has achieved a firm position as an analytical “paradigm” in the literature on twentieth-century music, and has found its way to the classroom, the theory of pitch-class sets has seldom been short of criticism. Some of it has already been mentioned. The theory has been criticized for its systematization of analytical practice. It has been criticized for the way in which it has constructed “autonomous” musical objects, for failing to consider the historical context of musical works of art, and for placing undue emphasis on their technical properties. But pitch-class set theory was not the only target of this criticism; music theory itself, as an independent academic discipline, was under attack, especially from those concerned with musical expression and immediacy, or from those who posited music as social discourse and sought to study it from a plurality of perspectives. I am referring to the members of the “New Musicology” movement of the 1980s and 1990s. The names most commonly associated with this movement are Lawrence Kramer, Susan McClary, and Gary Tomlinson. However, its progenitor was the earlier-mentioned Joseph Kerman, who, in Contemplating Music, had taken a stance against the narrow scope of mainstream music scholarship—against its unreflecting ideology, its uniform methodology, and its exclusive commitment to Western music in the high-art

17. The replies from these institutions to my inquiry (November 1996) included remarks to this effect, but it can also be inferred from the fourth column in table 1.1. At Simpson College and Memorial University, Stefan Kostka’s and Dorothy Payne’s Tonal Harmony was used as a course text; more specifically the last chapter of this book, “An Introduction to Twentieth-Century Practices,” only a few pages of which deal with “set theory.” Students at Oberlin Conservatory read Joel Lester’s Analytic Approaches to Twentieth-Century Music (1989), an elementary text discussing other topics as well. In San Diego they did not work with a prescribed course text.

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tradition. In this regard, Kerman made no distinction between theory-based analysis and historical musicology: Both were well calculated to thrive in the intellectual atmosphere of neopositivism. The appeal of systematic analysis was that it provided for a positivistic approach to art, for a criticism that could draw on precisely defined, seemingly objective operations and shun subjective criteria. (Kerman 1985, 73)

Whatever was new about “New Musicology,” it wasn’t the opposition to music theory and analysis as an area of specialization. Such opposition is the natural consequence of people claiming for themselves what others consider to be an integral part of their own discipline. The musicologist Richard Taruskin, someone not typically associated with the movement,18 was equally averse to an Alleingang of music theory, as appears from his polemic with Allen Forte about the latter’s analysis of The Rite of Spring (Forte 1978, 1985, 1986; Taruskin 1979, 1986). His remarks will concern us later (see chapter 7). And no spokesman for the New Musicologists has been more succinct than the composer-theorist George Perle, who characterized pitch-class set theory as “martian musicology” (Perle 1990).19 Pitch-class set theory also stands condemned for its failure to explain how music makes sense aurally. We often think that analyses of music should somehow reflect the way in which we hear it, or at least could learn to hear it. This is a concern of the music theory teacher, who helps students develop their hearing skills. But it is also a concern of those looking for a basis of scientific verification of analytical theories. A theory that tells us how we hear music can, in principle, be tested (that is, if we come to an agreement about who “we” are); a theory that tells us how it has been conceived cannot. Now, for a listener-based theory of music to be potentially testable, it should not merely produce interpretations of scores, but should also address the process through which such interpretations come into being. The composer and researcher Fred Lerdahl has pointed out why pitch-class set theory falls short in this regard. For one thing, “it provides no criteria for segmenting the musical surface into sets” (Lerdahl 1989, 66). Indeed, without such criteria there is no way of knowing with certainty which tones form a meaningful combination. We might only refer to incidental corroborating evidence, such as the resemblance between the opening of Schoenberg’s Opus 23, no. 3 and the exposition of a fugue. From this resemblance it follows that the combinations of example 1.1a stand out as important segments of structure. But how have we managed to identify the non-obvious combinations of example 1.1b and 1.1d as

18. Yet, Kerman included his work in an appraisal of some “radical” trends in American musicology around 1990, together with that of Susan McClary, Gary Tomlinson, and Carolyn Abbate. (Kerman 1991) 19. This expression had come to him through a remark of Richard Taruskin.

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pitch-class set theory: an overture

equally important segments? As said before, this involves a transformation, a reshuffling of the pitch material so as to match it with that of the fugal subject—the “melody”—and its answers. Why do we, as analysts, go to such lengths to show the importance of these non-obvious tone combinations? And in what sense are they important? The same questions apply to our analysis of Stravinsky’s “Pas de deux.” It is one thing to note and describe the motifs played by the violins in measures 412–13; to divide the accompanying sonic field into two segments that can be reduced to the same abstract forms—as in example 1.2—is quite another. Lerdahl would argue that a listener does not do this; the “hidden substance” of these segments reveals itself to the analyst alone. The latter’s work is met with suspicion: Practitioners [i.e., analysts] have in effect relied on two external criteria for set segmentation: its “musicality,” and its capacity to provide what the theory denotes as significant set relationships. The first criterion is unexplicated, and the second is self-reinforcing. (ibid.)

For this reason, Lerdahl wanted to describe music in terms of a “grammar” that the listener attributes to it. Such a grammar, he thought, was not likely to evolve from the abstractions of pitch-class set theory. The principal milestone in the history of pitch-class set theory was the publication, in 1973, of Allen Forte’s book The Structure of Atonal Music. The segmentation problem was immediately noted by William Benjamin in his review of it for Perspectives of New Music (Benjamin 1974), and it has remained a potentially fatal issue ever since.20 It loomed all the larger in view of the scientific spirit with which the theory was suffused. This incongruity must have been deeply worrisome to Lerdahl. The lack of consistent rules for segmentation was not the only thing that bothered him, however. He also questioned the analytical relevance of the concept of a pitch-class set, and of the concepts for relations between pitchclass sets. In his view, these concepts did not account for the way in which he believed music (whether it be tonal or atonal) was heard. For example, Lerdahl argued that a pitch-class set does not appear to the listener as such—that is, as a whole to which each member contributes in equal measure. “In a real context,

20. Benjamin’s criticism of Forte’s approach was diametrically opposed to Lerdahl’s. Whereas Lerdahl wanted clear criteria for segmentation in pitch-class set theory, Benjamin found Forte’s directions too restrictive: “Forte seems to regard pitches as associated only if they sound simultaneously, form non-overlapping, uninterrupted successions, or are otherwise in close proximity to one another. Notably absent is any basis for the association of [pitch-classes] over longer time-spans, by means of registral distribution, similarity of articulation, or the simple fact that each is somehow emphasized in its own context.” (Benjamin 1974, 178–79). Yet, both Benjamin and Lerdahl might be right, the former criticizing Forte’s own segmentations, and the latter the informal way in which he has presented this topic in his book.

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some pitches are heard as more or less structural than other pitches. Adjacent pitches and chords form relationships that tug and pull at one another” (ibid.). And he could cite a number of publications reporting experiments from which it had appeared that set-theoretical concepts do not inform our hearing, or only to a limited extent. Lerdahl’s proposal for a listener-based theory of atonal music—a theory along the lines developed in A Generative Theory of Tonal Music (Lerdahl and Jackendoff 1983)—will not concern us here. Rather, we should ask ourselves whether it is fair to want an analytical theory to be based on the musical intuitions of a listener. Should it pass a “reality check”? Should examples from manuals and textbooks of music analysis—or the observations of music students, for that matter—be rejected or modified when they fail such a check? Quite apart from the question of which form an inquiry into the empirical groundings of the theory should take—a very complex issue21—it might simply not be its purpose to match a verifiable reality. What, then, is an analytical theory of music? How does such a theory come into being, and how does it function? To which needs does it respond, and what kind of hold does it have on our musical imagination? In this study, these questions will be addressed regarding pitch-class set theory. If there is one thing that has raised these questions, it is that, in spite of serious criticism, pitch-class set theory has left such a big imprint on music scholarship and music teaching in the United States and beyond. Perhaps paradoxically, we can add the criticism as another measure of its institutionalization. This criticism betrays a deep engagement with the issues that pitch-class set theory has addressed, opening up new avenues of investigation that would have been unthinkable without it.

Aim, Scope, and Structure of this Study The questions raised above invite a historical and contextual account of pitchclass set theory. Such an account will be provided in this study, which can thus be seen as concerned with the history of music theory. Like the seminal works of this orientation—the narratives of François-Joseph Fétis (Esquisse de l’histoire de l’harmonie, 1840) and Hugo Riemann (Geschichte der Musiktheorie im IX.–XIX. Jahrhundert, 1898)—it looks at the history of music theory from the vantage point of a contemporary theory. However, it does not see the contemporary theory as the summit or logical end point of that history. Pitch-class set theory does not play the role of Fétis’s tonalité or Riemann’s theory of harmonic functions. It is true that a comparison of the pitch-class set with older musical concepts, such as chords, scales, motifs, or twelve-tone series, reveals 21. See for example Nicholas Cook’s summary of the critical reception of Lerdahl’s and Jackendoff’s A Generative Theory of Tonal Music ( Cook 1989b, 118–20).

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pitch-class set theory: an overture

certain similarities; music theory builds on its own legacy. For this reason, the present study describes the evolution of pitch-class set theory with occasional reference to sources from a past more remote than the beginnings of twelve-tone serialism. However, new musical concepts—or modifications of traditional ones—do not only result from a self-generating theoretical discourse on music. And similarities between past and present concepts of musical structure do not always signify lineal relationships. Such relationships can only be ascertained by a careful study of the use of these concepts. Apart from a conceptual history, then, this study offers a view of pitch-class set theory as a construction of its own time; a domain of musical competence that reflects contemporary concerns, interests, and perceptions. Earlier, I called pitch-class set theory a theory “for atonal music.” This does not mean the same as “atonal theory.” A more comprehensive body of theory, atonal theory includes, for example, the early attempts to establish a compositional method on the basis of properties attributed to the equal-tempered chromatic scale, by composers such as Josef Matthias Hauer (Vom Melos zur Pauke, 1925, and Zwölftontechnik, 1926) and Herbert Eimert (Atonale Musiklehre, 1924). It also includes the compendia of the harmonic and melodic resources contained in the twelve-tone universe (from the writings of the nineteenth-century French music theorist Anatole Loquin to Howard Hanson’s Harmonic Materials of Modern Music, 1960). And it includes the manuals and theories of twelve-tone serialism (e.g., Ernst Krenek’s Studies in Counterpoint, 1940, Eimert’s Lehrbuch der Zwölftontechnik, 1950, and Babbitt’s articles on combinatoriality). What all these theoretical works have in common is the idea of a tonal equilibrium that allows any combination of tones to be formed in both the horizontal and vertical dimensions. Any rule imposed on the combination of tones is contextual—that is, it pertains to a single work.22 (The most obvious example is a twelve-tone series serving as the referential structure for only one composition.) However, work-specific rules can be subsumed under general principles of organization (such as the principle of serial organization). The more elaborate atonal theories deal with such principles. This is a very broad delineation of the scope of atonal theory; it even allows us to analyze tonal music from an atonal perspective. In such a case, the tonal equilibrium is not what the music achieves, but what the theory takes as its starting point. David Lewin’s theory of transformations can be seen as a late outgrowth of atonal theory that has taken a portion of the tonal repertoire under its wings, especially music of the late nineteenth century (Generalized Musical Intervals and Transformations, 1987). 22. This use of the word “contextual” was introduced by Milton Babbitt: “Contextuality . . . has to do with the extent to which a piece defines its materials within itself” (Babbitt 1987, 167). Its antonym is “communal,” a word referring to materials that many musical works share, such as the triadic structures and progressions of tonal music.

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How should we distinguish pitch-class set theory from other varieties of atonal theory? This is not immediately obvious. First, as noted before, the concept of a “set” was already in use for the analysis of serial music. Second, pitch-class set theory was never introduced under this name. In 1964, in an article for Journal of Music Theory, Allen Forte presented what he called a “theory of set-complexes for music.” This theory was not designed for the mere description of the structure of atonal music in terms of pitch-class sets and their various associations, even though a considerable part of the article dealt with just that topic. It was the statement of an organizational principle connecting various, if not all, major pitch-entities in an atonal work: the “[pitch-class] set complex.” This is a special case of a “family” of pitch-class sets, something in which Forte took a particular interest. In The Structure of Atonal Music, he developed the idea further, convinced of its significance as a model for the pitch organization of movements or entire compositions. And in 1988 he advanced an alternative type of family: the pitch-class set “genus.” Almost one half of The Structure of Atonal Music deals with the pitch-class set complex. However, this has not proven the most durable part of this otherwise very influential book. In 1997, the composer and theorist Robert Morris, an advocate of the idea, noted that it had fallen, “if not by the wayside, at least in frequency of use” (Morris 1997, 275). The pitch-class set complex did not turn up in two successful, pedagogically inspired textbooks of pitch-class set theory: John Rahn’s Basic Atonal Theory (1980) and Joseph Straus’s Introduction to Post-Tonal Theory (1990), both of which are prominent in the fourth column of table 1.1. For the present study it is important to note that The Structure of Atonal Music only represents a phase in the development of pitch-class set theory. Not every part of the theory is contained in Forte’s book; nor has every chapter of this book been of lasting influence. Pitch-class set theory is represented by a literature stretching from 1945 to the present day, with the years between 1960 and 1990 forming a period of crystallization and consolidation. The focus of chapters 2 through 6 will be on pitch-class set theory’s conceptualization of musical structure. These chapters trace the path that led to the definition of musical elements, sets, operations, and relations. Notwithstanding its mathematical vocabulary, pitch-class set theory should be treated as the product of, and the basis for, a music-theoretical discourse. It is not a mathematician’s theory of post-tonal music, but an invention of composers and music theorists. Therefore, it is appropriate for someone undertaking an inquiry into the rise and development of pitch-class set theory to determine how its conceptual apparatus relates to music history and the history of music theory; to ask, first, what it has adopted from older theory and what it has added to it, and, second, which cues it has taken from the repertoire and which constraints it has imposed on its interpretation. It goes without saying that such an inquiry should also include influences from outside the realm of music, such as from mathematics; and

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pitch-class set theory: an overture

should critically address the interaction of mathematical and musical concepts. These principles guide the discussion in chapters 2 through 6. Having placed pitch-class set theory historically this way, we are left with the question of purpose and pragmatics. Chapter 7 (“Blurring the Boundaries: Analysis, Performance, History”) broadens the horizons to discuss music analysis, not as the cool, taxonomic study of musical artworks, but as an activity representing music in the here and now, like performance; an activity driven by its own contemporary agenda, which it finds mirrored in the music around which it revolves. As a theory for analytical practice, pitch-class set theory came about because it provided answers to questions of its time. What were these questions, and where did they come from? Chapter 8 (“Mise-en-Scène”) relates the theory to some crucial aspects and changes of its environment: to the impact of the computer on the study of music, and to the American university in its double role as protector of high culture and provider of mass education. And it acknowledges that, ultimately, individuals infused it with the sense of urgency that made it thrive.

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Chapter Two

Objects and Entities Pitch-class set theory is a musical application of mathematical set theory. As with the latter, it defines a set as a collection of things, and it refers to these things as the elements of the set. Sets are usually denoted by capital letters, like A, B, C, etc. If not specified, the elements are denoted by lower-case letters: a, b, c, etc. In pitch-class set theory, too, a set is defined by its elements only, not by a particular arrangement of these elements or by their quantities. When two sets A and B consist of the same elements, they are considered equal (A = B). What is special about pitch-class set theory is the type of element it deals with: the pitch class. Therefore, I will start this discussion of the theory with definitions of the terms “pitch” and “pitch class.”

Pitch and Pitch Class Pitch is one of the four distinguishing attributes of a musical tone (the others being loudness, duration, and timbre). Broadly speaking, pitch is the position of a tone on a spectrum that runs from low to high. In pitch-class set theory, the term “pitch” designates a particular value assigned to that position. Thus, it refers to what is often expressed by a letter name (C, D, E, etc.) together with an indication of the octave range. Pitch-class set theory does not usually apply these letter names. Under the postulate of equal temperament—the division of the octave into (twelve) equal parts—it associates pitches with integers. It arbitrarily assigns the number 0 to C4 (middle C), so that, by the equal distances between all successive pitches, C♯4 = 1, B3 = −1, D4 = 2, B♭3 = 2, E♭4 = 3, and so on (ex. 2.1). Thus, each pitch can be associated with a pitch number, which is a positive or negative integer. Of course, it is possible to assign the number 0 to a pitch other than C4. This may help to clarify specific musical contexts. However, unless otherwise stated, the pitch number 0 will represent C4 in this study. Enharmonic notes (e.g., B♭ and A♯, or E and F♭) are given the same pitch numbers. Another postulate of pitch-class set theory is that of octave equivalence. Under this postulate, each pitch has the same value as the twelfth pitch above and the twelfth pitch below it. The relation between two such pitches can be expressed as the congruence modulo 12 of their pitch numbers. In other words: these pitch numbers yield the same remainder when divided by twelve. For a relation in a set S to be an equivalence relation, it has to satisfy three conditions: it has to be reflexive (that is, for every element a in S, a is related to

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objects and entities 0

-14 -13 -12 -11 -10 -9

-8

-7

-6

-5

-4

-3 -2 -1

1

2

3

4

5

6

7

8

9

10 11 12 13 14

0

Example 2.1. Pitch numbers.

0

1

2

3

4

5

6

7

8

9

10

11

Example 2.2. Conversion of pitch numbers to pitch-class (PC) numbers. Pitches belonging to the same PC are arranged in columns.

itself), symmetrical (if a is related to b, then b is related to a), and transitive (if a is related to b and b is related to c, then a is related to c). This appears to be the case when S is the set of all pitch numbers and the open sentence “is related to” is replaced by “is congruent modulo 12 with.” The relation is reflexive, for each pitch number is congruent modulo 12 with itself; it is symmetrical, for the assertion that pitch number a is congruent modulo 12 with b can always be reversed; and it is transitive, for if a is congruent modulo 12 with b, and b, in turn, is congruent modulo 12 with a third pitch number, c, then the same holds true for a and c. It follows from this that an octave relation is an equivalence relation (see also chapter 4). The relation “is congruent modulo 12 with” partitions the set of all pitches into twelve equivalence classes determined by their common remainder modulo 12 of pitch number. Such an equivalence class is called a pitch class (PC; abbreviation of the plural: PCs). A pitch with pitch number a represents a pitch class consisting of all pitches equivalent to a, and is denoted “a mod 12.” In the following, this expression will be taken to mean the assignment to a of its remainder modulo 12. For example, E♭4 is denoted 3 mod 12 = 3, E♭5 is denoted 15 mod 12 = 3, and E♭2 is denoted −21 mod 12 = 3. Example 2.2 shows the same pitches as example 2.1, but now the pitch numbers are converted to PC numbers. The notion of octave equivalence has a long history. It already expressed itself in the equal letter names that medieval music theorists like Notker Labeo and Guido of Arezzo assigned to pitches separated by an octave. Much later, it enabled Baroque theorists like Johannes Lippius, Henricus Baryphonus, and Jean-Philippe Rameau to establish the concept of the invertible

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triad.1 Today, we often refer to the concept without even realizing it. Consider, as an example, this statement: “B is a chord tone in the II chord of D Major.” No particular B is indicated; any one could function as a member of the II chord in D major on the basis of octave equivalence. As far as the octave relation is concerned, the integer model reinforces a traditional conceptualization of pitch space. PC set theory has derived the postulate of octave equivalence from the practice of serial composition. The earliest reports of this practice stated that the octave displacement of one or more tones did not affect the identity of a twelve-tone series (see Greissle 1925, 64, and Stein 1925, 66). In Milton Babbitt’s wording, the elements of a twelve-tone series were “considered independent of register and durational values, and thus treated as equivalent if n octaves apart, where n is any integer.” (Babbitt 1992, 1). This meant that analysts could describe music without regard to its melodic contour. Serial composition thus went hand in hand with an abstract notion of the material of a composition: a common “substance” wholly or partly underlying the most diverse musical statements. This notion would become an essential—though by no means distinguishing—ingredient of PC set theory. Numerical pitch notations are distasteful to some. In a polemical article on PC set theory, the Dutch composer Peter Schat wrote: For the time being the notes will keep the names that thirty generations of composers and theorists have managed with. He who wants to deprive somebody of his name and his past should give him a number.2

An emotional outburst like this is unwarranted. Notes are not deprived of their past by numbering them instead of using the traditional letter names. The history of Western pitch notation and pitch nomenclature encompasses numerical systems as well. Aside from the numerical tablature notations (which indicated not pitches, but fingerings for lute or keyboard) and the figured-bass notation (which indicated chords to be played over a given bass line), several proposals for numerical pitch notations seem to have circulated in the past, especially during the seventeenth century. One of these, a notation devised by William Braythwaite in 1619, is still extant in print. It employs the numbers 1, 2, 3, . . .7; these represent the notes of the diatonic scale, with 1 = do, 2 = re, 3 = mi and so on (Gerads 1997/1998). A much more widespread numerical pitch notation was the one used for the Rousseau-Galin-Paris-Chevé method of teaching sight

1. A discussion of this topic requires more thoroughness than can be provided on these pages. I entrust the reader to Dahlhaus 1968 and Rivera 1984. 2. “Voorlopig houden de noten nog de namen waar dertig generaties componisten en theoretici het mee gedaan hebben. Wie iemand van zijn naam, van zijn geschiedenis wil beroven, moet hem een nummer geven.” (Schat 1998, 42, my translation)

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objects and entities

singing. Like Braythwaite’s notation it was based on the diatonic scale, its compass ranging from 1 (do) to 7 (si). It should be noted that all the numerical notation systems just mentioned, including the tablatures and the figured bass, served first and foremost practical purposes. They were intended for musical training and performance. The pitch notations were relative, not absolute. This means that the numbers were associated with intervals, and could be applied to different pitches (see below, “Pitch Interval and Pitch-Interval Class”). In the field of compositional theory and music analysis, the use of numbers is not without historical precedents either. Words like “fifth” (Quinte), “sixth” (Sexte), “second” (Sekunde) and “third” (Terz), as we encounter them in Heinrich Christoph Koch’s Versuch einer Anleitung zur Komposition (1782/1793), are numerals indicating scale degrees. In the fragment below, Koch describes how these scale degrees might assume the function of a new tonic in the course of a symphonic Allegro: The first and most usual construction of the first period of the second section begins in the key of the fifth with the theme, occasionally also with another main melodic idea, either note for note, in inversion, or also with other more or less considerable alterations. After that it . . . modulates back into the main key by means of another melodic idea, and from this [to the minor key of the sixth, or otherwise] to the minor key of the second or third.3

By numbering the keys according to their positions in a scale (Tonart der Quinte, Tonart der Sexte), Koch drew a hierarchical distinction between them. In that respect he was a forerunner of Heinrich Schenker, who carried the tonal hierarchy so far as to deny the existence of other keys than the main key at all. In Schenker’s analytical graphs scale degrees are indicated not verbally but by carated Arabic numerals (see ex. 1.5a). Probably the best-known instance of the use of numbers in music analysis is the practice of assigning Roman numerals to tonal harmonies, a practice dating from the early nineteenth century and still an important element of analysis courses today (Beach 1974). Again, it is a numbering of scale degrees. This means that in music analysis and compositional theory, too, as well as in the aforementioned music notation systems, numbers are used to refer to intervals rather than pitches. 3. “Die erste und gewöhnlichste Bauart dieses ersten Perioden des zweyten Theils bestehet darinne, das er mit dem Thema, zuweilen auch mit einem andern melodischen Haupttheile, und zwar entweder von Note zu Note, oder in verkehrter Bewegung, oder auch mit andern mehr oder minder beträchtlichen Abänderungen in der Tonart der Quinte angefangen wird, nach welchem . . . vermittelst eines andern melodischen Theils die Modulation zurück in den Hauptton geführt, und von diesem in die weiche Tonart der Sexte, oder auch in die weiche Tonart der Secunde oder Terz geleitet wird.” (Koch 1782/1793, IV, 307–8, transl. by Nancy Kovaleff Baker)

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The practice of measuring an interval in half steps rather than in diatonic steps ensued from the rise of what Jonathan Bernard has called “encyclopedic music theory”: the project of mapping out all the pitch or interval combinations that are possible within the twelve-tone universe, a project on which various composers and music theorists embarked during the second half of the nineteenth century and the first half of the twentieth (Bernard 1997). This project, in turn, was encouraged by the relaxing of tonal strictures and the enlargement of the harmonic palette in the music of that period, and by the acceptance of the equal-tempered chromatic scale as a basis for theoretical speculation. For the identification and tabulation of the vast amount of possible chords and scales a numerical notation of pitches and intervals using the semitone (the “1”) as a unit of measure was most appropriate. Early examples of this notation include the works of two nineteenth-century music theorists, the Austrian Heinrich Vincent and the Frenchman Anatole Loquin.4 Later it was applied by authors such as Ernst Bacon (“Our Musical Idiom,” 1917) and Joseph Schillinger (Kaleidophone: New Resources of Melody and Harmony, 1940; The Schillinger System of Musical Composition, 1946).5 Ernst Krenek recommended its use for intervals in his Studies in Counterpoint (1940)—the first real manual of twelve-tone serialism—though strangely enough without acting upon that recommendation himself.6 Babbitt’s integer notation for twelve-tone series—which he introduced in his 1946 thesis The Function of Set Structure in the Twelve-Tone System (published in 1992), and in the more widely read articles that he published in subsequent years (Babbitt 1955, 1960 and 1961a)—employed numbers from 0 through 11. Each number assumed the role of a “pitch number,” first called “set number,” which actually referred to a PC (Babbitt’s definitions invoke modulo 12 arithmetic), and that of an “order number.” Of a given twelve-tone series, the first tone received the “order number” and “pitch number” 0, and was referred to as “0,0”; “3,7” indicated the element with “order number” 3 (i.e., the fourth element of the series!), and “pitch number” 7. With respect to pitch, Babbitt’s notation was relative: “[the] initial assignment of any one of the integers to any one particular scale element is purely arbitrary, but once having decided upon, this allocation must remain fixed throughout the particular discussion” (Babbitt 1992, 1–2). It was devised for the sake of transparency. Each of the basic twelve-tone operations can be

4. Vincent’s writings were first discussed by Robert Wason in a study of progressive harmonic theory in the nineteenth century (Wason 1988). Catherine Nolan places the works of both authors in the context of the rise of a combinatorial paradigm for the teaching of harmony (Nolan 2003). 5. For accounts of the work of Bacon, see Neff (1991), Bernard (1997), and Nolan (2003). Bernard also discusses Schillinger’s treatises. 6. Krenek either maintained the traditional interval names (put between quotation marks, like “major third”) or he just mentioned the letter names of the notes involved.

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expressed as an arithmetical operation, involving either “pitch numbers” (transposition, inversion) or “order numbers” (retrogradation), or both (retrogradeinversion; see chapter 3). How should we view Babbitt’s use of numbers? We might put it on a par with the numerical music notations mentioned above, or with the analytical notations of someone like Schenker. However, it exceeded the merely practical, as will become clear from the chapters that follow. Babbitt took musical relations as manifesting principles of number, which is reminiscent of the medieval speculative theory of intervals. Numerical relations embodied an immutable logic that was very appealing to him. They served him, and other composers and theorists of the twentieth century, as a model of coherence. Babbitt’s twelve-tone theory was a renewal of the contract between music theory and number theory. Babbitt already prefigured the conversion from a relative to an absolute numerical pitch notation, with “0” representing C: It may be found to be helpful to always associate a given integer and a certain pitch name (0 for c, for instance) and maintain this correspondence throughout. This device must, however, be looked upon purely as an associational simplification, and in no sense as either a notational necessity, or as having any effect on the generality of the discussion. (Babbitt 1992, 2)

Allen Forte actually took this step “for convenient reference,” as he wrote in “A Theory of Set-Complexes for Music” (1964, 139). In Forte’s primary field of study—the non-serial atonal music of the second Viennese school—it was usually hard to determine a contextual “0.” In serial twelve-tone music, however, one could take the first note of the initial series to serve as a referential element. Practical or not, Forte’s decision meant a radical departure from the analytical tradition, for it was not customary to always assign the same number to a particular pitch. It is this aspect that may have provoked Peter Schat’s grumbling reaction. Fixed pitch numberings have been used most prominently in research into tuning systems.7 Therefore, they may seem inartistic and unduly “scientific” to many of those primarily interested in music as an art form. However, for composers of a modernist orientation, “scientific” was not an abusive term at all. Anton Webern said he could no longer see any difference between science and inspired creation (Webern 1960, 10). Edgar Varèse saw science and art collaborating “on the threshold of beauty” (Varèse 1967, 196). Karlheinz Stockhausen demanded a “consistency of the single part and the whole,”8 invoking not only an organicist poetics of music but also positivist criteria for 7. For example, the pitch numbering system known as “semitone count,” in which the keys of equally-tempered keyboard instruments are numbered consecutively, e.g., from 1 to 88 (piano). 8. “Die Sinnhaftigkeit einer Ordnung gründet in der Widerspruchslosigkeit zwischen Einzelnem und dem Ganzen.”

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scientific theories (Stockhausen 1963, 18). And the self-proclaimed logical-positivist Milton Babbitt insisted on a “scientific language” and a “scientific method” when music was discussed (Babbitt 1961b, 398). The boundary between art and science was permeable in such music journals as Die Reihe or Perspectives of New Music. The reason why will be addressed later (see chapter 8B). To summarize, numerical systems for the identification of pitches have arisen naturally in Western musical thought. The numbers can be movable or fixed, but in either case pitches are designated in terms of their intervallic relations. Though practical and appropriate in many cases, the use of interval measures as pitch labels has been subject to serious criticism, especially with respect to non-serial “atonal” music. This criticism was brought up by somebody who was not, like Schat, averse to PC set theory, but on the contrary has made important contributions to its development: the music theorist and composer David Lewin. Writing in 1977, Lewin judged the practice of numerical PC labeling “theoretically suspect,” “analytically confusing,” and “systematically unnecessary,” among other things (Lewin 1977a, 43). Principal among the problems he addressed was the underlying, if perhaps unintended, assumption of centricity. In example 2.1, D♯4 is labeled “3” because it is three steps in an upward direction from the referential C4 (“0”); B♭3 is labeled “−2” because it is two steps in a downward direction from the same C. But in “atonal” music there supposedly are no referential pitches or PCs. And if for a span of time a pitch does assume a referential function, it need not be a C (Forte), or represent the first PC of the most significant series-form (Babbitt). As a consequence, the association of pitches with numbers may obscure “the exact nature of the musical structure involved” (Lewin 1977a, 30). We shall see a few instances in due course.

Pitch Interval and Pitch-Interval Class In view of the preceding discussion, we can expect PC set theory to define intervals in a way that accommodates pitch numbers. A pitch interval (PI) is the number of semitones upward or downward that separates one pitch from another (ex. 2.3). Direction matters; the pitches involved form an “ordered pair.” One says that the pitch interval from a to b is b − a, in formal notation: PI(a,b) = b − a. The brackets indicate that a and b are considered in the given order. The pitch interval from C4 (0) to D♯4 (3), for example, is 3: PI(0,3) = 3 − 0 = 3

This is also the pitch interval from, say, F4 (5) to A♭4 (or G♯4) (8): PI(5,8) = 8 − 5 = 3

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The pitch interval from C4 to D♯5 (15) is 15, which is also the pitch interval from G♯4 to B5 (23): PI(8,23) = 23 − 8 = 15

The pitch interval from C4 to A3 is −3, and so is the pitch interval from F♯4 to E♭4: PI(6,3) = 3 − 6 = −3

Under the assumption of octave equivalence each pitch interval is equivalent to its congruent modulo 12, as is the case with each pitch. The pitch interval from C4 to D♯4 (three semitones upward) has the same value as the pitch interval from C4 to D♯5 (fifteen semitones upward). And the pitch interval from C4 to A3 (three semitones downward) has the same value as the pitch interval from C4 to A4 (nine semitones upward). From this it follows that an intervallic analogue of the PC can be conceived: the pitch-interval class (PIC). Like the set of pitches, the set of pitch intervals is partitioned into twelve such classes. Every pitch interval PI(a,b) represents a class of all pitch intervals equivalent to PI(a,b), denoted PI(a,b) mod 12. This expression signifies the assignment to PI(a,b) of its remainder mod 12. Thus, for the PIC represented by the pitch interval from G♯4 to B5, we write 3: PIC(8,23) = PI(8,23) mod 12 = (23 − 8) mod 12 = 15 mod 12 = 3

And for the PIC represented by the pitch interval from F♯4 to E♭4, we write 9: PIC(6,3) = PI(6,3) mod 12 = (3 − 6) mod 12 = −3 mod 12 = 9

Example 2.4 demonstrates the conversion of pitch intervals to PICs. The terminology concerning PICs varies in the literature. Milton Babbitt (1992, 6–7) and David Lewin (1959, 298–99), who established the concept as I presented it, just called them “intervals.” Others speak of “ordered PC intervals” (Rahn 1980a, Straus 1990a), or “directed PC intervals” (Regener 1974).9 The term “interval class,” however, usually functions as a concept reducing the number of intervallic entities even further than “PIC,” as we shall see further below. The above discussion may seem needlessly elaborate. However, this elaborateness is justified by a counter-intuitive aspect of PICs. From example 2.5 it appears that a minor third and a major sixth may represent the same PIC, while on the

9. These are not just different terms for, but actually different conceptions of, the same phenomenon. One is based on a classification of pitch intervals (PIC), the other on a classification of pitches (pitch-class interval).

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15 9

3

-3 -9 -15

Example 2.3. Pitch intervals.

9

3

Example 2.4. Conversion of pitch intervals to pitch-interval classes (PICs).

other hand two minor thirds (or two major sixths, for that matter) may represent two different PICs. This is the consequence of conceiving an interval as an ordered pair of pitches, a conception suggested by twelve-tone serialism. If the first two tones of a twelve-tone series are, say, an A and a C—and if octave equivalence is assumed—it is possible that they form an ascending minor third or a descending major sixth. However, in the same series they cannot possibly form an ascending major sixth and a descending minor third. Therefore, the concept of an “interval class” should involve order when it is used with reference to a twelve-tone series. The consequence of considering the order of pitches is that with each pair of PCs two PICs are associated, as appears from example 2.6. The two PICs associated with the PCs 0 and 9, for example, are 3 and 9; the two PICs associated with the PCs 7 and 5 are 2 and 10. In each case, both PICs are complementary with respect to twelve (the octave). Now, one might wish that there were only one intervallic category associated with each pair of PCs; in other words, one category for minor thirds, minor tenths, major sixths, and major thirteenths, irrespective of the order of the PCs involved. This is particularly relevant for PC set theory, which deals with relations that are independent of order. Allen Forte’s term “interval class” satisfied the need for such an intervallic category. This is not the “pitch-interval class” defined above, but a closely related concept, assuming the equivalence of PICs complementary with respect to twelve. Forte’s 1964 definition runs: Let S be the set of PC integers [0, 1, 2, . . . 11], with ordinary addition and subtraction (mod 12), absolute value differences, symbolized |d|, and x, y, any two

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3

3

9

9

Example 2.5. PICs are order-dependent. elements of S. Further, let a = |x−y| and b = x + y’, where y’ = 12−y and is called the inverse of y. Then, a ≡ b if and only if10 a + b = 0 (mod 12)

This definition is numbered (2.0) in Forte’s article. Forte continues: The letters a and b represent interval-classes, and the equivalence relation defined by (2.0) partitions S into 7 such classes comprising the familiar intervals of the chromatic scale. . . . By convention, the [smallest of a and b] represents the class in each case. Since the class containing 0 is trivial we usually speak of 6 instead of seven interval-classes. (Forte 1964, 140)11

In The Structure of Atonal Music, written nine years later, Forte confines himself to announcing that “inverse-related (modulo 12) intervals are defined as equivalent” (1973, 14). According to Forte’s system—a system that has been widely followed—an interval class is a class of unordered pairs of pitches denoted PI{a,b} mod 12, and is represented by the smallest of (a − b) mod 12 and (b − a) mod 12. The curly braces indicate that a and b are unordered. Thus, the pitch intervals from G♯4 to B5 and from F♯4 to E♭4—which represent the PICs 3 and 9 respectively—both represent the Fortean interval class 3. Example 2.7 shows the relation between PICs and the interval classes as conceived by Forte. A single twelve-tone series is not adequately represented by a succession of interval classes à la Forte. But a succession like that would express what the 48 forms of a series have in common. In an attempt to define that essence, Carl Dahlhaus called a twelve-tone series “a succession of interval classes comprising the twelve PCs of the chromatic scale, which constitutes the relational system of

10. In sentential calculus, a division of mathematical logic, one speaks of the equivalence of two sentences when these are connected by the expression “if and only if,” abbreviated “iff.” When a and b are sentences, the logical phrase “a iff b” means “a if b, and b if a.” It is imperative that both sentences be either true or false, and that the conversion of the two sentences does not change this condition. Mathematicians use sentential equivalence when they define a new concept. The definition takes the form “a iff b,” where a signifies an expression involving the new concept, and b an equivalent expression in already familiar terms (Tarski 1995). 11. The typography of the original text has been maintained in this quotation. This will be the policy for every quotation in this study.

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PCs

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0,10

0,11

PICs

0

1 11

2 10

3 9

4 8

5 7

6

7 5

8 4

9 3

10 2

11 1

39

Example 2.6. PICs associated with PC pairs. PICs

PICs

Forte:

0

1

2

3

4

5

0

11

10

9

8

7

PICs

1

2

3

4

5

6

0

1

2

3

4

5

6

Example 2.7. PICs and the interval classes according to Allen Forte.

a composition.”12 Dahlhaus’s use of the term “interval class” was in agreement with Forte’s. It united, for example, major thirds and tenths and minor sixths in both directions.13 Aside from all the advantages it offers, Forte’s definition of “interval class” is problematic. It is questionable whether an inverse relation of PICs is really an equivalence relation; in other words, whether it satisfies the three conditions of reflexivity, symmetry, and transitivity. Forte himself had summed them up in his article, in a note attached to the above definition (Forte 1964, 179, note 5). In the set of PICs the relation “is the inverse modulo 12 of” is not reflexive (a PIC is not, in general, its own inverse; only 0 and 6 are); nor is it transitive (when b is the inverse of a, and c is the inverse of b, c equals a, and is not, in general, the inverse of a, except when b = 0 or b = 6). Therefore, when inverse-related PICs are not equal, they are not equivalent either. Their relation is different from that between octave-related pitches or pitch intervals. Consequently, Forte’s interval class is not a “class” in the same sense as PCs and PICs are. Treatises and manuals of more recent years, like John Rahn’s Basic Atonal Theory, Robert Morris’s Composition with Pitch-Classes, or Michael Friedmann’s Ear-Training for Twentieth-Century Music do not speak of the equivalence of inverse-related intervals. However, Rahn and Morris still use the term “interval class” in the Fortean sense, as 12. “Eine die zwölf Tonqualitäten der chromatischen Skala umfassende Anordnung von Intervallklassen, die das Bezugssystem einer Komposition darstellt.” (Dahlhaus 1970, 507, my translation) 13. “Eine Intervallklasse . . . umfaßt z.B. außer der großen Terz (c-e und c-As) die große Dezime (c-e’ und c’-As) und die kleine Sexte (c-E und c-as).” (Dahlhaus 1970, 506)

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does Joseph Straus (Introduction to Post-Tonal Theory). Friedmann, however, prefers the term “unordered pitch-class interval.” This is a problem of vocabulary, a problem PC set theory has not been able to solve completely. “PIC” has correctly been defined above as signifying a class of intervals associated with ordered pairs of pitches. The almost identical term “interval class” (IC) refers to a different concept, a class of intervals associated with unordered pairs of pitches. In their contribution for the entry “interval class” in The New Grove Dictionary of 1980, Paul Lansky and George Perle called the Fortean “interval class” an “absolute interval class.” This term refers to the mathematical concept of the absolute value, by which positive and negative numbers of the same size are equated. Likewise, it is possible to equate inverse-related pitch-interval numbers—i.e., numbers assigned to ascending and descending intervals of the same size—and inverse-related PIC numbers. The absolute value of a minus b— notated |a − b|—is: a − b, when b is smaller than or equal to a,

or: b − a, when b is larger than a.

This means that the absolute value of PIC(a,b) equals that of PIC(b,a). For example, if a and b are the pitches G4 (7) and E4 (4) respectively, the absolute value is 3 in both cases: |(4 − 7) mod 12| = (7 − 4) mod 12 = 3

and: |(7 − 4) mod 12| = (7 − 4) mod 12 = 3

It can thus be said that 3 is the absolute pitch-interval class (APIC) represented by the pitch interval from G4 to E4 and the pitch interval from E4 to G4. Although the idea has met with no response, the absolute value can substitute for Forte’s somewhat problematic “interval class.”

Pitch-Class Set Pitch-class sets (PC sets) are specified according to the conventions of mathematical set theory. When a PC set consists of, for example, the PCs 1 (C♯), 2 (D), 5 (F), 6 (F♯), 7 (G), and 9 (A), its notation is:

{1,2,5,6,7,9}

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This notation does not take the quantity of an individual element into account. Moreover, the curly braces indicate that the elements are displayed in arbitrary order. Therefore, {1,2,5,6,7,9} does not mean anything different from {7,2,9,5,1,6}. The number of elements in this PC set is 6. This number is called its cardinal number, and is often represented by the sign “#” (“#A” means: “the cardinal number of set A”). The musical idea displayed in example 2.8 (a figure from Arnold Schoenberg’s Piano Piece, Op. 11, no. 1) is a “realization” of our PC set. From the accompanying table it appears that the PCs involved are the same. PC sets can represent any combination of tones—scales, motifs, chords, and harmonic-melodic progressions, for example. They do so regardless of the registral and temporal distribution of the notes, and of their harmonic and melodic functions. Tone combinations are represented, in other words, as mere collections. We say that they are equal if they realize the same PCs. Example 2.9 is an example from Joseph Straus’s Introduction to Post-Tonal Theory, which shows the reader “a single pitch-class set expressed in five different ways” (Straus 1990a, 27). The PC set is {1,4,5,7}; the excerpts have been taken from a serial work: the “Gavotte” from Schoenberg’s Suite for Piano, Op. 25. As a musical concept, the PC set originates from two traditions. The first (and oldest) of these is chordal theory. The PC set owes much to the concept of a “chord,” the members of which may appear in different octaves and exchange vertical order positions. A chord may also unfold linearly. In his Treatise on Harmony (1722), Jean-Philippe Rameau explains in which ways a melody can be ornamented above a bass: one way is by using “consonant tones”: Notes between beats which proceed by consonant intervals should be notes which are in the chord of the first beat. When the next beat is reached, notes of its chord should be used, and so on from one beat to the next until the end.14

The bass itself can be ornamented in the same way. Thus, a succession of “broken” or “arpeggiated” chords appears. Compositional instructions like these, involving the “prolongation” or “linear unfolding” of triads, can be traced until the 1630s (Rivera 1984, 74). So, what chords and PC sets have in common is a flexibility of arrangement. Some have actually referred to PC set theory as a “theory of chords” (Regener 1974). However, traditional chordal theory does make a qualitative distinction between the different members of a chord (the root, third, and fifth of a triad) and between the various possible bass positions in which it may appear (the root position and the first and second “inversions” of a triad). It also makes a distinction between chord tones and non-chord tones 14. “Lorsque l’on veut faire passer des Nottes entre les Temps par des Intervales consonans, on ne peut en faire entendre d’autres que celles qui sont comprises dans l’Accord du premier Temps, pour tomber ensuite sur une Notte de l’Accord du Temps qui vient immediament aprés, & ainsi d’un Temps à l’autre, jusqu’a la fin.” (Rameau 1722, 308, transl. by Philip Gossett)

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accel.

Pitch:

F4

D4

C 5 G3

A4

F 4 F5

D5

C 6

Pitch Nr.

5

2

13

-5

9

6

17

14

25

PC

5

2

1

7

9

6

5

2

1

Example 2.8. Arnold Schoenberg, Piano Piece, Op. 11, no. 1, mm. 49–50. Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272.

b. 7

a. 1

p

p

^

^

^

sfp sf

d. 24

c. 16 p5

1

dolce

5 4

5 3

^

2 1

rit

e. 26

f

ff ff

Example 2.9. An example from Joseph Straus’s Introduction to Post-Tonal Theory (1990). Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272.

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(like passing tones, neighbor tones, and suspensions). Furthermore, an important aspect of a traditional chord is that it operates within the system of a particular key; in other words, its function is always defined in relation to that of other chords. Finally, chords have always been more or less subject to the constraints of meter and rhythm. All this does not necessarily apply to the PC set. By the mid-nineteenth century the harmonic idiom of Western art music had grown more complex; chord tones and non-chord tones could no longer be so easily distinguished and the system of chordal relations gradually dissolved. Chordal theory could cope for a considerable time, but as the process continued some music theorists began to stop defining chords by their function, or imposing restrictions upon their structure, and classified them on the basis of their objective pitch or interval content. Many such classifications soon faded into oblivion, but in recent years interest in them has increased. Scholars now see them as having contributed to the “conceptual preparation of the pitch-class set.” The quoted phrase comes from the pen of Catherine Nolan (2003, 206). She and Jonathan Bernard (1997) have investigated several of these chord classifications.15 Bernard discusses the systems of Ernst Bacon, Joseph Schillinger, and Howard Hanson, which I mentioned earlier; and he evaluates similar efforts by composers like Fritz Heinrich Klein (“Die Grenze der Halbtonwelt,” 1925) and Alois Hába (Neue Harmonielehre des diatonischen-, chromatischen-, Viertel-, Drittel-, Sechstel-, und Zwölftel-tonsystems, 1927). One of his observations is that, in the process of enumeration, some of these authors let the traditional distinction between harmony and melody fade; in other words, they mixed the identities of chord, scale, and melodic pattern. Bernard concludes that they were drifting away from chordal theory in a strict sense, and entering the domain of PC set theory. Nolan traces the “prehistory” of the pitch-class set back even further, to the mid-nineteenth century. However, she relates the earliest comprehensive chord classifications—such as Loquin’s Tableau de tous les effets harmoniques de une à cinq notes inclusivement (1874)—to a general interest in the application of combinatorial techniques that goes back to the seventeenth century (Nolan 2003, 238). This, too, indicates that the pursuit of comprehensiveness does not go hand in hand with a concern for harmonic identity. “PC set” is neither a basically harmonic nor a basically melodic concept. It simply eliminates the distinction between the horizontal and the vertical. In that respect, the idea of the PC set was prepared not so much by the above-mentioned chord classifications as by twelve-tone serialism. We already have noted the proximity of PC set theory and the practice of serial composition (see chapter 1, p. 9). And we have mentioned the role of Babbitt as the serialist who pioneered the set-theoretical approach. Twelve-tone serialism, then, is the second tradition 15. I will again refer to their work in the section of chapter 4 that deals with some of these classifications (“A Little More History: Musical Statistics”).

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from which the PC set has sprung. Arnold Schoenberg stated that “the two-ormore-dimensional space in which musical ideas are presented is a unit.” This statement was made as a prefatory remark to the composer’s exposition of his “method of composing with twelve tones.” It referred to the twelve-tone series, which could generate melodic lines as well as sonorities.16 As we have seen, the elements of a twelve-tone series are considered independent from register and duration. In other words, a twelve-tone series consists of PCs. It is therefore a progenitor of the PC set. (Schoenberg himself called it a “twelve-tone set.”17) Needless to say, “twelve-tone series” is a far more restricted concept than “PC set,” since its cardinal number is fixed and its elements are ordered. But the fact that adjacent elements can also be projected harmonically implies that its ordering need not be musically manifest. Quite paradoxically, serial techniques have enabled the unordered PC set of less than twelve elements to stand out as an important analytical category. This is already evident from example 2.9, where the recurring association of the PCs 1 (D♭), 4 (E), 5 (F) and 7 (G) results from their order positions in a twelve-tone series. PC set theory claims that such sets have also been living a life of their own in music, independently from serial practice. Twelve-tone serialism is even claimed to be only a special case of a much more general structural principle, involving unordered PC sets of different magnitudes. As John Rahn has put it in the introduction to his Basic Atonal Theory, the first real manual of PC set theory: The relations taught here are basic to all atonal music, whether that music is serial or non-serial. Indeed, serialism arose partly as a means of organizing more coherently the relations used in the preserial “free-atonal” music. There are relations—those deriving from syntactical order—that are peculiar to serial atonal music. The theory of these particularly serial relations is an extension of the “basic atonal theory” given here; serial theory builds on basic atonal theory as its foundation. (Rahn 1980a, 2)

Any doubt cast on the generality of that hypothetical structural principle has not prevented PC set theory from considerably widening its scope in the past decades. Initially, in the 1960s and early 1970s, its focus was mainly on the nonserial atonal repertoire of the second Viennese school. Later, Stravinsky’s music 16. Schoenberg supplied this statement with the following explanation: “The mutual relation of tones regulates the succession of intervals as well as their association into harmonies; the rhythm regulates the succession of tones as well as the succession of harmonies and organizes phrasing. And this explains why . . . a basic set [sic] of twelve tones . . . can be used in either dimension, as a whole or in parts” (Schoenberg 1984, 220). Schoenberg’s method reflected concerns that were shared by people from outside the serial tradition as well. Joseph Matthias Hauer’s twelve-tone technique did not differentiate between the horizontal and the vertical either (Hauer 1925, 1926). 17. Milton Babbitt claims to have suggested this term to Schoenberg (Duckworth 1999, 63–64).

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45

came within its reach (Forte 1973 and 1978, Van den Toorn 1983), followed by the works of Debussy (Parks 1981 and 1989) and Scriabin (Baker 1986). By now, it covers a variegated repertoire, ranging from late-nineteenth-century music to contemporary jazz. This raises questions about its relation to other theories of musical structure. For example, how does it relate to tonal theory? John Rahn, continuing on the above-quoted line of thought, suggested that tonal theory could, in a sense, be regarded as a special case of atonal theory (Rahn 1980a, 19). Seen thus, PC set theory functions like an “umbrella” theory of music. Indeed, the quest for such a theory—which connects a multitude of musical idioms, including contemporary ones—has given much impetus to American music theory, and to PC set theory in particular, from the mid-1950s onwards. Now, to return to the claim that a PC set corresponds to a verifiable musical entity in non-serial music, how can such entities be found? Generally speaking, they should be recognizable as subjects of a musical argument: they should be repeated or transformed, or otherwise articulated. PC set theory has provided models for a considerable number of transformations and relations. This is territory to be explored in the following chapters, after we have finished this preliminary examination of the “set” concept. It is possible to cast the entire system of twelve PCs in terms of finite set theory, and to calculate the total amount of PC sets. This calculation calls in the terms “subset,” “superset,” “universal set,” “empty set,” and “power set.” Here is an overview: •

Set A is a subset of set B (A ⊆ B) when each element of A is also an element of B. Reversely, B is a superset of A (B ⊇ A) when it contains the elements of A. A and B might be equal (namely when, simultaneously, A ⊆ B and B ⊆ A). In PC set theory, it would seem more natural to consider proper subsets and supersets only. Indeed, this is what some prefer. Usually, A is called a proper subset of B (A ⊂ B) when it is not equal to B (A ≠ B). The reverse statement is that B is a proper superset of A (B ⊃ A).

•

The universal set is the fixed set of which all sets considered are subsets. Its counterpart is a set without elements, the empty or null set. This is a subset of each of the sets considered. In PC set theory, the universal set consists of integers, which, as we have seen, represent pitches under twelve-tone equal temperament. From now, I will call it PITCH, using capitals for quick identification. However, I will more often refer to PITCHCLASS, the partition of PITCH that was described in the first section of this chapter. This set consists of the integer values from 0 to 11 inclusive, which represent the twelve PCs.

•

The entire collection of subsets of a set A—including the empty set as well as the set equaling A—is the power set of A, or ℘(A). The number of elements in ℘(A) is 2n, where n is the cardinal number of A. ℘(PITCHCLASS)

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46



objects and entities

is the set of PC sets, henceforth designated PCSET. It contains 212 = 4,096 PC sets. Table 2.1 shows a partition of this amount into the quantities per cardinal number. The general expression of the sum total of distinct PC sets of a given cardinal number n is: 12! n!(12 – n)! Usually, PC set theory excludes certain sets from consideration when it comes to music analysis. Of course the null set meets this fate, but others do as well, for reasons to be discussed later. Partitioning PCSET into equivalence classes further reduces the number of distinct PC sets (see chapter 4).

The Interval Content of a PC Set Although a PC set is defined by its elements, one can say that the intervals between these elements are what really matters. Indeed, many of the more significant relations between PC sets boil down to similarities between their interval contents. Since “PC set” is an abstract concept, encompassing a large number of possible musical realizations, the interval content of a PC set should be defined so that it represents any of these realizations: first, it should be defined in terms of PICs— absolute or not—and second, it should involve all possible pairs of PCs. In 1959, in a contribution to the young Journal of Music Theory, David Lewin coined the term interval function, which indicated the frequency of a given PIC (which Lewin called an “interval”) among the ordered pairs of members of two PC sets: Let P and Q be any two collections of notes. We impose no a priori restrictions on P and Q. P may contain 2 notes and Q, 7; or P may contain 4 notes and Q, 12, etc.; possibly even Q may be the same collection as P.

We can define what I shall call the interval function between P and Q as follows: For every integer i between 0 and 11 inclusive, let m(i) be the number of pairs of notes [PCs] (x,y) such that x is a member of the collection P, y is a member of the collection Q, and the interval between x and y is i. The function m will be called the interval function between P and Q. (Lewin 1959, 299)18

It should be noted that the pairs (x,y) are ordered, which means that the interval function is always directed from one PC set to the other. This may seem inconvenient when there is no reason to assume a particular order of the collections

18. At the time it was not yet common usage to speak of “sets” in music theory, and the word “collection” was equally apt to refer to unordered combinations of tones.

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objects and entities



47

Table 2.1. The sum total of PC sets of successive cardinal numbers Cardinal nr.

Number of PC sets

0

1

1

12

2

66

3

220

4

495

5

792

6

924

7

792

8

495

9

220

10

66

11

12

12

1 4,096

P and Q. But the inconvenience is automatically removed in case P and Q are the same. The interval function is very useful in that case; it is an adequate measure of the interval content of a PC set. Lewin thus defined “interval content” as the interval function m(i) for every i between a PC set and its duplicate (Lewin 1960, 99). In the following, we will use the term PIC content when we refer to this property. Given a PC set of n elements, the number of possible ordered PC pairs is n2 (including the unisons) or n2–n (excluding the unisons). The PIC content is a partition of this number into the frequency rates of the successive PICs from 0 (or 1) to 11 inclusive. It can be determined by creating a matrix of the PICs represented by all PC pairs and adding up the occurrences of each PIC. Table 2.2 demonstrates this, using the PC set {1,2,5,6,7,9} (from ex. 2.8). The resulting numbers form an ordered array, which is called a vector—a PIC vector in this instance. Inverse-related PICs are represented equally. It was the composer Donald Martino, a former student of Babbitt, who started using ordered arrays of numbers to model the interval content of a PC set (Martino 1961, 27). In this he was followed by Forte, who assigned the term “vector” to such arrays (Forte 1964, 141). In mathematics and physics, row vectors are used to model spatial quantities. A row vector represents the co-ordinates marking a certain point in an x-dimensional space. Contrary to Lewin, Martino and Forte considered unordered pairs of PCs, of which there are ½(n2 + n) in each set of n elements, including the unisons. Their vector—commonly known as the “interval vector” or “interval-class vector,” but

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objects and entities

Table 2.2. PIC content of PC set {1,2,5,6,7,9} PIC matrix:

Multiplicity per PIC (PIC vector):

1 1 0 2 11 5 8 6 7 7 6 9 4

PIC:

(0) 1 (6) 3

2 5 6 7 1 4 5 6 0 3 4 5 9 0 1 2 8 11 0 1 7 10 11 0 5 8 9 10

9 8 7 4 3 2 0

2 2

6 2

3 2

4 4

5 3

7 3

8 4

9 10 11 2 2 3

Table 2.3. Absolute-PIC content of PC set {1,2,5,6,7,9} APIC matrix: 1 2 5 6 7 9

Multiplicity per APIC (APIC vector):

APIC: (0) (6)

1 0

2 1 0

5 4 3 0

6 5 4 1 0

7 6 5 2 1 0

9 4 5 4 3 2 0

1 3

2 2

3 2

4 4

5 3

6 1

more properly designated absolute PIC vector (APIC vector)—consisted of six “co-ordinates” only: the frequency rates of APICs 1 through 6. The frequency rates of the “unison” pairs (the 0s) were left out as being redundant. This changed the number of pairs to ½(n2 − n). The APIC vector of the PC set {1,2,5,6,7,9} and its derivation are shown in table 2.3. PIC and APIC vectors have both been used as representations of the intervallic content of a PC set. Although the choice between the two vectors is of limited importance, it should be noted that the entry of PIC 6 is twice as large as the entry of APIC 6. (Each PC pair represents two inverse-related PICs, and PIC 6 is its own inverse.) This forbids their exchange in at least one proposed application, which awaits discussion in the last section of chapter 5. In the preceding pages, various aspects of the PC set—both technical and historical aspects—have been touched upon. We have introduced the PC set as a representation of musical entities. The ensuing discussion will address the nature of these entities more thoroughly. Among other things, this involves the musical relations in which they are found, and the history of these relations.

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Chapter Three

Operations Each combination of tones can be identified on the basis of its PC set. However, for such a combination to be considered of structural interest—that is, worth identifying at all—it is a necessary (though not a sufficient) condition that it bear a relation to other combinations. A relation between two combinations of tones can sometimes be conceived of as a transformation. PC set theory has defined several operations that transform one PC set into another, the most important of which are transposition (T), inversion (I), and multiplication (M). No doubt, transposition and inversion are backed by the longest history. The discussion of them in this chapter will reveal the strong bonds that tie PC set theory to the history of music theory. Multiplication is considerably younger as a concept of a musical transformation, but it, too, antedates PC set theory and is rooted in compositional practice.

Notes on the Term “Operation” The term “operation” has been borrowed from mathematics. Some confusion may arise over its correct use. A mathematical operation is defined with respect to a collection of elements. Often, it is a protocol (a “function” or “mapping” in mathematical language) assigning to each pair of elements of a collection one element of the same collection. This is called a “binary operation.” If we think of the collection of integers, ordinary addition, subtraction, and multiplication are examples of binary operations. Under addition, for example, 7 is assigned to the pair consisting of 3 and 4. There are similar protocols for collections consisting of sets. For example, the union of two sets A and B (denoted A ∪ B) is a set consisting of all elements belonging to A and all elements belonging to B. The intersection of A and B (denoted A ∩ B) is a set containing the elements that A and B have in common. The difference of A and B (denoted A − B) is the set consisting of those elements of A that do not belong to B. A special case of the difference of two sets is the complement. This has been an important concept in PC set theory, for reasons to be discussed in chapters 4 and 6. Given a set of things that I shall call ELEMENT, ℘(ELEMENT) is the set of subsets of ELEMENT. One member of ℘(ELEMENT) is the set that equals ELEMENT. Now, when A is another member of ℘(ELEMENT), the complement A' of A is the difference of ELEMENT and A:

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50



operations A' = ELEMENT − A

In the case of PC set complementation, PITCHCLASS substitutes for ELEMENT, and PCSET for ℘(ELEMENT). The complement of the PC set {1,2,5,6,7,9}—the PC set taken from Schoenberg’s Opus 11, no. 1 (see ex. 2.8)—is {0,3,4,8,10,11}. The latter results when we “subtract” {1,2,5,6,7,9} from PITCHCLASS. PC set theory conceives of an operation in a slightly different way, though. We shall see that it refers to a type of protocol assigning to each single element of a collection a unique element of the same collection. If this collection is called S, the operation is said to be an “operation on S” (Lewin 1987, 3). Suppose S consists of five elements: a, b, c, d, and e. Then table 3.1a shows such an operation. Each element must appear once in the left-hand column and once in the righthand column. In other words, the operation is a bijective function. (The term “bijective” implies that the arrows in table 3.1a can also be reversed.) This definition of an operation does not exclude complementation. For each set that is a member of PCSET there is a unique set in the same collection that is its complement. However, it has particularly served as a model for transposition, inversion, and multiplication. Some historical background may help explain why. PC set theory has adopted its definition of an operation from twelve-tone theory. The rise of twelve-tone serialism invited a description of compositional techniques in terms of formal operations. Milton Babbitt defined the familiar four forms of a twelve-tone series—prime, retrograde, inversion, and retrogradeinversion—as resulting from such operations. Each of these series-forms can be seen as a one-to-one mapping of PITCHCLASS onto itself, corresponding to table 3.1a. As we saw in chapter 2, Babbitt identified the elements of PITCHCLASS by their pitch and order numbers relative to the actual twelve-tone series he was using (see chapter 2, p. 33). He let both categories of numbers run from 0 through 11. The retrograde resulted from the mapping of each order number i onto 11 − i, whereas the inversion resulted from the mapping of each pitch number j onto 12 − j. The retrograde-inversion was the result of a composite operation: the mapping of each pitch number j onto 12 − j, and the subsequent mapping of each order number i onto 11 − i. Rather than combining two elements in order to obtain a third one, these operations affect each element individually. They thus seem to fall into the category of the “unary operations,” such as—if again we think of integers—squaring. It would match with the concept of a binary operation if the protocol itself were viewed as the element of another collection, that is, a collection G of operators. In that case, an element of collection S is associated with each pair of elements from G and S (table 3.1b). But the moot point here is that, in mathematics, operations, irrespective of whether they are unary or binary operations, need not be bijections. This requirement applies to a category of functions called “group actions.” When every element of G combines with each element of S to yield another (or the same) element of S—and the latter is the one and only

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operations



51

Table 3.1. Operations as defined by PC set theory (a)

(b) a b c d e

→ → → → →

b d e a c

(g,a) (g,b) (g,c) (g,d) (g,e)

→ → → → →

b d e a c

Table 3.2. Milton Babbitt’s “group multiplication table” S I R RI

I S RI R

R RI S I

RI R I S

element associated with that combination, as in table 3.1b—G as a whole can be said to “act” on S. Now, for this to be a “group action,” G should qualify as a “transformation group” with respect to S. This means two things. First, if a is an element of S, G should contain an element that combines with a to yield a (as when we add 0 to an integer). Second, if one element of G combines with a to yield b (g(a) = b), and another combines with b to yield c (h(b) = c), then the composition of these two elements of G should combine with a to yield c (hg(a) = h(g(a)) = c, where a, b, and c are all elements of S). The reader may be wondering what sense to make of this. Basically, the above is a way to express the symmetry of S under the application of G. As a consequence of this symmetry, the distinction between “element” and “transformation” fades. The essence of a transformation group’s “acting” on a set is that every element of the latter becomes itself a transformation. Babbitt understood this very well when he entered the four forms of a twelve-tone series in a “group multiplication table” (table 3.2; Babbitt 1960, 252). Here, the word “multiplication” refers to the combination of two or more serial operations (or transformations). In the table, each entry is the result of combining the left-most entry of the same row and the uppermost entry of the same column. The table enables us to verify that the prime (“S”), retrograde (“R”), inversion (“I”), and retrograde-inversion (“RI”) satisfy the criteria of a transformation group. We will return to this topic later (in chapter 4). For now, it suffices to say that the transposition, inversion, and multiplication operations of PC set theory, too, combine in a number of ways to form group actions with respect to sets of pitch values, such as PITCHCLASS. Apart from their historical backgrounds, this is another reason why PC set theory presents them as operations.

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operations

Transposition Transposition can be defined on PITCH or PITCHCLASS. To start with the former, the integer value of a pitch corresponds to that of a pitch interval. The value a of a pitch corresponds to PI(0,a). Therefore, a pitch interval value can be added to a pitch value. For example, when the pitch interval 4 is added to pitch 9 (A4 in example 2.1), the latter transforms into 13 (C♯5). However, PC set theory commonly conceives of transposition as the addition of PICs. It thus defines the operation on PITCHCLASS. This means that a PIC value is added to a PC value (mod 12). When PIC 4 is added to PC 9, the latter transforms into 1. This leaves the octave ranges unspecified: (9 + 4) mod 12 = 13 mod 12 = 1

The transposition of a PC set A is the addition (mod 12) of an integer n to each of the PC integers of that set. This operation is designated Tn(A), with the index n representing a PIC.1 When the PC set {0,6,7,9} is transposed with n = 4, the result is {4,10,11,1}: T4({0,6,7,9}) = {(0 + 4) mod 12, (6 + 4) mod 12, (7 + 4) mod 12, (9 + 4) mod 12} = {4,10,11,1}

This new PC set has the same interval content as the original set. The APIC vector (see chapter 2, p. 48) is 112011 in both cases. The operation is defined for unordered PC sets, but can also be applied to ordered ones (for example, a twelve-tone series). A definition from Heinrich Christoph Koch’s Musikalisches Lexicon (1865 edition) captures what transposition is still commonly understood to mean today: The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.2

Example 3.1 is the example that illustrates the entry in Koch’s Lexicon. Traditionally, the term “transposition” belongs to the spheres of music theory 1. For labels of operations I use a separate font to avoid confusion with labels of objects and measurements (like PC, PI and PIC). 2. “Das versetzen einer Melodie, Harmoniefolge, resp. eines ganzen Tonstückes in eine andere Tonart unter Beibehaltung derselben Tonordnung, das heisst, derselben Aufeinanderfolge der ganzen und halben Töne und übrigen Tonschritte.” (Koch 1865, 879, my translation) This concise definition is lacking in Koch’s original article in the 1802 edition of the lexicon. However, it accurately summarizes Koch’s lengthy explanation. The original musical example consisted of just the first two measures.

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53

Example 3.1. The transposition of a melody. A musical example from Koch’s Musikalisches Lexikon (second edition, 1865).

(e.g., the transposition of scales and modes), music notation (e.g., transposing instruments), and performance practice (e.g., the transposition of a song to accommodate different voice ranges). A compositional device to which it might refer is the “real” answer to the subject of an imitative piece—say, a fugue, motet, or ricercare. Here is a description of this device, matching the above arithmetical definition of transposition in all its precision, from Rameau’s Treatise on Harmony: If one part begins or ends on the tonic note, the other should begin or end on the dominant, and so on for each note thus related within the octave of the key in use. The notes between the tonic and its dominant should also correspond in each part; i.e., the second note which is immediately above the tonic should correspond to the sixth which is immediately above the dominant; the same holds for each note a third, a fourth, or a fifth above or below the tonic and that note which is the same degree above or below the dominant, following the direction of the melody, which may ascend or descend. The conformity which we say should be observed by the notes which begin and end the fugue should also be observed by the entire phrase making up this fugue.3

It is important to note that real answers were not straightforwardly called “transpositions” in fugal treatises. Characteristically, when the term “transposition” turns up in the context of fugal theory in Christoph Bernhard’s Tractatus Compositionis Augmentatus (1650), it refers to the shift from one key to the other (transpositio modorum) rather than to the shift of the fugal subject itself (Bernhard 1926, 97). In English translations of fugal treatises, however, the term is 3. “Si une partie commence ou finit par la Notte tonique, l’autre doit commencer ou finir par la Dominante; & ainsi de chaque Notte qui se répond dans l’étendue de l’Octave du Ton que l’on traite, faisant en sorte, que les Nottes qui se trouvent entre la Tonique & sa Dominante se répondent également dans chaque partie, c’est-à-dire, que la Seconde Notte qui est immediatement au-dessus de la Tonique doit répondre à la Sixième, qui est immediatement au-dessus de la Dominante, selon le progrès du chant, qui peut monter ou descendre; car la conformité que nous pretendons devoir être observée dans ces Nottes qui commencent & terminent, la Fugue doit être également observée dans toute la suite du Chant dont cette Fugue est composée.” (Rameau 1722, 333. Transl. by Philip Gossett)

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54



operations Rasch (

= 80)

5

6

7

8

9

2

10 11

3

5

6

7

9

10

11

12

8 12

Example 3.2. Twelve-tone transposition. This is the beginning of the prelude from Schoenberg’s Suite for Piano, Op. 25. The left hand realizes an imitative texture by playing a transposition (T6) of the twelve-tone series in the right hand. The melodic contour of the series is largely retained, with rhythmic alterations. The last four notes of the transposition (9–12), however, have been used to create a counterpoint against the four preceding notes (5–8). Furthermore, in this section of the transposed series the “original” downward leap of a major seventh (B4–C4; m. 3, right hand) is transformed into a rising minor second (F2–F♯2; m. 2, left hand). Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272.

used quite often, not only with reference to real answers, but also with reference to tonal answers and sequences.4 With the introduction of twelve-tone serialism in the early 1920s, the concept of transposition started to play a more prominent role in compositional theory, and hence in music analysis too. It was listed among the canonical transformations of a twelve-tone series. However, a twelve-tone series is a much less tangible unit than a melody, a harmonic progression, or a musical composition. Its elements are not pitches but PCs, and these may generate both melodies and harmonic progressions. A twelve-tone series may not even be presented as a single musical unit. This means that in the context of twelve-tone serialism the term 4. In English, the terms “to transpose” and “transposition” may signify other procedures than the one described by Koch. For example, “transposition” oddly signifies a diatonic sequence in Alfred Mann’s translation of Friedrich Wilhelm Marpurg’s Abhandlung von der Fuge: “The restatement of a subject by use of different tones in the same part is called transposition.” It is clear from the accompanying musical example that Marpurg is talking about a diatonic sequence (Mann 1987, 142). Consultation of the original German edition reveals that Marpurg does not use the word transponi(e)ren but versetzen (Marpurg 1753, 6). The term Versetzung is found as a distinct entry in Koch’s Musikalisches Lexicon, and it is reported there to differ from Transposition in that it allows the position of the semitones to vary according to the degree of the scale on which the subject appears (Koch 1865, 919–20). Since transponi(e)ren is defined by Koch as a special case of versetzen, the latter term must have been considered a more general one, referring to any kind of moving of a subject in pitch space. Therefore, it is recommended to use an English equivalent for versetzen other than “to transpose.” “To shift” would be an appropriate alternative.

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operations



55

“transposition” had a much more flexible meaning than in Koch’s Musikalisches Lexicon. According to Koch, it was essential to preserve the melodic contour of the transposed unit (“the same succession of whole tones and semitones and remaining intervals”); by contrast, the transposition of a twelve-tone series might involve a considerable change of its intervallic profile (ex. 3.2). Milton Babbitt defined transposition in arithmetic terms. In his 1946 thesis The Function of Set Structure in the Twelve-Tone System he wrote: By applying the transposition operator (T) to a [twelve-tone] set we will mean that every p of the set P is mapped homomorphically (with regard to order) into a T(p) of the set T(P) according to the following operation: To(pi,j) = pi,j + to where to is any integer 0–11 inclusive, where, of course, the to remains fixed for a given transposition. The + sign indicates ordinary transposition. (Babbitt 1992, 10)

Recall that in Babbitt’s notation every element (p) of the twelve-tone series (P) is indicated by its position in that series (here the “order number” i) and its pitch value modulo 12 (which is not yet called “pitch-class” but “set number,” in this case j). The last sentence (“The + sign. . .”) designates addition as the standard protocol of PC(-set) transposition. In other words, the PIC integer o is added to, not subtracted from, the pitch value j. The expression “is mapped . . . into” can be read, “is transformed into” or “transforms into.” In 1946, however, Babbitt called transposition not a “transformation” but a “translation,” a term borrowed from geometry and mechanics referring to a motion without rotation. He used the term “transformation” with reference to the inversion and retrogradation of a twelve-tone series only. Under transposition, the succession of “interval numbers” (that is, of PICs) remained invariant; for this reason, Babbitt considered transposition a “secondary” operation, an operation with “very little independent significance” (Babbitt 1992, 15– 16). This was in keeping with earlier treatises on twelve-tone serialism, in which transposition used to be mentioned only, and with little emphasis, after the mirror operations (inversion, retrogradation, and retrograde-inversion).5 Babbitt’s later writings mark a shift in the valuation of twelve-tone transposition. In his article “Twelve-Tone Invariants as Compositional Determinants,” published in the Musical Quarterly in 1960, he put it on a par with inversion, arguing that it likewise resulted in a permutation of all twelve “pitch numbers” (Babbitt 1960, 249) or “pitch class numbers” (ibid., 253). Transposition thus rose considerably in rank as a musical operation.

5.

Schuijer.indd Sec1:55

See, for example, Stein 1925, Krenek 1940 or Schoenberg 1984.

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56



operations 12

25

A: [2, 3, 7, 8, 9]

B: [0, 1, 5, 6, 7]

Example 3.3a. PC-set transposition in the fifth of Anton Webern’s Five Movements for String Quartet, Op. 5. An example from Allen Forte’s The Structure of Atonal Music. In the first excerpt, the first beat should be divided into three. The triplet sign is missing. Furthermore, it should be noted that Forte notates unordered PC sets with square brackets instead of curly ones. Copyright © 1973 by Yale University. Sehr langsam ( 3

12 arco

= 40) rit.

Sehr langsam (

= ± 40)

25

rit.

verlöschend

verlöschend

verlöschend

verlöschend

Example 3.3b. A less contracted version of Forte’s example. The boxes mark the elements singled out by Forte. © 1922 by Universal Edition A.G., Wien/UE 5888, UE 5889.

Babbitt’s definition of transposition showed a correspondence between certain musical operations and operations on numbers. Babbitt used this correspondence to explain and predict the effects, in terms of PC relationships, of serial manipulations. This was the beginning of a new, distinctly American branch of music theory. In European treatises on twelve-tone serialism numbers were used merely as labels. Besides, no standard had emerged or would crystallize in the years to follow, as a few examples will illustrate. The numerals of Arnold Schoenberg (“Composition with Twelve Tones,” the written version of a lecture dating from the 1930s, first published in Style and Idea, 1950) and Josef Rufer (Die Komposition mit Zwölf Tönen, 1952) strangely

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57

enough indicated diatonic intervals.6 Ernst Krenek (Studies in Counterpoint, 1940),7 René Leibowitz (Schönberg et son école, 1947), and Hanns Jelinek (Anleitung zur Zwölftonkomposition, 1952) numbered the twelve transpositions of a series from 1 to 12 (instead of from 0 to 11). Herbert Eimert (Lehrbuch der Zwölftontechnik, 1950) called the transpositions of a twelve-tone series by the traditional letter names of their first PCs. Thus, “U(cis)” meant: the inversion (“Umkehrung”) beginning on C♯. A similar rule was introduced many years later by George Perle (Twelve-Tone Tonality, 1977). Instead of Eimert’s bracketed letter names, Perle added absolute subscript numerals to the series-form labels “P” (“prime form” or “prograde”), “I” (“inversion”), “R” (“retrograde”) and “RI” (“retrograde-inversion”)—with “0” representing C, as in Forte’s notation. He also used this protocol in his and Paul Lansky’s contributions to the New Grove Dictionary of 1980.8 Allen Forte extended Babbitt’s definition of transposition to unordered PC sets of cardinal numbers other than 12. In his 1964 essay “A Theory of SetComplexes for Music” he spoke of the mapping of a PC set “P” into the universal set “S” (designated PITCHCLASS in chapter 2) under transposition, which he defined as: “the addition mod 12 of any integer k in S to every integer p of P.” Thus there are “12 transposed forms of P” (Forte 1964, 149).9 Any reference to order is omitted in this definition, which basically equals the one given at the beginning of this chapter. After the abandonment of contour preservation this meant a significant further generalization of the concept of transposition. George Perle had already enforced this generalization in his book Serial Composition and Atonality (1962), which was the standard American text on the analysis of twentieth-century music in the 1960s and early 1970s (that is, before the publication of Forte’s The Structure of Atonal Music). It is noteworthy that “serial composition” preceded “atonality” in the title. This was indicative of Perle’s approach, which laid stress on those aspects of “free” atonal music that anticipated the serial twelve-tone technique: Microscopic elements are transposed, internally reordered, temporally or spatially expanded or contracted, and otherwise revised, in a fluctuating context that constantly transforms the unifying motif itself. (Perle 1962, 19) 6. For an explanation of Schoenberg’s protocol, see Schmidt 1988, 19. Schoenberg used the signs “+” and “−” to differentiate between major (in case of a fourth: augmented) or minor (in case of a fifth: diminished) intervals. Furthermore, he counted the transpositions of the original series in upward direction and those of the inversion in downward direction. 7. I regard Krenek’s manual as a European contribution to twelve-tone theory, although it was originally published in the United States. 8. See the entries “Serialism” and “Twelve-Note Composition.” 9. It should be noted that certain PC sets—for example the PC sets of augmented triads—may transform into duplicates under transposition. The number of distinct transposed forms is smaller in these cases.

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(a)

13 Tempo I = ± 60

14 rit.

= 48

arco

pizz.

espr. pizz.

= ± 60

rit. 2

mit Dämpfer

3

pizz.

arco

(b) [pizz.]

pizz.

pizz.

arco 11

arco 15

espr. [am Steg]

pizz.

3

pizz.

( ) pizz.

pizz.

am Steg

pizz.

3

Example 3.4. Instances of T9 in Webern’s Opus 5, no. 5. © 1922 by Universal Edition A.G., Wien/UE 5888, UE 5889.

In Perle’s view, transposition was an operation applied to unordered collections of any number of tones. For him, the meaning of the word “set”—which he had borrowed from Babbitt—included such collections. Oddly enough, he regarded them as attributes of “serial composition,” even when the compositions in which he had found them did not feature any series, twelve-tone or otherwise: In a strict sense the term “series” denotes an ordered succession of elements, such as the Schoenbergian twelve-tone set . . . [In] the work of Scriabin and some other composers the set is a collection of pitches the specific ordering of which is purely compositional. The term “serial composition” is used in the present study as a general designation for music based on any of these types of set. (Perle 1962, 37n)

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From the last part of this quote it appears that Perle regarded the twelve-tone series as the reference point for a PC set. Forte, however, considered the PC set to be independent from the twelve-tone series. And whereas Perle, the composer, spoke of transposition as a variation technique, Forte, the analyst, used transposition to identify relations between groups of tones, regardless of whether these relations were consciously used by the composer. In The Structure of Atonal Music, Forte provided three examples of the transposition operation. One of these is reproduced here as example 3.3a (Forte 1973, 5). It concerns two excerpts from the last of Anton Webern’s Five Movements for String Quartet, Op. 5. In Forte’s book, musical examples are usually very contracted, so as to enable maximal focusing on the properties discussed. As a consequence, they veil a serious problem: when is a transpositional relationship important? An examination of the context from which example 3.3a has been extracted reveals this problem. Although the PC set of the second excerpt—the “chord” (m. 25)—is irrefutably a transposition (T10) of that of the first—the “melody” (m. 12)—the excerpts do not seem to interact very strongly. They are separated by almost thirteen eventful measures, and the “chord” is not as clearly articulated as the “melody.” As example 3.3b shows, the F♯4 in measure 25 is part of a short line played by the first violin. This line refers back to the opening of the piece played by the cello. Since tones cannot simply be classified as “chord tones” or “non-chord tones” in the atonal idiom of this piece, it is hard to prove that the first violin’s F♯4 also forms a meaningful entity with the sustained tones of the other instruments in measure 25. It is surprising, by the way, that Forte did not refer to the earlier and more marked appearance of the “chord” on the second beat of measure 24. The “melody” is noticeable, to be sure. It is played over a single harmony, as we can see in example 3.3b, and it provides a strong timbral and textural contrast with the measures that precede and follow it. What is more, an almost literal transposition of it (T8) appears in measures 17–18. Only the rhythm has changed a bit. Although this transposition is not particularly interesting from the perspective that Forte’s definition offers,10 it is precisely its being “almost literal” that qualifies the “melody” as contextually meaningful. PC sets cannot convincingly be shown to function as transpositions of other PC sets without invoking criteria other than the transpositional relationship alone—criteria such as textural salience, the preservation of a melodic contour, or some significant number of such relationships. One wonders why Forte did not select two other excerpts from Opus 5, no. 5 to show PC set transposition. The operation T9 links the PC set of the cello’s opening line to that of the pizzicato gesture in measure 13. Furthermore, it links the latter to the next five notes in the piece, four of which are also played by the cello alone (mm. 13–14; ex. 3.4a). These tone combinations stand out texturally 10.

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Forte calls it an “ordered transposition.” (Forte 1973, 62)

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and timbrally. Also, measures 13–14 refer back to the opening by featuring the sound of the cello’s lower strings, solo. Besides, it marks the return of the initial tempo (eighth-note: M.M. 60), which is slowed immediately afterwards, just as it was in the two first measures (to eighth-note: M.M. 48). The melodic interval of the perfect fourth, the opening line’s first interval (F♯2–B2), is also prominent in the tone combinations of measures 13–14 (as A♭2–E♭2 and C2–F2 respectively). Finally, the operation T9 defines the relationship between two other important excerpts as well, and it does so much more obviously: measures 10–11 (sehr ruhig) and measures 14–15 (ex. 3.4b). This is an “almost literal” transposition, like the transposition of the “melody” in measures 17–18. Forte’s book The Atonal Music of Anton Webern (1998) contains an analysis of this movement that does not mention the transpositional relationship of example 3.3 anymore. Nor does it mention the transpositional relationship presented in example 3.4a—that is, the relationship between the cello’s opening line and the events in measures 13–14. Forte provides a reading that rules out the introductory cello solo as a point of comparison. He detaches the perfect fourth F♯2–B2 from the rest of that solo, reasoning that this interval is not “a conspicuous contributor to the motivic fabric of the movement” (Forte 1998a, 85)—a challengeable position. On account of the PC sets he has found articulated further on in the piece, Forte singles out the successions B2–G3–G♯2–C3–E2–C♯2 (mm. 1–3), B2–G3– G♯2–C3–E2 (mm. 1–2), and C3–E2–C♯2 (mm. 2–3) (ex. 3.5; ibid.). Forte’s selection of PC sets is, of course, open to dispute. However, there is a much bigger problem than its being right or wrong. The above discussion has shown how hard it is to prove the importance of a selection of PC sets—or to disprove it, for that matter. Unordered PC sets are quite malleable: there are no criteria for their verification, how they are projected musically, and how they connect to form larger structures. For example, from the standpoint of musical phrasing one might object to Forte’s pasting the cello’s ostinato motif E2–C♯2 in measures 3ff. on to its introductory solo. The ostinato motif obviously defines a new section. However, PC sets are not necessarily defined by phrases or sections. They are not necessarily defined by any obvious musical shape or Gestalt. What, then, does it mean to say that the PC set represented by one combination of tones is the transposition of the PC set represented by another? It suggests that the musical piece in question is coherent in an inconspicuous way; that configurations of tones, disparate though they may seem, are based on similar intervallic structures. And this, in turn, may serve in a debate on the assessment of that piece. From the very beginning of his involvement with atonal music, it has been Forte’s concern to defend the early atonal repertoire against the criticism that it lacked a structural principle of its own, that it was either distorted tonal music, or half-baked serial music (Forte 1963, 72). The claim that musical coherence resides in the consistent use of abstract intervallic patterns obviously takes its cue from the paradigm of twelve-tone serialism. However, unlike the concept of a twelve-tone series, that of an unordered

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61

(a)

(b)

21

Vl.I

Vc.

am Steg

5

Example 3.5. Excerpts from Webern’s Opus 5, no. 5, selected by Allen Forte in The Atonal Music of Anton Webern. (a) Forte’s segmentation of the opening line of Webern’s Opus 5, no. 5. (b) A motif transpositionally related to the succession B2–G3–G♯2–C3–E2 in m. 1–2 (m. 21). Copyright © 1998 by Yale University.

PC set lacks a clear technical or procedural basis, due to the process of generalization from which it results. This process of generalization has continued after the publication of The Structure of Atonal Music. Robert Morris (Composition with Pitch-Classes, 1987) and David Lewin (Generalized Musical Intervals, 1987) specified multiple “spaces” to which this and other operations could be applied. (The word “space” denotes a set, usually a set delimiting a world of specific musical objects.) The twelve PCs under equal temperament provide but one such space. For example, the transposition operation can be applied to objects forming a (modular or non-modular) diatonic space,11 or to the space outlined by a chord. Spaces can also consist of objects other than pitches or PCs. For example, objects can be chords, durations, or time-points. In a diatonic space, or in a space consisting of triads (like the spaces pertaining to Richard Cohn’s “hexatonic systems”; Cohn 1996), a minor triad can be a transposition of a major triad, and vice versa, which is impossible in spaces like PITCH or PITCHCLASS. Joseph Straus (1997) introduced the term “near-transposition” (along with “near-inversion”), which he later replaced by “fuzzy transposition” (and “fuzzy inversion”).12 He conceived of transposition as a voice-leading event, the “sending” 11. The objects of a modular diatonic space are PCs; the objects of a non-modular diatonic space are pitches. 12. Joseph N. Straus, “Voice Leading in Atonal Music,” unpublished lecture for the Dutch Society for Music Theory, delivered on April 11, 2003. Royal Flemish Conservatory of Music, Ghent, Belgium.

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of each element of a given PC set to its Tn-correspondent. This view enabled him to relate PC sets of two adjacent chords in terms of a transposition, even when not all the “voices” participated fully in the transpositional move. The “forces” determining successions of chords in atonal music were originally beyond the scope of PC set theory. Straus’s invocation of voice leading can be seen as an attempt to bring up the question of what these “forces” are—without questioning the theory.

Inversion In the context of PC set theory, inversion is most appropriately defined as an operation applied to pitch intervals or PICs. The inversion of a pitch interval (x, x + a) yields (x, x − a). The latter is called the inverse of (x, x + a). The position x on the pitch scale is the axis of inversion. According to this definition each interval inverts around one of its own elements. For example, the inversion of (G4,F5) or PI(7, 7 + 10), is (G4,A3), or PI(7, 7 − 10). Inversion of PICs is carried out modulo 12. It relates a larger number of pitch intervals and it involves no “real” axis of inversion. The pitch interval (G4,F5) may transform into (G3,A5) or into (G4,A4), to mention just two possibilities. It is customary to speak of the inversion of single PCs as well. This use matches the above definition of inversion if a PC a mod 12 is conceived as a PIC of the type PI(0,a) mod 12. Then, this PC inverts to −a mod 12. The inversion of a PC set A means the application of this protocol to each of the members of that set, and is designated I(A). The entire PC set A is thus, in an abstract sense, inverted around 0. When A = {0,6,7,9}: I({0,6,7,9}) = {0 mod 12, −6 mod 12, −7 mod 12, −9 mod 12} = {0,6,5,3}

Inversion does not affect the APIC vector of the original PC set (112011). Like transposition, it can also be applied to ordered PC sets. “Inversion” was an important subject in treatises dealing with fugal technique. Nicola Vicentino’s L’antica musica ridotta alla moderna prattica (1555) and the third part of Gioseffo Zarlino’s Le Istitutioni harmoniche (1558) constitute early milestones in this field. The authors pointed out that subjects of fugues could be imitated alla riversa (Vicentino) or per mouimenti contrarij (Zarlino), and that it was of particular interest when they were imitated strictly, so that each falling semitone in a subject would correspond to a rising one in the answer and vice versa. Vicentino and Zarlino supplied their examples with notes that are descriptive rather than explanatory. At first sight, the examples of both authors are without sharps or flats. From Vicentino’s example it can be deduced that he exploited the symmetrical division of the Dorian octave (in diatonic steps: 1 ½ 1 1 1 ½ 1),

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Example 3.6. The inversion of a fugue subject. An example from Nicola Vicentino’s L’Antica Musica ridotta alla moderna prattica. The interval measures are retained strictly. Documenta Musicologica. Erste Reihe: Druckschriften-Faksimiles XVII. © Bärenreiter-Verlag, Kassel.

Example 3.7. The inversion of a fugue subject around B. An example from Gioseffo Zarlino’s Le Istitutioni harmoniche. A strict inversion of the subject requires that one of two corresponding Bs be flattened in either guida or consequente. Facsimile of the 1558 edition, Monuments of Music and Music Literature in Facsimile, II/1 (New York: Broude Brothers Limited). Reproduced by arrangement with Broude Brothers Limited.

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in accordance with the directions of later theorists (ex. 3.6; Vicentino 1555, 89). In the first of Zarlino’s examples, however, the subject appears to invert around the double-faced B of the Renaissance tonal system. The desired inversional symmetry of guida (i.e., the subject) and consequente (the answer) arises only when the b quadrato in one voice is answered by a b molle in the other and vice versa (ex. 3.7; Zarlino 1558, 215). The rules of musica ficta do not always give a decisive answer to the question which B should be taken as b quadrato and which as b molle. For example, when the progression A–B–C is answered by C–B–A, it is only reasonable to think that the latter B was flattened. This is not certain, though. In any case, the Renaissance tonal system allowed D (and, by transposition, G) and B to serve as centers of strict inversion. It was for theorists of later generations, like Giovanni Maria Bononcini (Musico Prattico, 1673), Johann Joseph Fux (Gradus ad Parnassum, 1725), and Friedrich Wilhelm Marpurg (Abhandlung von der Fuge, 1753) to explain inversional relationships in more general terms, and to establish a systematic procedure that would be of service to teachers of counterpoint for many years.13 The first step of this procedure was to write down a diatonic scale spanning an octave and showing the same succession of intervals in ascending and descending direction. Of the seven diatonic octave species, only the Dorian had this property. Next, one paired off the notes that were equally far removed from the opposite sides of the scale: D and D, E and C, F and B, and G and A (ex. 3.8a). Finally, to produce the answer to a subject in strict contrary motion, one had to replace each note of the subject by the one it formed a pair with (ex. 3.8b; Bononcini 1673, 84–85). According to Aloysius—the teacher in Fux’s dialogized treatise—it was “inverted” to that other note: Strict inversion arises if the progression of notes is so inverted that the relation of mi to fa is always retained . . . If you compare the notes of the ascending scale [from D4 to D5] on the left with those of the descending scale [from D5 to D4] on the right, you will find that the d remains d by inversion, e is inverted to c, f to b, and g to a.14

Theorists using the viewpoint of major-minor tonality, to whom speaking of the Dorian scale must have been something of an anachronism, put this differently. Marpurg’s instruction in his Abhandlung von der Fuge was to place the ascending octave of the first degree of a major key against the descending octave of

13. Reference to this procedure, and to its application by Bononcini, is made in Mann 1987, 144. That book also includes the relevant passages from the works of Fux and Marpurg. 14. “Contrarium autem reversum efficitur ita invertendis notis, ut ubique mi contra fa eveniat . . . Aequiparentur notae à sinistris ascendendo, cum illis à dextris descendendo, & reperies, D. per inversionem nihilominùs D. manere: E. in C., F. in B.mi, G. in A. inverti.” (Fux 1725, 204–5; English translation in Mann 1987, 130)

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(a)

(b)

Example 3.8. The inversion of a fugue subject according to Giovanni Maria Bononcini. (a) Symmetrical relationships between the elements of the Dorian scale. (b) The employment of these relationships in the answer. Facsimile of the 1673 edition, Monuments of Music and Music Literature in Facsimile, II/78 (New York: Broude Brothers Limited). Reproduced by arrangement with Broude Brothers Limited.

the third (that is, C–C/ascending versus E–E/descending), and the ascending octave of the first degree of a minor key against the descending octave of the minor seventh (that is, A–A/ascending versus G–G/descending; Marpurg 1753, 6). As example 3.9 shows, Marpurg thus arranged the notes in the same pairs as Bononcini and Fux (C/E, D/D, E/C, F/B, and G/A), but he described the derivation of these pairs through modern concepts of tonality. In modern tonal practice, however, strict inversion was hardly, if at all, pursued. Inversions of fugue subjects, such as in The Well-Tempered Clavier, or in The Art of Fugue, left the framework of the key (consisting of tonic, mediant, and dominant) intact. Intervallic strictness was subordinate to tonal firmness.15 In 15. For the same reason, “tonal” answers were preferred to “real” ones when the latter affected the main key’s framework right from the beginning. This would happen if in the exposition of a fugue the subject started on the fifth. A “real” answer would then give the main key’s supertonic as its first note. In a “tonal” answer that supertonic is replaced by the tonic.

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Example 3.9. Marpurg’s derivation of inverted forms of the major and minor scales. Nederlands Muziekinstituut, Den Haag. NMI 6 E 22–23. ∧ 1

(a)

[]

∧ 5

(b)

∧ 5

∧ 1

(c)

∧ 1

∧ 5

∧ 5

∧ 1

Example 3.10. Tonally modified inverted forms of subjects in fugues by Johann Sebastian Bach. (a) The Well-Tempered Clavier I: Fugue in D minor, mm. 21–23. (b) The Well-Tempered Clavier II: Fugue in B♭ minor, mm. 1–2 and 42–43. (c) The Art of Fugue: Contrapunctus V, mm. 1–6.

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an inverted subject, the tonic was more likely to correspond to the dominant than to the third (in major) or minor seventh (in minor). Example 3.10 presents three rectus/inversus pairs from The Well-Tempered Clavier and The Art of Fugue. None of them can be classified as an instance of strict imitation. Not until the twelve-tone system established itself as an alternative to the major-minor system did the principle of strict inversion reappear.16 Unlike the diatonic collection, the fully circular twelve-tone collection allows each tone to function as an axis of strict inversion. When the operation is applied to a twelve-tone series, that series as a rule inverts around its first tone. Twelve-tone serialism has affected the concept of inversion in the same way as it has affected the concept of transposition. Since a twelve-tone series is basically an abstract configuration, an inversional relationship between two such series may not be obvious. The registral distribution of PCs is not essential to either of them, although a composer could choose to observe the exact intervallic relationships in the series. Like twelve-tone transposition, twelve-tone inversion was defined arithmetically by Milton Babbitt in his 1946 thesis: Given a certain prime set, with general element pi,j; under the inversion operation, pi,j → I(pi, 12 − j); that is, each element of the prime set is mapped into an element with identical order number but with set number the complement (mod.12) [sic] of the original set number. (Babbitt 1992, 16)

Apart from the reference to an “order number,” this equals the definition of PC set inversion with which this section opened. The “set number” is the PC integer, and for “complement (mod 12)” one may read “inverse modulo 12.” As with most texts on twelve-tone technique, Babbitt’s definition presumes the axis of inversion to be the first PC of the prime form. For each twelve-tone series there is one inversion that can be transposed eleven times. This means that it requires a composite operation, designated TnI (or In), to get from the prime form to an inverted form at a different pitch level (different, that is, with respect to PC). Thus a hierarchy of series-forms suggests itself. Each series-form is at some “distance” from the prime form, a “distance” measured in operations. This hierarchy may or may not be exploited in musical works. It has been put to use in an early serial work like Schoenberg’s Wind Quintet, Op. 26, which plays with the four basic series-forms—the prime, the inversion, the retrograde, 16. For a discussion of symmetrical inversion in the context of late-nineteenth- and early-twentieth-century harmonic dualism, see Bernstein 1993. Bernstein also discusses studies of symmetrical inversion by Georg Capellen (Fortschrittliche Harmonie- und Melodielehre, 1908) and Bernhard Ziehn (Harmonie- und Modulationslehre, 1887; “Über die symmetrische Umkehrung,” in Gesammelte Aufsätze zur Geschichte und Theorie der Musik, 1927), both of whom used it as a means to stretch the confines of harmonic tonality, and, in that respect, prefigured the practice of twelve-tone composition. See also chapter 6, pp. 215–16.

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and the retrograde-inversion (first movement, mm. 1–24)—before it introduces a series much further “removed” from the prime form: the T5 of the retrogradeinversion (m. 24). In his analysis of this piece Schoenberg wrote: While a piece usually begins with the basic set itself, the mirror forms and other derivatives, such as the eleven transpositions of all the four basic forms, are applied only later; the transpositions especially, like the modulations in former styles, serve to build subordinate ideas. (Schoenberg 1984, 227)

Schoenberg saw the twelve-tone system as hierarchical. The above statement betrays a parallel between his hierarchy of series-forms and the hierarchy of tonal keys. However, the basic serial operations often served just to generate potential material for a composition. The composition itself could impose new hierarchies on that material. In Schoenberg’s Variations for Orchestra, Op. 31, the musical argument is dominated by series-forms containing dyads of B♭ and E and of C♯ and G.17 Still, serial nomenclature conveys the notion of a hierarchy of twelve-tone series in a musical work, relating them all to one “prime form.” And this notion involves considering one axis of inversion only. Now, even if there is a PC functioning that way, it need not be the first PC of the prime form. In the second movement of Anton Webern’s Variations for Piano, Op. 27, another axis clearly announces itself. The movement is based on a succession of pairs of inversionally related series-forms set almost note-againstnote. Moreover, the series-forms of each pair are transposed with inversionally related values (for example, 5 and 7), so that they keep forming the same pairs of PCs.18 Example 3.11 shows the entire pitch material of this “classic” of axial symmetry. The piece projects most PCs into a single octave range. Therefore, its structure is best described in terms of pitches, not PCs. And the pitches, in turn, are most appropriately defined in terms of pitch intervals measured from A4, since each of them is audibly associated with its inverse about A4. Thus, G♯3—that is, PI(A4,G♯3) or −13—is associated with B♭5—that is, PI(A4,B♭5) or 13—on the upbeat of measure 1 and further in measures 5, 8–9, 11, 15, 18 and 22. Likewise, C♯4—that is, PI(A4,C♯4) or −8—is associated with F5—that is, PI(A4,F5) or 8—in measures 2, 3–4, 8–9, 10, 16–17, 17 and 19–20.19 The inverserelated PCs D and E are projected in various octave ranges, but always in such a way that they invert to one another around A4 (cf. mm. 2–3, 6, 8, 15, 17 and 22). The same is true for the PCs B and G (cf. mm. 2, 5–6, 8, 8–9, 12, 12–13, 19–20 17. Since the pitch levels of the two intervals of PIC 6 in the prime series-form of Opus 31 are separated by an interval of PIC 3, a succession of B♭and E (or of E and B♭) will necessarily occur with a succession of C♯ and G (or of G and C♯) in the same series. In sum, there are 16 series-forms containing dyads of these PCs. This obviously has to do with the symmetrical properties of the diminished seventh chord (C♯–E–G–B♭). 18. Perle (1977, 6) called such pairs P/I dyads. 19. The pitches G♯3 and C♯4 are sometimes spelled A♭ and D♭ in the score.

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and 20). The occurrences of E♭ (or D♯) are arranged symmetrically around A4 as well (mm. 6, 15 and 21). Finally, A4 inverts to itself, which accounts for the striking tone repetitions in this piece (mm. 2, 9, 13 and 19). When Babbitt analyzed this movement in his 1960 article “Twelve-tone Invariants as Compositional Determinants”—so as to show that “the IT [sic] operation . . . effects a categorization into [even and odd] interval classes”20—he labeled the series-form starting on G♯3 in the right hand “prime form.” The inversion, which starts on B♭5 in the left hand, was labeled “T2I,” in accordance with notational conventions (Babbitt 1960, 254). This means that Babbitt considered his prime form as inverted around G♯ and subsequently transposed with n = 2.21 Babbitt’s labeling is not incorrect, but in view of the pitch structure’s symmetry around A4 it is not too accurate either. The problem is the use of a labeling system that describes relationships between entities in only one, sometimes circuitous way, whereas the music itself may simply take a “shortcut” from one entity to the other. Here the reader may be reminded of David Lewin’s objections against the assignment of pitch values in terms of distance from an arbitrary reference point (like G♯3 in ex. 3.11). As Lewin pointed out, this involves the risk of arbitrarily attributing a centric, or “tonic,” status to the latter. And it may lead to expressions that obscure rather than clarify the actual musical structure (like “T2I” in Babbitt’s analysis of the same example).22 The problem looms larger when we consider Allen Forte’s definition of inversion. According to Forte “each element p of [a given set] P is associated with one and only one inverse element s = p’ in [the universal set] S” (Forte 1964, 144). Order and magnitude of PC sets are no longer relevant in this definition. A PC set is the inversion of another if it consists of all the inverse-related PCs, no matter what kind of structure they form. In this respect, the concept of inversion has developed in the same way as the concept of transposition. What should be noted, however, is Forte’s preliminary decision to assign the integer 0 to C as a rule—so as to let this PC become a general axis of inversion—and for the rest to retain the serial nomenclature, including the underlying notion of transposed 20. A sample of Babbitt’s characteristically dense prose. This sentence communicates to the reader that the classes of intervals between corresponding notes of inversionally related series are either all odd—e.g., minor seconds (1), minor thirds (3), major sixths (9) and major sevenths (11)—or all even—e.g., major seconds (2), major thirds (4), minor sixths (8) and minor sevenths (10). 21. The index n of the operation TnI is the sum of the integer values of a PC a and its correspondent under that operation, i.e., (−a + n) mod 12: n = a + (−a + n) mod 12 Thus, the PC values of each pair of inverse-related PCs in Webern’s movement add up to 2 (mod 12)—that is, if the value 0 applies to G♯. The index n of transposition (Tn), on the other hand, represents the difference between the integer values of a PC (a) and its correspondent (a + n). 22. Lewin 1977a; see also chapter 2, p. 35.

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operations

(a) = ± 160

(b)

Example 3.11. The symmetrical arrangement around A4 of the pitch material in the second movement of Anton Webern’s Variations for Piano, Op. 27. © 1937 by Universal Edition A.G., Wien/UE 10881.

inverted forms of PC sets. As a consequence of this decision, discrepancies such as those arising in Babbitt’s Webern analysis are sanctioned by PC set theory. We can find such a discrepancy in Forte’s analysis of measure 30 of Arnold Schoenberg’s Piano Piece, Op. 23, number 3, which also discusses structural aspects other than inversional relationships. Example 3.12 shows the fragment (which has been enlarged to include measure 31) with a segmentation that Forte may have borrowed from George Perle (1962, 45). This segmentation isolates the dyad C4–G4 that the two sonorities of measure 30 have in common. Forte obtains the PC set of the sonority labeled β by the operation T7I on the PC set of the sonority labeled α (Forte 1973, 74–75). The same operation transforms the PC set of γ into that of δ. This formulation does not make clear that the pitch structure of this fragment is as symmetrical as that of the second movement of Webern’s Opus 27. The “axis” lies between E♭4 and E4 this time. The dyad C4– G4 is part of both sonorities, and is retained under the operation because C4 and G4 invert to one another around this axis. However, since the axis is not a pitch, it is no use trying to show axial symmetry by an assignment of pitch values; there is no pitch qualifying for the label “0” more obviously than any other. On the positive side, it is significant that the interval retained under inversion is a perfect fifth. This provides a link with the practice of tonal inversion, where the tonic inverts to the dominant and vice versa (see ex. 3.10). Such links mattered to Schoenberg. The analysis, in chapter 1, of the opening measures of Opus 23, no. 3, showed a similar preponderance of the fifth relation. (In these measures, the PC sets of α and δ underlie the fugal subject and its answer; see ex. 1.1a.) Seen thus, the label T7I—“invert and transpose with seven semitones” (i.e., a fifth)—may not be that bad after all. But its use is merely coincidental. In

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operations

30



71

=

α

31

γ

[ ] β

δ

Example 3.12. A symmetrical arrangement of PCs. Two measures from the closing section of Schoenberg’s Piano Piece, Op. 23, no. 3. The segmentation was proposed by George Perle and adopted from him by Allen Forte. An imaginary axis lies between E♭4 and E4 FÜNF KLAVIERSTÜCKE, OP. 23, by Arnold Schoenberg. Copyright © Edition Wilhelm Hansen AS. International Copyright Secured. All Rights Reserved. Used by Permission.

Forte’s system, a fifth other than that between C and G would have required a different label; and Schoenberg did not think of C as a kind of “fixed do.” Although it is not a pitch, there still is an imaginary axis of inversion in example 3.12. Sometimes, however, it is impossible even to imagine an axis, like in example 3.13. This is a fragment of the third of Webern’s Six Pieces for Orchestra, Op. 6. The first flute, the horn, and the glockenspiel present two inversionally related PC sets in measures 5–6 (ex. 3.13a). There are no pitches inverting around the same axis (ex. 3.13b). In a case like this, we can view the inversional relationship in terms of an operation TnI about the general axis C (0), and just accept that this may not reflect the actual musical situation in every detail. The expression “T4I” serves well to describe the progression in example 3.13a. However, we can produce a more precise description of this fragment. Another segmentation emphasizes the voice leading rather than the harmonic entities. Example 3.13c shows that the progression outlined by the highest tones of the flute (E♭5 and F5) is inversionally related to the progression outlined by the lowest tones of the horn and the glockenspiel (F♯3 and E3). The axis of inversion lies between E4 and F4. The highest tones of horn and glockenspiel (B3 and B♭3) and the flute’s lowest (C5 and D♭5) form “inner parts” that invert to one another about an axis between F4 and F♯4. Therefore, they seem to be displaced upward by a semitone with respect to the “outer parts.” This reading of the passage reveals a complex symmetry that is not apparent in example 3.13b. Some models of inversional relationships can do more justice to the musical context than TnI, as in David Lewin’s “Generalized Interval System” (GIS), an algebraic structure representing various spaces of musical objects.23 The following definition appeared as part of his presentation of the GIS: 23.

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For a formal definition of a GIS, see Lewin 1987, 26.

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72

operations

(a) fl.

(b)

hrn. glsp.

(c)

Example 3.13 (a) Two measures from Anton Webern’s Piece for Orchestra, Op. 6, no. 3. (b) The passage is based on a succession of two inversionally related PC sets. An axis must bedefined for each pitch and its “inverse.” (c) An alternative reading of these measures, stressing the symmetrical voice leading. © 1956 by Universal Edition A.G., Wien/UE 12012. PI (5, 14)

PI (5, 9) PI (9, 14)

PI (5, 14)

PI (10, 5) PI (14, 10)

Example 3.14. Inversion applied to the division of a major sixth. For each u and v in S (v may possibly equal u), we shall define an operation Iv/u, which we shall call “u/v inversion.” . . . . . . [W]e conceive any sample s and its inversion I(s) [. . .] as balanced about the given u and v in a certain intervallic proportion. I(s) bears to v an intervallic relation which is the inverse of the relation that s bears to u. (Lewin 1987, 50)

S refers to “space,” the collection of objects of which u and v are members. It should be noted that Lewin defined inversion with respect to a referential pair of objects. This meant a generalization of the axis of inversion, smoothing out the problem of its identification. Imagining a referential pair of pitches or PCs improves our understanding of inversion, and is more sensitive to the musical context than TnI. To show this, I will modify my initial definition of inversion as an operation applied to pitch intervals or PICs. My starting point is the division of an interval. For example, it is possible to divide the major sixth (F4,D5)—i.e., PI(5,14)—into a major

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operations PI (5, 14)



73

PI (5, 14)

PI (14, 16)

PI (5, 3)

Example 3.15a. Inversionally related pitch intervals stretching a referential interval (major sixth). PI (5, 14)

PI (5, 16)

PI (5, 14)

PI (14, 3)

Example 3.15b. Inversionally related pitch intervals breaking out of the referential interval.

third and a perfect fourth ascending. Inversion divides the same major sixth into a major third and a perfect fourth descending (ex. 3.14). To identify the initial division it suffices to specify the dividing pitch by measuring the interval separating it from one side of the major-sixth compass: PI(5, 5 + 4), or PI(5,9). Then, for the inverted division we write: PI(14, 14 − 4), or PI(14,10). This means that the inversion of the division of an interval can easily be stated in terms of pitches. The pitches 9 and 10 are inversionally related with respect to the major sixth (5,14). No axis is involved. This concept of inversion is reminiscent of Bononcini and Fux, who applied it to the Dorian octave (see ex. 3.8 above). Generally speaking, inversion is defined with respect to a referential pitch interval—that is, an ordered pair of pitches (x,y). To determine the inverse of a pitch interval (x, x + a), the latter is measured from y in the opposite direction. As a consequence of this general statement, inversionally related pitch intervals may stretch as well as divide the referential interval (ex. 3.15a). Furthermore, they can break out of it (ex. 3.15b). The pitches x and y themselves are inversionally related of necessity. My initial definition of inversion—that is, I(PI(x, x + a)) = PI(x, x − a)—is thus modified to read: IPI (x,y) (PI(x, x + a)) = PI(y, y − a)

where x may or may not be equal to y; and where a may be 0. When the operation is applied to single pitches the definition reads: IPI (x,y) (x + a) = (y − a)

This definition adjusts to different situations. For example, when there is one pitch obviously serving as an axis, y = x. This means that the referential pitch

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operations

interval is a perfect unison (0). The second movement of Webern’s Opus 27 (see ex. 3.11) is a case in point. The referential pitch interval is (A4,A4), or (9,9). For example, the pitch F5 (17, or 9 + 8) can be shown to invert to C♯4 (1) as follows: IPI (9,9) (9 + 8) = 9 − 8 = 1

Similarly, G♯4 (8, or 9 −1) can be shown to invert to B♭4 (10): IPI (9,9) (9 − 1) = 9 + 1 = 10

When the axis lies between two adjacent pitches, x and y may represent these pitches. Depending on the musical situation, they may represent another symmetrical pair as well.24 In the closing section of Schoenberg’s Opus 23, no. 3 (see ex. 3.12), the best referential pitch interval is (C4,G4) or (0,7). It is contained in all the sonorities. With respect to this pair, D♭5 (13) inverts to F♯3 (−6): IPI (0,7) (0 + 13) = 7 − 13 = −6

Similarly, A♭5 (20) inverts to B2 (−13): IPI (0,7) (0 + 20) = 7 − 20 = −13

When the operation applies to PCs instead of pitches, it is carried out modulo 12. Each represented pair of inversionally related PCs can be referential. Thus, in the passage from Webern’s Opus 6, no. 3 (see ex. 3.13), the inversional relationships can be defined with respect to the PICs represented by the PC pairs (E♭,D♭)/(3,1), (C,E)/(0,4), (B,F)/(11,5), and (F♯,B♭)/(6,10). For example, if we take (E♭,D♭)—that is, (3,1)—as referential, the PC F (5) can be shown to invert to B (11) as follows: IPIC(3,1) (3 + 2) = (1 − 2) mod 12 = −1 mod 12 = 11

However, with respect to PIC(F♯,B♭) the same relationship is notated: IPIC (6,10) ((6 + 11) mod 12) = (10 − 11) mod 12 = −1 mod 12 = 11

The alternative reading of the passage, shown in example 3.13c, considers pitches as such, not as PC representatives. There are two sets of referential pairs:

24. David Lewin has pointed out that two objects of a system may invert to one another with respect to various pairs, and that it is possible to consider each of these pairs as referential (Lewin 1987, 52).

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operations



75

{(E♭5,F♯3), (F♯3,E♭5), (F5,E3), . . . (E4,F4), . . .}, about which the “outer voices” invert, and {(C5,B3), (B3,C5), (D♭5,B♭3), . . . (F4,F♯4), . . .}, about which the “inner voices” invert. In Luciano Berio’s 1967 tribute to Martin Luther King: O King, for voice and five instrumentalists, there is an inversional relationship around the referential pitch interval (F4,D5), which I used in example 3.14 and 3.15.25 The key feature of this piece is a recurring melody of three phrases, which is sung on a set of vowels gradually developing into the syllables of the subject’s name. The instruments play along with bits and pieces of this melody, or they echo them softly. Some notes receive a fortissimo accent. These accents, given by the piano and one or more other instruments, and following one another ever more closely, outline a large-scale projection of the melody. The melody of O King employs seven PCs, all of which occur within the range F4–D5 (ex. 3.16a). The remaining five PCs are represented by pitches spanning the registers above and below the melody’s range (see ex. 3.16b). Initially, their occurrences provide a faint accompaniment to the melody. The higher pitches are G5 and E♭6. In the first half of the piece, they are played by the piano. In the second half, the G5 is taken up by other instruments, and gradually becomes more manifest—a process that continues until the climax (measure 4 after rehearsal letter “F” in the score) and parallels the transformation of the vocal sounds into the name “Martin Luther King.” The lower pitches are C4, E3 and F♯2. After rehearsal letter “C” in the score, E, E♭ and F♯ begin to appear in other registers as well. G and C, however, remain in theirs. Now, the pitches G5 and C4—and, in the beginning, E♭6 and E3 as well— are inversionally related with respect to the interval (F4,D5), which defines the range of the melody of O King. This is a significant relationship, for the PCs G and C mark the climax of the piece, the moment the dramatically sustained G5 of singer, flute and clarinet is suddenly contrasted with the C2 of the piano (a pitch not heard before). The inversional relationship of G5 and C4 can only be defined with respect to the interval (F4,D5), not only because it is a structurally important pitch interval, but also because the melody’s pitches do not yield a symmetrical subdivision of that interval. This means that the inversional relationship between G5 and C4 is not part of an overall symmetry. There is not always a pair of inversionally related PCs that is so obviously referential as the pair (F4,D5) in O King. A context-sensitive formula like IPI (x,y) is of little use when the context does not give any clue to what x and y might be. The TnI-protocol still has the advantage of providing a single referent for any inversional relation between PCs, but it may be at odds with the musical context.

25. An orchestral arrangement of this piece serves as the second movement of Berio’s Sinfonia (1968/1969).

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operations

Example 3.16a. The central melody of Berio’s O King. © 1970 by Universal Edition Ltd. London/UE 13781.

Example 3.16b. The initial projection of the remaining PCs. G5 and C4, and E♭6 and E3 are inversionally related with respect to the range (F4–D5) of the melody.

Multiplication It is most convenient to define multiplication on PITCH first. This shows us what the basic musical meaning of this protocol is. As said before, each member a of PITCH corresponds to a pitch interval PI(0,a). Multiplication is conceived as the enlargement of this interval by a factor n. PI(0,a) thus transforms into PI(0,(n · a)). This protocol can be applied to pitches and sets of pitches. For example, the multiplication of the pitch value 5 (or F4) by a factor of 7 yields the pitch value 35 (B6). The multiplication of the pitch set {1,−3} (or {D♭4,A3}) by a factor of 2 yields {2,−6} ({D4,F♯3}). The multiplication of {−3,4,−5} (or {A3,E4,G3}) by a factor of 3 yields {−9,12,−15} ({E♭3,C5,A2}). And the multiplication of {3,−2,−7} (or {E♭4,B♭3,F3}) by a factor of −4 yields {−12,8,28} (or {C3,A♭4,E3}; ex. 3.17). Applying multiplication with respect to PITCHCLASS means carrying it out modulo 12. As a consequence, the relationships are no longer obvious. Multiplication is thus abstracted from a basic musical intuition in much the same way as transposition and inversion. It is important to note that not all factors n (mod 12) result in a one-to-one mapping, which is a prerequisite for an operation as defined by PC set theory (table 3.3). For example, when the factor is 6, all twelve PCs are mapped onto two PCs: 0 and 6 (representing the tritone). When the factor is 3 or 9, the twelve PCs are mapped onto four PCs: 0, 3, 6, and 9 (representing the diminished seventh chord). And when the factor is 2 or 10, the twelve PCs are mapped onto six PCs: 0, 2, 4, 6, 8, and 10 (representing the wholetone scale). The only factors that do effect a one-to-one mapping on PITCHCLASS are 1, 5, 7, and 11. Of these, 1 and 11 are not commonly used, since multiplication by the former maps each element onto itself (like T0), and multiplication by the latter maps each element onto its inverse mod 12 (like I). What

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operations 5

1

4 -3

x7

77

3

-3

x2



-5

-2

-7

x3

x-4

12

8

35 2 -9

-6

-15

28

-12

Example 3.17. Multiplication in PITCH.

Table 3.3. The multiplication (mod 12) of PC values by factors from 1 through 11

PC

1

2

3

4

5

0

0

0

0

0

0

1

1

2

3

4

5

2

2

4

6

8

3

3

6

9

0

Times 6

7

8

9

10

11

0

0

0

0

0

0

6

7

8

9

10

11

10

0

2

4

6

8

10

3

6

9

0

3

6

9

4

4

8

0

4

8

0

4

8

0

4

8

5

5

10

3

8

1

6

11

4

9

2

7

6

6

0

6

0

6

0

6

0

6

0

6

7

7

2

9

4

11

6

1

8

3

10

5

8

8

4

0

8

4

0

8

4

0

8

4

9

9

6

3

0

9

6

3

0

9

6

3

10

10

8

6

4

2

0

10

8

6

4

2

11

11

10

9

8

7

6

5

4

3

2

1

remains, then, are the factors 5 and 7. For these, the same is true as for any pair of factors n and 12 − n in table 3.3: M5(a) is the inverse mod 12 of M7(a). The multiplication of a PC set A by a factor 5 is designated M5(A). When this operation is applied to the PC set {0,2,3,7}, the latter transforms into {0,10,3,11}: M5({0,2,3,7}) = {(0 · 5) mod 12, (2 · 5) mod 12, (3 · 5) mod 12, (7 · 5) mod 12} = {0,10,3,11}

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When the same PC set is multiplied by a factor 7, the result is {0,2,9,1}: M7({0,2,3,7}) = {(0 · 7) mod 12, (2 · 7) mod 12, (3 · 7) mod 12, (7 · 7) mod 12} = {0,2,9,1}

Since M7(A) is the inverse of M5(A), most authors use the labels M (denoting M5) and IM (denoting M7). Although I will follow this practice, I will keep the indexed labels in reserve for convenient reference to the history of the PC set multiplication. Interval expansion, the intuitive basis of PC multiplication,26 plays a constructive role in works of Béla Bartók and Alban Berg. In his Harvard Lectures of 1943, Bartók described how he used it: The working with these chromatic degrees gave me another idea, which led to the use of a new device. This consists of the change of the chromatic degrees into diatonic degrees. In other words, the succession of chromatic degrees is extended by leveling them over a diatonic terrain. (Bartók 1976, 381)

Thus, the successive intervals of the chromatic fugue theme of Bartók’s Music for Strings, Percussion, and Celesta (PI 1,3,−1,−1,−2,1, etc.) are transformed into diatonic ones, as example 3.18 shows.27 In the Third String Quartet, such an “extension in range” (as Bartók called it) connects the opening held chromatic tetrachord {C♯,D,D♯,E} and the closing fifths chord {C♯,G♯,D♯,A♯} (Antokoletz 1993, 260). When we assign the value 0 to C♯, which is the bass common to both sonorities, (C♯,G♯,D♯,A♯) is the M7-transform of (C♯,D,D♯,E). In the orchestral introduction of Berg’s orchestral song Seele wie bist du schöner—the first of a song cycle on picture postcard texts by Peter Altenberg, Op. 4—each instrumental part, or group of instrumental parts, provides sequences of a different motif. In some instrumental parts the sequential intervals increase. The first violins, piccolo, glockenspiel, and xylophone repeat their motif at the PIs 1, 3, 6, and 9, and the second violins and flutes repeat theirs at the PIs 1, 3, 6, 10, 15, 21, and 28.

26. Pierre Boulez provided a different musical interpretation of the concept of multiplication, defining it as an increase of number rather than size. Very basically, Boulez “multiplies” a PC set A—say, a segment of a series—by a PC set B by transposing A to the level of each pitch element of B (Boulez 1963, 35). The product AB is thus a complex consisting of as many transpositions of A as there are elements in B. Boulez’s idea started to reverberate in the American music-theoretical discourse in the 1990s through the work of, among others, Heinemann (1998). 27. Michael Friedmann (1990, 118) used the somewhat confusing term “modal transposition” to denote this technique of musical derivation. In Bartók’s Music for Strings, Percussion, and Celesta, diatonic themes are similarly transformed into chromatic ones.

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operations Andante tranquillo ( 1



79

= 116) 2 etc.

Molto moderato (

= 144) etc.

molto espress.

Example 3.18. An instance of interval expansion: Bela Bartók, Music for Strings, Percussion and Celesta, first movement, mm. 1–5, and fourth movement, mm. 204–9. © 1937 by Universal Edition A.G., Wien/UE 34129.

However, the actual antecedent of PC multiplication was a twelve-tone technique that Berg employed in his unfinished opera Lulu. The music critic Willi Reich—Berg’s friend and, later, biographer—described this technique in an article which appeared in the Musical Quarterly shortly after the composer’s death. Berg had derived a new series from the prime series-form by taking every seventh tone (starting with the first). Each tone with order number n—that is, the nth note after the first28—was thus replaced by the tone with order number (n times 7) mod 12. Taking every fifth tone of the prime series created another new series. Each tone with order number n was thus replaced by the tone with order number (n times 5) mod 12 (ex. 3.19a; Reich 1936, 394–95). Both operations yield the total chromatic, since the lowest common multiples of 5 and 12, or 7 and 12, are their products. Berg multiplied 5 or 7 by the order numbers of the tones in his series. Whose idea was it to use their PC numbers instead? In 1936, Ernst Krenek, who was deeply involved in a study of the relationship between music and mathematics, described the same operations in one of six lectures he delivered in Vienna. These lectures were published shortly thereafter under the title Über neue Musik. When introducing the “five-series” (Fünferreihe) and “seven-series” (Siebenerreihe) of a given twelve-tone series (Grundreihe), Krenek did not refer to Alban Berg or Willi Reich, but to one “Ingenieur [engineer] Konrád in Brünn [i.e., Brno],” who apparently had done some research in the field of twelve-tone serialism that provided a slightly different perspective on the subject. Krenek pointed out that the “n-times-five” and “n-times-seven” operations, when applied to the chromatic scale, resulted in a cycle of fourths and a cycle of fifths, respectively (ex. 3.19b).

28. As a consequence, the first tone should have the number 0. Babbitt (1992, 1960), too, let order numbers run from 0 through 11.

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operations etc.

etc.

Example 3.19a. Two order operations on the prime series-form of Alban Berg’s Lulu, as demonstrated by Willi Reich. Willi Reich, “Alban Berg’s Lulu,” Musical Quarterly 22/4 (1936): 394–95. Used by permission of Oxford University Press. 8va

14

etc.

9

5

7

etc.

Example 3.19b. The transformation of a chromatic scale into cycles of fourths and fifths by the same operations (Krenek 1937, 77). The “9” should be replaced with a “10” (A♯). f g

Example 3.19c. The same transformations as in (b), now presented as the result of mirror operations (Eimert 1950). © Breitkopf & Härtel, Wiesbaden. Used with kind permission of the publisher.

Conversely, these one-interval cycles transformed into a chromatic scale— another one-interval cycle—under the same operations (Krenek 1937, 77). The next step was to replace each tone, as numbered according to its position in the chromatic scale, by its cycle-of-fourths or cycle-of-fifths correspondent irrespectively of its order position in the actual twelve-tone series. This step was taken by the German composer and music theorist Herbert Eimert, who sanctioned these operations by including them in his manual on twelve-tone composition of 1950 (Krenek had not mentioned them in his Studies in Counterpoint). With a keen sense for historical connections, Eimert spoke of the “eight modes” of a twelve-tone series. He presented all these “modes” as mirror forms of one another. The inversion was obtained from the prime form through a horizontal mirror. The retrograde and retrograde-inversion reflected the prime

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operations



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form and the inversion through a vertical mirror. Pursuing the analogy further, Eimert described his “cycle-of-fourths-transform” (Quartverwandlung) and “cycleof-fifths-transform” (Quintverwandlung) as resulting from a slanting mirror: Furthermore, one can sort of move the mirror at an angle, that is, the “angle” of a fourth or fifth, so that the chromatic row is reflected in both cycles . . . In this way, one obtains the cycle-of-fourths transform and the cycle-of-fifths transform of the row.29

Eimert’s illustration is shown in example 3.19c. In turn, one could project the cycle-of-fourths and cycle-of-fifths transforms through a vertical mirror, obtaining their retrogrades. The mirror analogy served as a common denominator of all the twelve-tone operations, but it was hardly enlightening in the case of these new operations. (What is “the angle of a fourth”?) Eimert added instructions for realizing the cycle transforms, which involved the concept of multiplication: In general, the procedure is like this: to transform G [Grundreihe, i.e., the prime series-form] into IV [“cycle-of-fourths-transform”] or V [“cycle-of-fifths-transform”], one multiplies the interval numbers with 5 or 7, respectively. When the products are larger than 12 . . ., their difference with the next-smaller number in the multiplication table of 12 should be determined.30

Thus defined, the multiplication operations came to the notice of Princetonbased composers and music theorists. James K. Randall, Godfrey Winham, and Hubert S. Howe were the first to discuss and adopt them, not only with regard to twelve-tone series. At the time, the unordered PC set had just established itself in the literature, although different names were in use to denote it. In his 1965 paper “Some Combinational Properties of Pitch Structures,” published in Perspectives of New Music, Howe defined the multiplication operation with respect to a “PC collection.” (Others, including Forte, used to call this a “pitch set”). Apart from this broader field of application, Howe’s definition was very close in meaning to Eimert’s. He wrote: By a multiplicative operation Mn [the correct notation would be Mn] performed upon a PC coll x, we mean the substitution, for each PC m in x, of the PC obtained by multiplying m by n. (Howe 1965, 50) 29. “Ferner kann man den Spiegel gewissermaßen in Winkelstellung bringen, und zwar in den ‘Winkel’ der Quarte und der Quinte, so daß sich die chromatische Reihe in den beiden Zirkeln spiegelt . . . Auf diese Weise erhält man die Quartverwandlung und die Quintverwandlung der Reihe.” (Eimert 1950, 29, my translation) 30. “Generell sieht das Verfahren so aus: Um G in IV bzw. V zu verwandeln, multipliziert man die Intervallziffern mit 5 bzw. 7. Bei den Zahlen, die größer sind als 12 . . . muß man die Differenz zur nächstniedrigen Zahl der Multiplikationsreihe von 12 feststellen.” (Eimert 1950, 31, my translation)

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adding in a footnote that “n” is a modulo 12 integer. (ibid., note 19)

Howe defined inversion as a multiplicative operation. He mapped a PC set onto its inverse by M11. Thus, it was possible to present a coherent system of four multiplicative operations similar to Schoenberg’s canon of twelve-tone operations: M1 (“identity”), M11 (“inversion”), M5 (“cycle-of-fourths equivalence”), and M7 (“cycle-of-fifths equivalence”).31 Each composition of these operations equaled a single operation: M11M5 = M7, M7M5 = M11, M5M5 = M1, M7M11M5 = M1, etc.32 Judging from his references, Howe had borrowed much from an unpublished monograph by Randall. This monograph, entitled Pitch-Time Correlation, was completed in 1962.33 In it, Randall had listed several properties of PC multiplication. For example, he pointed to the identical behavior of PCs and interval classes under this kind of operation. When two PCs a and b are multiplied by n, so is the value of the interval class associated with a and b: PIC(M(a,b)) = M(PIC(a,b))

A transposition of the same pair does not affect the PIC, while under inversion it is the APIC that remains invariant. Multiplication, however, affects PCs and APICs alike. More specifically, it causes the APICs 1 and 5 to swap with one another: PIC(M(0,1)) = PIC(0,5) = 5; PIC(M(0,5)) = PIC(0,25) = 1 PIC(M(6,7)) = PIC(30,35) = 5; PIC(M(−2,3)) = PIC(−10,15) = 1

This was a reason for Winham (1964, 110–11; 1970, 63–67) to question the inclusion of multiplicative operations in the set of canonical twelve-tone operations. These operations did not preserve the “succession of interval classes” that Carl Dahlhaus considered fundamental to a twelve-tone series, the property remaining invariant under inversion, retrogradation, and any combination of these operations (see chapter 2, p. 38). In view of the unordered PC set, it was argued that multiplication could cause a change of interval content. In other words, it could affect the “sound” that all realizations of the PC set, its transpositions, and the transpositions of its inversion supposedly had in common.

31. Howe’s terminology. 32. The system exhibits the properties of a transformation group. More on this in chapter 4. 33. According to a reference in one of the author’s published papers (Randall 1965, 92).

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On the other hand, certain PC sets map onto themselves under M, while this operation equals a transposition or transposed inversion when applied to other PC sets. These varying results are dependent on the original set’s interval content. If the APICs 1 and 5 are equally represented, M yields a PC set with the same interval content. In a relatively small number of cases, this new PC set is not a transposition or a transposed inversion; it just has the same interval content. It has been a special attraction of PC multiplication that it provides a link between some of these intriguing “Z-related” pairs of PC sets. John Rahn’s manual Basic Atonal Theory made explicit mention of this property.34 The inclusion of a section on PC multiplication in a college textbook like Rahn’s—Forte had ignored the topic in The Structure of Atonal Music seven years earlier—was a response to the interest certain American composers had been taking in the operation, rather than a token of its success as an analytical tool. The composer Robert Morris had adopted multiplication as a basic twelve-tone operation in a study he had written with the computer programmer Daniel Starr. This study (“A General Theory of Combinatoriality and the Aggregate”) was published in two installments in Perspectives of New Music, in 1977 and 1978. Starr and Morris wanted to enlarge “the combinatorial possibilities of the [twelve-tone] row.” In other words, they wanted to gain a larger number of derivatives of the row that could be combined with it without doubling PCs. (This was an issue of importance in twelve-tone theory; see also chapter 4.) They did not worry about the possibly less audible relationship between a PC set and its M-transform. As they argued, the “classical” twelve-tone operations were equally abstract, since they were applied to PCs; why should a melodic contour only expand or contract under multiplication, if it may dissolve entirely under transposition or inversion (Starr and Morris 1977/1978, 5)? Although this was a valid argument in itself, it wrongly suggested that audibility was a criterion of relatedness in “classical” twelve-tone serialism. From a Schoenbergian viewpoint, the only possible objection one could raise against multiplication concerned the changing order of interval classes. This objection was of a conceptual rather than an empirical nature.

34.

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See Rahn 1980, 104; “Z-related” pairs will be discussed in chapter 4.

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Chapter Four

Equivalence The operations discussed in chapter 3 are generalized representations of compositional techniques whereby PC sets are derived from each other. However, not all relations between PC sets are based on derivation. This chapter and the following ones will deal with other relations, seen within a historical framework. In this chapter I will discuss the evolution of the concept of PC set equivalence. Like the term “operation,” the term “relation” evokes the world of mathematics. In mathematics, more specifically in algebra, a relation is commonly conceived as an open sentence, designated P, connecting the elements of two collections, S and T. This open sentence is true or not true for each ordered pair of these elements. If s is an element of S, and t is an element of T, a relation can be defined as the collection of ordered pairs (s,t) for which the open sentence P is true. This is a subcollection of the entire collection of these pairs. A relation like this is called a relation “from S to T.” We will deal here with the special case in which S = T. Then, the relation is called “a relation in S.” In chapter 2, we defined the relation “is congruent modulo 12 with” for the collection of pitch values under twelve-tone equal temperament (PITCH). This relation singles out specific ordered pairs of pitch values from this collection, such as (14, −10), (5,29), or (10, −2). For these ordered pairs the relation obtains: 14 is congruent modulo 12 with −10, 5 is congruent modulo 12 with 29, and 10 is congruent modulo 12 with −2.

Definitions of Equivalence The relation “is congruent modulo 12 with,” when it is defined on PITCH, is what algebra refers to as an “equivalence relation.” In chapter 2, we have seen that this relation satisfies all three conditions of equivalence: reflexivity, symmetry, and transitivity. The concept of equivalence helps us bring large collections of musical elements down to reasonable proportions. PITCH can be mapped onto PITCHCLASS; and the collection of PC sets (PCSET) can be compressed by defining equivalences on the basis of certain transformations or common properties, as this chapter will show. This may sound rather trivial. It is common practice to assign equal names or labels to pitch values that are separated by one or more octaves, or to a chord and its transpositions, or to the transpositions and inversions of the

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subject of a fugue. Indeed, an informal notion of equivalence has always been part of music theory and analysis. PC set theory, however, has adhered to formal definitions of equivalence. Many such definitions invoke the three conditions mentioned above. However, the same kind of relation can be stated in another way, as follows. Equivalent elements in a collection form subcollections that are called “equivalence classes.” In turn, these equivalence classes form a “partition” of the original collection. This means two things: first, the sum of all equivalence classes exhausts the original collection; and, second, two equivalence classes do not share any elements. Thus, a relation in S that partitions S is an equivalence relation in the algebraic sense.1 Which relations between PC sets satisfy this concept of equivalence? Theorists have answered this question in different ways. It has always been obvious that they did not want to apply the concept whenever it was possible, but considered it appropriate to specific musical relations. In the 1960s it became customary to call pitch-class sets “equivalent” when they were related as transpositions or transposed inversions, which may reflect musical intuitions molded by traditional training. However, does this use of “equivalence” satisfy the algebraic definition of equivalence? It does, when the relation is defined in PCSET, and is cast in the sentence: “PC set A is transpositionally related to PC set B.” This relation is reflexive, for any PC set is transpositionally related to itself by T0. It is also symmetrical: when A is transpositionally related to B by Tn, B is transpositionally related to A by T−n mod 12. And it is transitive: when A is transpositionally related to B by Tn, and B is transpositionally related to C by Tm, then A is transpositionally related to C by T(n + m) mod 12. Once we specify an index of transposition—that is, a PIC, represented by n— these three properties vanish. The sentence “PC set A is related by the operation Tn to PC set B”—in formal notation: “B = Tn(A)”—does not imply that A and B might be equal. This means that the relation is not reflexive. Furthermore, the sentence cannot be reversed; in other words, the expression “B = Tn(A)” does not imply that A = Tn(B). This is the case only when n = −n mod 12 (that is, when n = 0 or 6). Therefore, the relation is not symmetrical. Nor is it transitive: when B = Tn(A), and C = Tn(B), it cannot be stated that C = Tn(A). This is the case only when (n + n) mod 12 = n (that is, when n = 0). In chapter 2, we already have seen that a relation between inverse-related PICs does not satisfy the three conditions of algebraic equivalence. Since PC values correspond to PIC values, the same holds true for PC sets and their inversions. The relation “is the inversion of,” as defined in PCSET, is not reflexive; most PC sets are not their own inversions. It is symmetrical: B = I(A) implies that 1. This partitional view of equivalence can be traced back to Richard Dedekind’s edition of Dirichlet’s papers on number theory (cf. Lejeune Dirichlet 1871, 136), and even further to Carl Friedrich Gauss’s Disquisitiones arithmeticae (cf. Gauss 1863, 222–23.).

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A = I(B). However, it is not transitive. When B = I(A), and C = I(B), it follows from the symmetrical property of the relation that C = A; and since the relation is not reflexive, C ≠ I(A). Transpositional and inversional relations between pitch-class sets do, however, satisfy a concept of equivalence defined with respect to relations between sets. According to this set-theoretic definition, two sets A and B are equivalent if there is a protocol defining a one-to-one correspondence between them—in other words, if sets A and B have the same number of elements, and each element of A is assigned to a different element of B. The relation “A is equivalent to B” is a possible relation in a family of pitch-class sets. What is more, it defines an equivalence in that family—that is, an equivalence in the algebraic sense—for it is reflexive (A is equivalent to itself), it is symmetrical (the sentence “A is equivalent to B” implies that B is equivalent to A), and it is transitive (the sentences “A is equivalent to B” and “B is equivalent to C” imply that A is equivalent to C). It may seem odd to say that the relation “A is equivalent to B” is itself an equivalence. However, this statement involves two different meanings of the term “equivalence,” one being a specific relation between sets (set-theoretic equivalence), and the other a property of that relation (algebraic equivalence). To summarize, Tn and I define relations between PC sets that are equivalence relations in the set-theoretical sense only. The set-theoretic equivalence relation, in turn, defines an algebraic equivalence relation in PCSET. Despite its aura of precision, the term “equivalence” can cause confusion. It has different mathematical meanings, and these meanings have been subject to change. The partitional view of algebraic equivalence gained general currency relatively recently, that is, in the 1930s and 1940s, through the algebra textbooks of Van der Waerden (Moderne Algebra 1930/1931), Albert (Modern Higher Algebra, 1937), and Birkhoff and Mac Lane (A Survey of Modern Algebra, 1941).2 And common-sense notions of musical equivalence must be taken into account as well. In view of this complex etymology, it is not surprising that music theorists have different concepts of equivalence, as I will show later. It is easy to wave this problem aside, reasoning that we are dealing with music, and that musical notions of equivalence differ from mathematical ones. Mathematical definitions can be adjusted to musical conditions, provided that this is stated explicitly. Therefore it is perhaps unfair to judge definitions of musical relations by purely mathematical standards. However, if this is unfair, then why have these standards been

2. The concept of an equivalence class is described but not named in these books. In an unfinished paper on the history of this concept, David Fowler (University of Warwick, UK) has reported that the English term “equivalence class” appeared in print for the first time in November 1941, in an article by Samuel Eilenberg and Saunders Mac Lane (“Infinite Cycles and Homologies,” Proceedings of the National Academy of Sciences of the United States of America, 27/11, 535–39). I am grateful to the late Dr. Fowler for sharing his information with me.

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invoked by PC set theorists? Mathematics served them as a model of clarity, as a badge of academic respectability, or as an authority encompassing music as well as many other fields of human knowledge and activity. This is why, in a study of the evolution of PC set theory, we have to deal with these mathematical definitions.

Equivalence and Equality Milton Babbitt’s definition of equivalence (1946) at first sight seems peculiar and arbitrary: Two sets, P and P', will be considered equivalent if and only if, for any pi,j of the first set and any p'i',j' of the second set, for all i’s and j’s, if i = i', then j = j'. (= denotes numerical equality in the ordinary sense). (Babbitt 1992, 8–9)3

In Babbitt’s vocabulary, a “set” was a twelve-tone series. As he saw it, the equivalence of two such series stipulated that they showed the same succession of PCs (or “js”).4 This means that he considered the term “equivalent” as synonymous with “equal.” Why did he speak of equivalence here? Perhaps he was guided by musical considerations. A twelve-tone series can take many different shapes. Therefore, a music theorist or analyst may prefer a word other than “equal” to account for the relation between two realizations of it, a word referring to a shared substance while providing scope for variation. “Equivalent” may have been that word for Milton Babbitt. It is noteworthy that he reserved it for such an exclusive relation. Apparently, he did not yet think of transpositional or inversional relations as “equivalence” relations.5 Babbitt’s use of the term “equivalent” is informal, in spite of the formal air his prose assumes (NB his use of sentential equivalence: “if, and only if”). However, it does not require much effort to “improve” his definition, to fill the gap that frustrates its full validation in terms of mathematical accuracy. From an algebraic

3. In this passage “P'” does not mean “the complement of P.” 4. When two series showed the same succession of “interval numbers,” or PICs, Babbitt called them “dependent.” “Dependent” series are necessarily related as transpositions. 5. This betrays a much more Schoenbergian view of twelve-tone serialism than is usually ascribed to Babbitt. He is known as a composer and theorist less inclined than Schoenberg to make hierarchical distinctions between the various forms of a twelve-tone series. He articulated this position in a review of René Leibowitz’s Schönberg et son école: “The functionality of a twelve-tone composition is defined by the specific twelve-tone set. A functional norm is stated, and deviations from this norm appear; but there is no degree of deviation, no hierarchy of deviations such as presented in tonal music, to make possible progress and growth—stated in terms of the functional context—through various stages of compositional expansion.” (Babbitt 1950, 59) Characteristically, in Babbitt’s compositions the “functional norm”—i.e., the prime form—is not necessarily stated at the beginning.

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perspective, equality is an equivalence relation. This is what Babbitt may have had in mind when writing the passage quoted above. It renders clear what is now so easily overlooked: that it is possible to conceive other equivalencies than the equality relation. For example, “giving F the same order position” is a relation that is reflexive, symmetrical, and transitive in the collection of all twelve-tone series, and can thus be called an equivalence. On that basis, (B♭,E,F♯,D♯,F,A,D, C♯,G,A♭,B,C)—the prime series-form of Schoenberg’s Variations for Orchestra, Op. 31—is equivalent to (C,B,D,C♯,F,F♯,E♭,E,A♭,G,B♭,A)—the prime seriesform of Anton Webern’s String Quartet, Op. 28. Equivalence, in the algebraic sense, is only a value assigned to a relation; it is not the relation itself. Moreover, the equivalence of two things is contingent upon a specific sentence expressing their relationship. This sentence does not actually express the relationship, but is an integral part of it. For example, in the collection of all people the relation “a is the sister of b” is not an equivalence, since it is not reflexive (nobody is her own sister) or symmetrical (one may be the sister of a brother). However, the phrase “is the sister of” can be replaced by “has the same parents as.” Then, the relation between a and b fully complies with the algebraic definition of equivalence. Babbitt’s later writings confirm that he did not care too much about the exact meaning of the term “equivalent.” In “Twelve-tone Invariants as Compositional Determinants” (1960) he rightly equates “octave-equivalence” with “congruence mod.12” (248), but in footnote 3, which discusses the fundamentals of group structure, the term “equivalence relation” is applied to equations (249). In “Set Structure as a Compositional Determinant,” an article from 1961, Babbitt calls complementary intervals “equivalent,” in a sentence that must have inspired Donald Martino to develop the APIC vector (see chapter 2, p. 47).6 Speaking of the relations between complementary hexachords, he defines the “total intervallic content” of a hexachord as: “the 15 intervals . . ., specified most conveniently in the form of the six possible intervals—since complementary intervals are equivalent—with their associated multiplicities.” (Babbitt 1961a, 80). The use of the word “equivalent” in relation to complementary (that is, inverse-related) intervals is deserving of criticism, which was expressed in chapter 2. For the next step in the evolution of the concept of PC set equivalence we turn to an article from 1963 by Allen Forte, entitled “Context and Continuity in an Atonal Work: A Set-Theoretic Approach.” This article—which presents an analysis of Schoenberg’s Six Little Piano Pieces, Op. 19—was the first to make explicit reference to set theory as an analytic tool, specifically for the non-serial atonal music written by Schoenberg between approximately 1908 and 1921. Forte began his argument by defining a universal set of twelve PCs—numbered from 0 (C) to 11 (B)—and specifying several subsets that he considered sig6. Martino paid tribute to Babbitt’s articles, including this one, as main sources of inspiration for his contribution to PC set theory. (Martino 1961, 224)

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nificant in Opus 19 (ex. 4.1). The abstract relations between these subsets were his first object of study. He described them in terms of naive set theory, identifying sets as the subsets or complements of other sets, or as the result of set union or set intersection (see chapter 3, p. 49). Set theory meant to him a “system,” a common tool for understanding all atonal music. His deductive approach was clearly inspired by Milton Babbitt’s formulation of the (serial) twelve-tone system. And it has become typical for the generations of American theorists after Forte’s to speak of musical structure as the realization of a coherent set of general principles. The system that Forte described in 1963 seems exclusively based on his analysis of Schoenberg’s Opus 19. Forte did not refer to a totality of PC sets extending beyond the ones he had selected from the score of this work. The deductiveness of his argument was largely a matter of rhetoric; in fact, the argument was circular. The term “equivalent” appears in the first section of Forte’s article, and is taken by him to mean “identical” (Forte 1963, 76). Indeed, Forte considered two subsets as equivalent when they consisted of the same elements. In such a case, mathematical set theory speaks of the “equality,” not the “equivalence,” of sets. The critique that can be leveled against Babbitt’s definition of equivalence holds true for Forte’s as well. In any collection of sets, equality is, of course, an equivalence relation; but it is pointless to say, without further qualification, that two sets are “equivalent.” In example 4.1, the PC sets A4 and B5 are equivalent on the basis of their identity relation (“A4 is equal to B5”). However, other equivalence relations can be defined as well. For example, the PC sets A1 and B4 are equivalent on the basis of their equal cardinal numbers (“A1 has the same number of elements as B4”), and so are A2, B1, and B2.

The Study of Interval Content Less than two years separate “Context and Continuity” from “A Theory of SetComplexes for Music” (1964), a period of mushrooming growth in PC set theory. The system described in the latter article is much more elaborate than the one presented in the former, and—above all—it is much more general in scope.7 Forte maps out a large number of possible musical relations. It is significant that the role of music analysis, the study of one or more compositions, is considerably reduced. The analytical section covers only five pages out of forty-two. In 1963, it was five pages out of eleven. Meanwhile, the proposed field of analytical inquiry exceeds the early atonal repertoire of the Second Viennese School, Forte’s main focus in the earlier article. Pointing out the relevance of the PC set, Forte mentions the music of Liszt, Debussy, and Scriabin before that of Schoenberg and 7. Forte acknowledged that “it was the deficiencies of [the] analysis [in “Context and Continuity”] which led to the present article” (Forte 1964, 182, note 18).

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Perspectives of New Music, 1/2 (Spring 1963): 74–75. Used by permission of Perspectives of New Music.

Example 4.1. PC sets extracted from Schoenberg’s Six Little Piano Pieces, Op. 19. An example from Allen Forte’s article “Context and Continuity in an Atonal Work: A Set-Theoretic Approach.”

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his pupils.8 He also refers to precedents in theoretical writings by FrançoisJoseph Fétis, Carl Friedrich Weitzmann, Arnold Schoenberg, Bernhard Ziehn, and Charles Ives (Forte 1964, 136–37). Thus, he places the use of set theory in music analysis historically, with reference to a variety of musical styles and theoretical concepts from the past. Strangely enough, the musical examples do not reflect the broadening of Forte’s perspective. Forte took them from the fourth of Anton Webern’s Five Movements for String Quartet, Op. 5, one of the most accessible pieces from the Viennese early atonal repertoire.9 During this early phase of its development, PC set theory was still marked by a major commitment to the works of Schoenberg and his pupils, notwithstanding Forte’s lip service to other composers. In 1965, shortly after the publication of “A Theory of Set-Complexes,” Forte read a paper at a New York conference on computer applications in musicology, discussing the implementation of his theory in computer-aided analysis. In this paper—with the same title as his later book, The Structure of Atonal Music—he was concerned with “a structural description of the revolutionary and enigmatic non-tonal music of Schoenberg, Berg, and Webern composed during the period 1908–23, so-called atonal music” (Forte 1970, 10). The example he used was Webern’s Three Pieces for Cello and Piano, Op. 11.10 The actual mission was still the one he had stated in 1963, in “Context and Continuity”: to show that the non-serial atonal music of the Second Viennese School satisfied criteria of musical logic and coherence as much as tonal or serial music. Forte’s claim that the concept of a PC set could be applied to other repertoire may have been gratuitous at the time, but he had in any case compiled an ingenious list of PC sets representing all possible combinations of tones under twelve-tone equal temperament without regard to functional context. He had reduced the sum total of 4,096 PC sets (see chapter 2, p. 47) by partitioning it into

8. The idea that these composers applied “non-tonal sets” (structurally significant groupings of tones that do not constitute tonal chords) in their music, which can thus be related to the atonal and serial repertoire of the Second Viennese School, had probably been suggested to Forte by George Perle’s book Serial Composition and Atonality, to which he casually refers. Perle, in his discussion of the work of composers like Debussy, Scriabin, and Roslavets, speaks of “nondodecaphonic” sets (structurally significant groupings of tones that are not part of a twelve-tone series). I believe that both authors refer to the same class of things (structurally significant groupings of tones that are neither tonal chords nor twelve-tone series), which they view from different historical perspectives. 9. Forte was not the first to deal with this piece analytically. For example, Perle— again—had included an analysis of it in his Serial Composition and Atonality. 10. None of the important papers following on Forte’s—papers by Howe (1965), Teitelbaum (1965), Clough (1965) and Chrisman (1971)—dealt with music from another repertoire either, if they dealt with other works than Webern’s Opus 5 at all. Howe and Teitelbaum examined this work once again; the papers of Clough and Chrisman discussed conceptual issues without reference to written music.

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200 equivalence classes, or set-classes (example 4.2 shows the classes of cardinal numbers 5 and 7). This time, Forte’s definition of equivalence fully complied with the algebraic standard: We now define an equivalence relation for arbitrary pitch-sets A and B. Let v(A) represent the interval-vector of set A, and v(B) the interval vector of set B. Then, A ≡ B if and only if v(A) = v(B)11

By “interval vector” Forte meant the APIC vector. He claimed that “the defined equivalence [had] the reflexive, symmetric, and transitive properties” (Forte 1964, 143). This is true. The equal-vector relation obtains when the PC sets A and B are equal; it also obtains when they are reversed; and if B has its APIC vector in common with a third PC set C, then so does A. Ironically, it was this new, sound basis of algebraic equivalence that would soon come under attack, and would be withdrawn by Forte afterwards. This is not difficult to understand. First, however, it has to be explained why Forte had decided to consider interval content the prime distinguishing property of a PC set. As we have seen, Lewin (1960), Babbitt (1961a), and Martino (1961) had been pondering a way to encode the total interval content of a PC set. The issue cropped up in relation to the serial technique of Arnold Schoenberg, especially his use of hexachords. Schoenberg’s twelve-tone series are usually so constructed that the two halves—the hexachords—have no PCs in common with their inverted forms at some transpositional level. Each hexachord and its transposed inversion thus fill out the total chromatic. In Babbitt’s words, they constitute the twelve-tone “aggregate.”12 This is only possible if the first six PCs of a twelve-tone series form a set that has a TnI-related complement. Then, reordering this complement, i.e., the last six PCs, yields the first half of a transposed inversion of the series (ex. 4.3a, inside the boxes). Of course, a similar relation exists between the first half of the prime form and the second of the transposed inversion. Babbitt called this property “hexachordal inversional combinatoriality.” The term “combinatoriality” refers to combinatorics, a branch of mathematics dealing with the permutations and combinations of elements of finite sets. The property has long been regarded as peculiar to Schoenberg’s series, probably because Schoenberg himself has written something about it. Here is that famous statement from his paper “Composition with Twelve Tones”: 11. Among the 200 equivalence classes resulting from this definition there were three that Forte considered trivial: the unique classes of sets with one, eleven and twelve elements. 12. Babbitt used the word “aggregate” when he considered the total chromatic without regard to order.

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Example 4.2. A part of Forte’s 1964 table of set-classes, representing the PC sets of cardinal numbers 5 and 7. Each class has an ordinal number, which is called a “set number,” and can be found in the first column of each table. (The “Z”s will be dealt with in a later section of this chapter.) This order number is based on the magnitudes of the successive entries in the APIC vector. The first set-classes are those with the highest magnitudes of APIC 1: 4 in the case of set-classes of cardinal 5, and 6 in the case of set-classes of cardinal 7. When the magnitudes of APIC 1 are equal, those of APIC 2 are considered, and so on. The vectors are specified in the third column of each table. The second column contains one representative of each set-class, the PCs of which are given in “normal order” (another term awaiting discussion). Finally, the fourth column gives the number of PC sets per class. Equivalence classes of cardinal 5 and 7 with corresponding order numbers contain PC sets related as complements with respect to PITCHCLASS. Allen Forte, “A Theory of Set-Complexes for Music,” Journal of Music Theory, 8/2 (1964): 147. Used by permission of Duke University Press and Yale University.

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Later, especially in larger works, I changed my original idea, if necessary, to fit the following conditions: the inversion a fifth below of the first six tones, the antecedent, should not produce a repetition of one of these six tones, but should bring forth the hitherto unused six tones of the chromatic scale. Thus, the consequent of the basic set, the tones 7 to 12, comprises the tones of this inversion, but, of course, in a different order. (Schoenberg 1984, 225)

It is not so much the combinatorial property itself that is typical of Schoenberg’s series—for example, Webern’s series have this property as well—as rather the use he made of it. Schoenberg’s idea was to simultaneously unfold two series-forms while not allowing octave-related tones to coincide. Babbitt’s mission was to attain a more general level of understanding of this idea. He had noted that the interval of a perfect fifth below (PIC 5) had no special meaning in this regard, apart from the symbolic meaning Schoenberg may have attributed to it.13 Even in Schoenberg’s own works there are instances of hexachordal inversional combinatoriality at other transpositional levels, such as the series of the Variations for Orchestra, Op. 31 (T9I), or that of Moses und Aron (T3I). Furthermore, Babbitt wanted to prove that the combinatorial property was not just typical for inversionally related twelve-tone series (or for the simple case of a series and its retrograde), but could also involve transpositionally related series, or series related as a prime form and its transposed retrograde-inversion. In these cases, hexachordal combinatoriality stipulated conditions other than TnI-related hexachords: for the two halves of a series to complement the corresponding halves of a transposition of that series, they need to be transpositionally related to one another, and for the two halves of a series to complement the corresponding halves of a transposed retrograde-inversion of that series, they need to be inversionally symmetrical. When all these conditions were met—that is, when the two halves of a series were TnI-related, inversionally symmetrical, and hence transpositionally related as well—Babbitt called the series “all-combinatorial,” indicating that its two halves complemented the corresponding halves of all other series-forms at one or more transpositional levels. Example 4.3b shows how the second half of such a series can be reordered to yield the first half, not only of its retrograde but also of a transposition (lower staff, T6) and an inversion

13. It was a token of the past. It served to link Schoenberg’s music to that of the tonal tradition, of which the composer could not stop seeing himself as a successor. (See the references in chapters 1 and 3 to the role of the fifth in his Piano Piece, Op. 23, no. 3, pp. 7 and 70). Schoenberg’s remarks on another piece, the Wind Quintet, Op. 26, prove that, at the time, hexachordal inversional combinatoriality mattered to him especially when it involved two series related as prime form and T5I. What he claimed only is that the corresponding hexachords of the prime and T5I forms do not fill out the total chromatic (Schoenberg 1984, 225). He failed to mention that the relation of the prime and T11I forms do satisfy this condition. Significantly, in the piece itself this relation is not exploited.

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(0)

{C , D , D , E , G , G }

Example 4.3a. Hexachordal inversional combinatoriality; Arnold Schoenberg, Piano Piece, Op. 33a. The first hexachord of the prime series-form is related to the second by T5I. This means that the second hexachord can be reordered to yield the first hexachord of an inversion.

(0)

{F, F , A , B , D , D}

Example 4.3b. Hexachordal all-combinatoriality; Schoenberg, Modern Psalm, Op. 50c. The first hexachord of the prime series-form is related to the second by T2, T6, T10, T1I, T5I, and T9I. The second hexachord can thus be reordered to yield first hexachords of all four series-forms.

(0)

{C , E , E , F}

{F , G , A , B}

Example 4.3c. Tetrachordal combinatoriality; Schoenberg, Fourth String Quartet, Op. 37. The second tetrachord of the prime series-form is related to the third by T7I. The three tetrachords of this series-form can thus be reordered to yield those of a retrograde-inversion.

(middle staff, T5I). Finally, it was Babbitt’s aim to show that tetrachords, trichords, dyads, and even single PCs could be sources of combinatoriality as much as hexachords (ex. 4.3c). Combinatoriality, as it is generally understood, involves unordered PC sets underlying non-corresponding sections of different series-forms. When Babbitt considered a twelve-tone series in terms of such PC sets, he called it a

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Example 4.4. The source set of Schoenberg’s Piano Piece, Op. 33a.

“source set.”14 Example 4.4 shows what this means for the twelve-tone series of Schoenberg’s Piano Piece, Op. 33a (cf. ex. 4.3a): it is reduced to a succession of two hexachords defined with respect to PC content only, ignoring order. The “source set” can be seen as providing the historical link between twelve-tone theory and PC set theory: observing that two sections of different series-forms contain exactly the same PCs comes close to stating that these sections represent different orderings imposed on one, unordered PC set. This shows how closely related the concepts of a PC set and a twelve-tone series actually are. The actual step toward emancipation of the PC set as a musical concept was taken when theorists started to define groupings of PCs in their own right, without reference to a twelve-tone series. Since no particular intervallic relationships among the PCs prevailed, and since the familiar twelve-tone operations did not preserve the PCs themselves, these groupings were defined in terms of their total interval content. This property, for which David Lewin and Donald Martino had given formulas in 1960 and 1961 (see chapter 2, p. 47), is independent from permutation, transposition, or inversion. Being less susceptible to change, it was considered to be more fundamental to a PC set than the PCs themselves. In music theory, it is customary to define a musical entity, not by its constituents (such as pitches), but by the relations between its constituents (such as pitch or time intervals). Entities thus defined include modes, hexachords, chords, fugue subjects, and sonata themes. The study of interval content had opened up new musical relations that intrigued composers and theorists. And this study was all the more appealing now that the tedious labor it involved could be left to computers.15 One particularly exciting result was the discovery that any two hexachords complementing one another with respect to PITCHCLASS are equal in terms of interval content, even if they are not related by transposition and/or inversion. The statement of this relationship has become known as the “hexachord theorem,” and is commonly attributed to Babbitt. Lewin (1960) presented it as a special case of a

14. The term “source set” was introduced in Babbitt 1955, 57n. Babbitt had described the concept already in his 1946 doctoral thesis (Babbitt 1992, 104–16), occasionally referring to it as a “basic set.” Schoenberg, too, used the term “basic set,” though with reference to the non-transposed prime series-form. 15. The influence of the computer on the formulation of research questions and the valuation of research results in the early 1960s can hardly be overestimated. Chapter 8A will give an account of the euphoria induced by the possibilities of computer-aided research among American composers and musicologists.

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more general statement to the effect that the difference between the frequency rates of a PIC in two complementary PC sets equals the difference between the cardinal numbers of these sets. (This is called the “complement theorem.” In the case of two complementary hexachords the difference is, of course, 0.)16 Lewin related the hexachord theorem to the typical way in which Schoenberg partitioned his twelve-tone series. In the following quotation, the term “interval function” refers to Lewin’s concept of the inclusive intervallic relation between two PC sets: If a piece is based on a hexachord P and is so constructed that, in general, some form of P is always being played against the complementary form—P being distinguished from its complement at any given point, e.g. by instrumentation, melody-accompaniment, etc.—and if the order of notes within the hexachords is not too strictly constrained, then the interval function m(i) between P and [its complement] P' will probably exert an aural effect in the long run. Since m(i) represents the “total ‘contrapuntal’ relation” between P and P', of course it also represents the “total ‘contrapuntal’ relation” between any transposed or inverted form of P or P' and the corresponding form of P or P'. In this sense the “total potential counterpoint” in any section of the piece is always the same. And . . . this “total potential counterpoint” uniquely characterizes P, P', and their transposed and inverted forms. This may explain in what sense a hexachord has an aural “identity” in such a piece.” (Lewin 1960, 100)

In brief, the combinatorial properties of the twelve-tone series on which a work is based place certain constraints on the harmonic idiom of that work. (Since “total contrapuntal relation” implies linear statements of the two sets, which do not necessarily occur, I prefer speaking of “harmony” instead of “counterpoint” here.) Whereas Lewin thought of “harmonic idiom” only in terms of aural effect, Milton Babbitt invoked the composer’s intentions. In his words, “this characteristic of singular harmonic definition [suggested] one important reason for Schoenberg’s compositional concern with hexachords as combinational units” (Babbitt 1961a, 80). The claim that Schoenberg’s technique was informed by an awareness of the equal distribution of intervals in complementary hexachords is probably too bold; Schoenberg’s writings give no evidence of such awareness. However, what is important here is the rise of interval content to the status of an analytical category, a determinant of musical structure. Forte’s decision to choose interval content as a basis of equivalence was a logical result of this development.17

16. Apart from Lewin, other theorists set out to prove the hexachord theorem mathematically as well. Proofs were published by Kassler (1961), Regener (1974), and Wilcox (1983). 17. He was not even the first to classify PC sets accordingly. Donald Martino, whose research in the field of combinatoriality built on Babbitt’s, had already provided tables of hexachords, tetrachords, trichords, and pentachords in his 1961 article “The Source Set and its Aggregate Formations,” defining them in terms of interval content.

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“Operational Equivalence” As mentioned, Forte’s interval-oriented definition of PC set equivalence was soon challenged. In 1965, the equivalence problem was raised in an article that appeared in the same Journal of Music Theory that had published Forte’s “A Theory of Set-Complexes” the year previously. The author was John Clough, a budding specialist in the field of music theory and mathematics. His main objection to Forte’s definition of PC set equivalence was that Forte had invoked a relatively unobvious relation—equality of interval content—whereas most PC sets involved in analyses resembled each other in a more obvious way as well: as a set and its transposition, inversion, or transposed inversion (a fact that Forte had not failed to observe either). And the word “most” was the symptom of another, even more acute problem: some PC sets with equal interval contents were not in such an obvious relation. The set-classes that contain these PC sets are marked with a “Z” in Forte’s table (for “zygotic,” a term originally referring to a pair of animals under a yoke, but currently in use in the field of biology to denote the fusion of two reproductive cells; see ex. 4.2); this peculiar relation between PC sets has become known as the “Z-relation” (ex. 4.5).18 Clough raised the question of whether the “equality-of-intervallic-content” relation partitioned the collection of PC sets in a way that was musically meaningful and unambiguous. He proposed to define the equivalence of PC sets A and B as follows: A ≡ B if and only if A and B are related by IT (Clough 1965, 169)

Clough had taken the label “IT” from Babbitt 1960 and Forte 1964. This label denoted the same operations as TnI. (“Invert and then transpose A” and “transpose the inversion of A” have the same results.) For Clough, however, the exact protocol was not an issue at all. He used the label “IT” as an abbreviation of “transposition and/or inversion” signifying a relation between PC sets that was based on one of the two operations, or on both (165n). Clough’s article was the first in the field of PC set theory to reveal the mathematical background of the term “equivalence,” and to come up with other, less apparent, maybe less useful, but equally valid equivalencies among PC sets.19 18. The discovery of the relation—reported by Lewin (1960)—was another result of the study of interval content. Some “Z-related” pairs are connected by M (or IM), a relation depending on an APIC vector with identical entries for the interval classes 1 and 5 (see chapter 3, p. 82). Forte revealed the meaning of the “Z” to the 25 participants of a seminar on the history of serialism at the Orpheus Institute in Ghent, Belgium, on November 17, 2004. 19. For example, Clough defined a property which he called the odd-even characteristic of a PC set: “the absolute value difference between the number of notes in one whole-tone scale and number of tones in the other whole-tone scale” (Clough 1965, 164). The relation between two PC sets sharing this characteristic is an equivalence in the algebraic sense.

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99

4 4

APICs:

5

3 1

1

{ 0, 1, 3, 4, 8 } 4

2 3 5 4 5

APICs:

4

4 3

1

{ 0, 3, 4, 5, 8 } 3

1 2 5

Example 4.5. Two PC sets not related as a set and its transposition, inversion or transposed inversion, but nevertheless having the same interval content (Forte’s “Z-relation”). With their common APIC vector 212320 they both belong to the set-class of cardinal 5 listed as number 17 by Forte in his 1964 table of set-classes (see ex. 4.2).

The relation that he proposed as an alternative to Forte’s equal-vector relation, based on T and/or I, would ultimately become the paradigm of PC set equivalence. Having first rejected it in a rather haughty reply to Clough’s article in the same issue of Journal of Music Theory, Forte adopted it later, without reference to Clough, in The Structure of Atonal Music (Forte 1965a, 1973).20 To define an algebraic equivalence on the basis of transposition and inversion, as Clough did, seems problematical. Being not reflexive or transitive (in general), an inversional relation is not a proper equivalence of this kind, and a transpositional relation satisfies the criteria of algebraic equivalence only when no index of transposition is specified. The term “equivalence” applies to these relations in the way of Hubert S. Howe’s “operational equivalence”: The PC coll[ection] y obtained by performing an operation o upon a PC coll[ection] x will be called the operational equivalent of x under o. (Howe 1965, 50)

Yet it was the standard algebraic definition that Clough invoked. Transpositional and inversional relations are consistent with this definition as long as it is immaterial which one of them obtains. Although Clough’s statement is valid in itself, it may strike one as odd that each of the operations Tn, I, and TnI instantiates 20. It was necessary for him to do so, since the equal-vector relation contradicted the relational concept that was crucial to Forte’s theory: the PC set complex (Clough 1965, 169; see also chapter 6.).

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PC set equivalence, for none of them can serve as a formal basis for equivalence without the other. Is the algebraic concept of equivalence actually suited to represent these musical relations? In any case, assigning objects to equivalence classes accurately models the way in which we speak and write about music. When we call something a “major triad,” a “pentatonic scale,” a “triplet,” or an “F♯,” we identify it as a member of a class. These names refer to properties the objects concerned have in common with other objects. And “having property x in common with” is an equivalence in any collection of objects. However, when we define a class of transpositionally and/or inversionally related PC sets, we are not, at first sight, dealing with a property they share, but with a set of operations transforming one into another. Since operations are not an obvious basis of algebraic equivalence, can we state their relation in terms of a common property, a property invariant under transposition and inversion and not shared by PC sets of other classes? There is at least one property invariant under transposition. The composer and theorist Richard Chrisman drew attention to it in 1971, in an article entitled “Identification and Correlation of Pitch-Sets.”21 Chrisman stated that two PC sets with the same interval content would not yield two equal PIC successions if they did not relate as transpositions. In other words, what transpositionally related PC sets have in common is a set of PIC successions. It is possible to reduce the number of these successions by arranging the PCs in ascending order and subjecting them to cyclic permutation. A common application of this procedure is the derivation, in manuals of tonal harmony, of the 6/3 and 6/4 chords from a root position triad by successively transferring the lowest notes to a higher octave. Chrisman would define these chords, not in terms of pitch intervals (or PICs) measured from the bass—as in figured-bass practice—but in terms of adjacent pitch intervals bottom up, including the interval completing the range of an octave. Thus, he would identify the three bass positions of a major triad as “4–3–5,” “3–5–4,” and “5–4–3” respectively (ex. 4.6a). Chrisman called these successions of pitch intervals “successive-interval arrays.” Each PC set generates a number of these arrays equal to its cardinality. Chrisman called two successive-interval arrays “equivalent” when one was a cyclic reordering of the other. Apart from PC sets that were equal, this equivalence involved transpositionally related PC sets. It did not involve PC sets related as (transposed) inversions. Ultimately, however, Chrisman assigned PC sets and their inversions to the same set-classes, which roughly equal the set-classes identified by Forte in 1964 (see ex. 4.2. Chrisman even used Forte’s “set numbers” to label these classes. See his table 1 on p. 79). He did not include the inversions on the basis of a defined equivalence. The inversional relation was described by him as follows: 21. The article was derived from Chrisman’s dissertation: A Theory of Axis-Tonality for Twentieth-Century Music (Yale University, 1969).

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101

(a)

4

3

5

3

5

4

5

4

3

3

4

5

4

5

3

5

3

4

(b)

Example 4.6. Richard Chrisman’s successive-interval array. An application to the major and minor triads. “The interval array of one [PC set is] equivalent by cyclic reordering to the retroversion of the other.” (Chrisman 1971, 70; my italicization)

For example, the retroversion of the successive-interval arrays pertaining to the major triad—i.e., “4–3–5,” “3–5–4,” and “5–4–3”—yields those pertaining to the minor triad: “5–3–4,” “4–5–3,” and “3–4–5” (ex. 4.6b). However, the successiveinterval array of a PC set is not equivalent by cyclic reordering to that of its inversion. This means that Chrisman’s set-classes do not satisfy the definition of an equivalence class. Chrisman’s article can be seen as an attempt to define PC sets (or “pitch sets,” as they were called at the time) in terms of a more exclusive concept of their interval contents than the APIC vector. An algebraic equivalence in PCSET, involving only a PC set and its transpositions, can be defined on the basis of a common set of successive-interval arrays. This equivalence did not involve inversionally related PC sets. Notwithstanding the obvious relation between their successive-interval arrays, it proved difficult to pin down a property held in common between PC sets and their inversions.

Problems of Representation Useful though it may have been, Chrisman’s concept of a PC set has not established itself in the literature.22 A common set of successive-interval arrays is not a property lending itself to concise modeling. For the sake of conciseness, we can single out one of them as a referential array, but this requires a fixed protocol. 22. Eric Regener (1974) and Robert Morris (1987, 40, 82; 1995) have proposed concepts that are similar to Chrisman’s successive-interval array. Morris coined the term “cyclic interval succession,” or “CINT.” His “pitch-class” interval succession” (PCINT) will receive brief coverage later in this chapter.

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There already was such a protocol when Chrisman published his article. It did not relate to successive-interval arrays, but to PC sets. One used to speak of the “normal form” or “normal order” of a PC set, a concept that can be traced back to the teaching and writings of Milton Babbitt, like so many other concepts of PC set theory. The following definition appeared in “Set Structure as a Compositional Determinant”: It is suggested that a given H[exachord] be ordered in “normal form” wherein the interval determined by the first p.c.−last p.c. is greater than any interval determined by p.cs in H. (Babbitt 1961a, 77)

Babbitt’s language is obscure here. In order to clarify his definition, let us consider the example given by him: the first hexachord of the series-form opening the third movement of Schoenberg’s Fourth String Quartet, Op. 37 (ex. 4.7). Babbitt identifies (0,1,4,5,6,8) as its normal form. Two things are important: first, we can obtain these numbers from the first six notes of Schoenberg’s movement if we assign the value 0 to the PC G; second, in this order they represent the most compact arrangement of the PCs involved. Apparently, what Babbitt means to say is this: in any cyclic ordering of these six PCs—like (G,A♭,B,C,C♯,E♭), (B,C,C♯,E♭,G,A♭), or (E♭,G,A♭,B,C,C♯)—the adjacent PCs E♭ and G yield the largest difference modulo 12 of all adjacencies, namely 4. When this adjacency is not included—in other words, when G is the first PC and E♭ the last—the PC set is represented in normal, that is, most compact form. From Babbitt’s rather casual definition, we can infer that the concept of normal form had already been in use for some time. Otherwise, Babbitt would have introduced it formally. Applying rules of scientific discourse in statements about music was very important to him (as we shall see in chapter 8B). Another, even more casual reference to normal form seems to confirm that the reader’s familiarity with the concept was taken for granted. This reference comes from Donald Martino’s article “The Source Set and its Aggregate Formations,” which was published in the same year and in the same journal as Babbitt’s article (i.e., Journal of Music Theory): Given a [hexachord] whose combinatorial properties are unknown, first reduce it to normal form, then count up the fifteen intervals and consult the table. (Martino 1961, 227)

Someone who had devised an algorithm for obtaining the normal form of a PC set was James K. Randall, another composer from Babbitt’s circle at Princeton. Hubert Howe provided the following summary of this description, with reference to Randall’s unpublished study Pitch-Time Correlation from 1962. The term “pitchclass collection” (“PC coll”) refers to a PC set. The term “pitch-structure” designates a collection of transpositionally related PC sets:

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poco accel. 3



103

a tempo 3

Example 4.7. The opening of the third movement of Schoenberg’s Fourth String Quartet, Op. 37. The first hexachord of the twelve-tone series is marked. STRING QUARTET, NO. 4, OP. 37, by Arnold Schoenberg. Copyright © 1939 (Renewed) by G. Schirmer, Inc. (ASCAP). International Copyright Secured. All Rights Reserved. Used by Permission.

(a) Select from among the pitch-class collections xi contained in a pitch-structure z the PC collections which contain the PC 0 (zero) and arrange the PCs in numerical order. (For example, the pitch structure containing the PC coll 0127 and its transpositions yields the following four sets: 0127, 016 11, 05 10 11, and 0567.) (b) Select from these PC colls the set or sets which have the smallest final number. (In the example above, this step yields the sets 0127 and 0567.) (c) If step (b) yields more than one set, select from among these sets the set with the smallest second, third, etc., number until only one set remains. (Continuing the above example, this step yields 0127.) The unique ordered number-set thus obtained is defined as the normal form of the pitch-structure z. (Howe 1965, 49)

The normal form corresponds to one of the successive-interval arrays of a PC set. It represents all possible realizations of this PC set and its transpositions. Allen Forte adopted the concept in “A Theory of Set-Complexes,” with a reference to Martino’s article. It appeared in his table of set-classes, where it was designated “normal order” (see ex. 4.2, the second column). However, Forte did not adopt it without changes. The normally ordered PC sets in his table—all of which shared the same first element (0, or C)—represented set-classes, but these set-classes did not consist only of transpositionally related PC sets. Since the APIC vector served as the basis of equivalence, they also contained the inversions of these PC sets. And in a few cases they contained PC sets that were neither transpositionally nor inversionally related to the normally ordered reference set (the “Z-related” sets). This means that the normally ordered reference set represented many more PC sets than just its transpositions. Forte had thus extended the range of PC sets covered by the concept of a normal form. I believe he did this inadvertently; it was the consequence of using a common representation (the normally ordered reference set) based on a more exclusive relation (transposition, or a common set of successive-interval arrays) than the relation that actually determined the set-classes (equal interval content). This discrepancy between representation

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and content of Forte’s set-classes required clarification, which Forte eventually provided in The Structure of Atonal Music. In “A Theory of Set-Complexes,” Forte had not indicated by which protocol he obtained the normally ordered reference set of a set-class. This was an omission, for if “normal” means “most compact,” there may be several normal forms from which to choose. First, among the cyclic permutations of a PC set there may be more than one with the smallest difference modulo 12 between the first and last PC values. Second, in most cases a PC set and its inversion have different normal forms. And third, if a set-class contains two members not related as a set and its transposition, inversion, or transposed inversion—that is, a Z-pair— one needs to take into account the normal forms of each. Forte had only casually referred to an “optimal” normal order (Forte 1964, 144). In The Structure of Atonal Music, however, he described its derivation. Here is a summary: Step 1 Consider a PC set. Put the PC values in ascending numerical order. Step 2 Consider all cyclic permutations of this ordered set. Select the permutations with the smallest PIC value determined by the ordered pair of the first and last PCs. Proceed, if more than one permutation remains. Step 3 For each selected permutation, consider the ordered pairs of the first, and each of the remaining PCs. Select the permutation with the smallest non-common PIC value determined by these ordered pairs. If there is no such permutation among the ones that remain, each of them may serve as normal order. An example will be helpful here. The PC set (1,4,6,7,8,11) is already ordered by step 1. Step 2 yields two other permutations: (4,6,7,8,11,1) and (11,1,4,6,7,8). The ordered pairs of the first and last PCs of these permutations yield the smallest possible PIC value, namely 9. Since step 2 singles out two permutations, it is necessary to proceed with step 3. In both permutations, the smallest PIC value determined by an ordered pair involving the first PC and one of the other PCs is 2: PIC(4,6) and PIC(11,1). The smallest-but-one such value is 3 in the first permutation (PIC(4,7)), and 5 in the second (PIC(11,4)). Therefore, 3 is the smallest non-common PIC value, and (4,6,7,8,11,1) is the optimal, or “best,” normal order of this PC set (cf. Forte 1973, 4). One additional step should be taken: Step 4 Consider the remaining permutation of our PC set. Rename the PCs in terms of the PIC values pertaining to the ordered pairs they form with the first PC. The result is an ordered PC set the first element of which is 0, like the sets in the second column of the table in example 4.2. The PC set (4,6,7,8,11,1) thus

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transforms into (0,2,3,4,7,9). In The Structure of Atonal Music this ordered PC set is called “prime form” (Forte 1973, 5). The term “prime form” originally belonged to the vocabulary of twelve-tone serialism. “Prime set”—or simply “prime” (Babbitt 1955, 1992; Perle 1962)—was an alternative designation of what Schoenberg had called a Grundgestalt, or “basic set.” In “A Theory of Set-Complexes” Forte had used the term “prime form” as the antonym of “inversion,” calling to mind the traditional pair of rectus and inversus. (“Normally each PC set has two basic and distinct forms: a prime form and an inversion”; p. 144.) But it ended up referring to the final outcome of the above reductive protocol. How did Forte deal with the conflict between the representation and the content of his set-classes? Only PC sets that are related as transpositions will produce the same prime form. How can one prime form represent a PC set and its inversion at all transpositional levels, let alone two “Z-related” PC sets? In The Structure of Atonal Music, as will be recalled, set-classes consist only of PC sets that are related transpositionally and/or inversionally. Here, Forte had finally met John Clough’s criticism of his earlier definition, which proceeded from the interval vector; he had abandoned equality of interval content as a basis of PC set equivalence. However, there was still a need for a representational format applying equally to the transpositions of a PC set and those of its inversion. Forte’s proposal was to consider the PC set and its inversion in normal order, and to repeat steps 2, 3, and 4 above (cf. Forte 1973, 12–13). One of these normally ordered PC sets should prevail over the other (unless they were inversionally symmetrical, of course), and this set was to be reduced to its prime form. To return to our example: the inversion of (4,6,7,8,11,1) is (8,6,5,4,1,11). The normal order of this new PC set is (4,5,6,8,11,1). The PIC value of its initial pair—that is, (4,5)—is smaller than that of the normally ordered original set. It is, in other words, the smallest non-common PIC value of step 3. Therefore, (4,5,6,8,11,1) prevails over (4,6,7,8,11,1). Step 4 reduces it to (0,1,2,4,7,9), the Fortean prime form of the setclass to which these inversionally related PC sets belong. We can also obtain the prime form without first determining the normal order. We should then apply Randall’s algorithm to the transpositions of a PC set and its inversion. Given an arbitrary set, which I call A: Step 1 Consider all transpositions of A and I(A). Put them in ascending numerical order. Step 2 Select the PC sets beginning with 0, and having the smallest last integer. Step 3 Select the PC set with the smallest non-common integer. The PC set that remains is the ultimate prime form. Table 4.1a and 4.1b show how, according to this protocol, (0,1,2,4,7,9) is identified as the prime form of the PC set {1,4,6,7,8,11}.

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Table 4.1a. The transpositions of the PC set {1,4,6,7,8,11} and its inversion 1

4

6

7

8

11

11

8

6

5

4

1

2

5

7

8

9

0

0

9

7

6

5

2

3

6

8

9

10

1

1

10

8

7

6

3

4

7

9

10

11

2

2

11

9

8

7

4

5

8

10

11

0

3

3

0

10

9

8

5

6

9

11

0

1

4

4

1

11

10

9

6

7

10

0

1

2

5

5

2

0

11

10

7

8

11

1

2

3

6

6

3

1

0

11

8

9

0

2

3

4

7

7

4

2

1

0

9

10

1

3

4

5

8

8

5

3

2

1

10

11

2

4

5

6

9

9

6

4

3

2

11

0

3

5

6

7

10

10

7

5

4

3

0

Table 4.1b. The same PC sets, put in ascending order . . . 1

4

6

7

8

11

1

4

5

6

8

11

0

2

5

7

8

9

0

2

5

6

7

9

1

3

6

8

9

10

1

3

6

7

8

10

2

4

7

9

10

11

2

4

7

8

9

11

0

3

5

8

10

11

0

3

5

8

9

10

0

1

4

6

9

11

1

4

6

9

10

11

0

1

2

5

7

10

0

2

5

7

10

11

1

2

3

6

8

11

0

1

3

6

8

11

0

2

3

4

7

9

0

1

2

4

7

9

1

3

4

5

8

10

1

2

3

5

8

10

2

4

5

6

9

11

2

3

4

6

9

11

0

3

5

6

7

10

0

3

4

5

7

10

Note: Twelve sets begin with 0, four of them having the smallest last integer. The boldly printed set is the prime form.

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107

It is unfortunate that, in most cases, the PIC succession of the prime form is not obtainable from all members of a set-class. Forte did not actually resolve the conflict between representation and content.23 However, the prime form figured prominently in Forte’s new definition of PC set equivalence: Two pc sets will be said to be equivalent if and only if they are reducible to the same prime form by transposition or by inversion followed by transposition. (Forte 1973, 5)

There is a subtle difference between this statement and the one John Clough had proposed eight years before. Clough spoke of operations transforming PC sets into other PC sets (“related by IT”). Forte spoke of a property that PC sets may have in common (“being reducible to the same prime form. . . .”). It is the same subtle, but not unimportant, distinction as that between “being a sister of” and “having the same parents as.” The elements are the same (PC sets, or people), but their relation is conceived of in two different ways. One way tells us what the first element is with respect to the second, and what, in turn, the second element is with respect to a third (for example: “is related by Tn to,” “is a sister of”). This may not be what the second element is with respect to the first (“is related by T−n mod 12 to,” “is a brother of”), and what the first element is with respect to itself, or to the third element. (Clough could solve this problem for the PC sets by not being too specific about their relationship. As he saw it, they were mutually related just by T.) The other way tells us what these three elements are with respect to something else (prime form, parents), thus putting a lesser strain on the conditions of algebraic equivalence. The substitution of the equal-vector relation by a set of relations based on transposition and/or inversion resulted in 23 more equivalence classes than Forte had listed in 1964. The 23 equivalence classes containing PC sets not related by transposition and/or inversion (the “Z-”pairs) had to be split up. On the other hand, the new list that Forte had published as an appendix to The Structure of Atonal Music did not contain the twelve classes of PC sets with two or ten elements. Forte had abandoned them, probably because he considered them too common to generate meaningful relationships. Example 4.8 shows a part of the new list: the part with the set-class representatives of cardinal number 5 and 7, which can be compared to the corresponding part of the 1964 list shown in example 4.2. 23. It would be an improvement if the normal order of a PC set would always clearly reflect the prime form. For example, it is possible to define a normal order that is an arrangement of the elements corresponding to that of their images in the prime form. (The term “image” refers to the element to which a given element is assigned by some function; in this case by transposition and/or inversion.) But this would contravene a by now common practice in two ways: first, the normal order would be derived from the prime form, and not, as usual, the other way around; and second, this alternative protocol would sanction descending normal orders. For example, the normal order of the PC set {1,4,6,7,8,11} would be (8,7,6,4,1,11).

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Example 4.8. A part of Forte’s 1973 table of set-classes, representing the PC sets of cardinal numbers 5 and 7. Of each set-class, the name, prime form, and APIC vector are given. The name is composed of the cardinal and ordinal numbers (the latter of which is the “set number” from the table of 1964; see example 4.2.) In this table there is no separate column for the number of PC sets per class. When it is not the “normal” 24, this number appears in brackets in the “Name” column. On each side of the table—that is, for each of the cardinal numbers 5 and 7—there are three more PC sets than in the table of 1964. The new classes with ordinal numbers 36, 37, and 38 were originally part of those with ordinal numbers 12, 17, and 18 respectively. The “Z”s now refer to their common APIC vectors. Allen Forte, The Structure of Atonal Music. Copyright © 1973 by Yale University.

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109

The number of items in the new list is greater, because the “Z-related” PC sets now belong to different classes. Apart from this, the set-classes have “names.” A set-class is not only identified by its prime form, but also by its hyphenated cardinal and ordinal numbers.24 In spite of the criticism that the latter are arbitrary identifiers (Regener 1974), they have been in general use since the publication of The Structure of Atonal Music. Single PC sets are commonly identified by the names of the Fortean set-classes to which they belong. This has the advantage of quickly revealing transpositional and inversional relations. Some of Forte’s names have become very familiar among music theorists, because of the peculiar sets to which they refer, or because of the role they play in the repertoire: “4-Z15” ({0,1,4,6}; one of the so-called “all-interval” tetrachords), “6-20” ({0,1,4,5,8,9}; the “Ode-to-Napoleon” hexachord), “5-Z17” ({0,1,3,4,8}; the “Farben” chord), and “8-28” ({0,1,3,4,6,7,9,10}; the octatonic scale). The establishment and rapid spread of this new vocabulary in the English-speaking world has been one of Forte’s major successes.

Class Types In the preceding sections I have more than once referred to the concept of a set-class: an equivalence class of PC sets. An equivalence in a collection of things (like PC sets) stipulates that this collection divides into equivalence classes. This aspect of PC set equivalence was hardly mentioned until after the publication of The Structure of Atonal Music. Forte did not use the term “set-class” in his treatise. He did use the term “equivalence-class,” but not with reference to collections of PC sets. In earlier publications, the concept of a set-class did not play an important role either. To be sure, it was sometimes mentioned. For example, Howe’s term “pitch-structure” referred to it (Howe 1965, 48). Usually, however, equivalence was defined as a relation between individual PC sets satisfying the conditions of reflexivity, symmetry, and transitivity. As we have seen, it is difficult to state transpositional and inversional relations in a way that meets these three conditions. On the other hand, this set of relations divides PCSET into non-overlapping subcollections, thus complying with the criteria of an equivalence class. It is possible to discuss PC set equivalence from this angle, and avoid mentioning the relation itself altogether. What is crucial for an equivalence relation between PC sets is not its being reflexive, symmetrical, and transitive, but its partitioning the collection of all PC sets. John Rahn took this approach in his Basic Atonal Theory. Rahn presented several partitions of PCSET, each of which resulted in distinct “types” of PC sets. 24. Forte already used these names in his 1964 article, but the corresponding list only provided the ordinal numbers of the set-classes.

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The first of these partitions was based on cardinal number, and resulted in 13 “cardinality types” (from cardinal number 0 to cardinal number 12). Rahn proceeded by partitioning these cardinality types on the basis of transposition and of transposition and/or inversion. These partitions yielded “Tn-types” and “Tn/TnI-types,” respectively. This is how Rahn defined the concept of a “type”: “Types” are “equivalence classes” . . . Equivalence classes . . . must meet two main criteria: 1.

“exhaustivity”: the set of equivalence classes (the “partition”) must exhaust the domain (i.e., the sum of the equivalence classes must equal the domain) and

2.

“exclusivity”: no two equivalence classes may have a member in common (i.e., the intersection of every pair of the equivalence classes must equal the null set). (Rahn 1980a, 74–75)

Any collection consisting of PC sets related as transpositions and (transposed) inversions fulfills these criteria. Therefore, Rahn considered any two PC sets equivalent that belonged to one such collection. PC sets could be equivalent “under Tn,” or “under Tn or TnI, or both.”25 Furthermore, Rahn contemplated equivalence classes of ordered sets (“lines”) as well as of unordered sets, and of pitch sets as well as of PC sets. Of course, pitch sets and ordered sets are members of much larger collections than PCSET. Now that the partitioning of a collection was presented as the prime distinguishing property of an equivalence relation, it was easier to accept other sets of relations as equivalencies, even though they were less common and often less obvious than transpositional and inversional relations. One such set was determined by transposition, inversion, and/or multiplication. As we have seen in chapter 3, multiplication (by 5 or 7) transforms one PC set into another that may not be a transposition or transposed inversion. Therefore, equivalence classes resulting from this set of relations tend to be larger than those resulting from transposition and inversion alone. More specifically, a number of these equivalence classes comprise two classes of the “Tn/TnI-type.” Rahn did not pay much attention to the “Tn/TnI/M5/M7-type” in his manual, which nevertheless included a section on multiplicative operations; he only mentioned it in an appendix (Rahn 1980a, 139). In Rahn’s treatise, equivalence is not a specific relation, like “having the same interval content” or “being reducible to the same prime form,” but a property of various possible relations in various possible collections. As well as being more

25. It should be noted that in Rahn’s notation the index “n” does not represent a specific integer value, but one from a given set of such values. Seen thus, B = “Tn”(A) is an equivalence in the collection of all PC sets, for there is always a value “n” such that A = “Tn”(A) (reflexivity), A = “Tn”(B) (symmetry), C = “Tn”(B), and C = “Tn”(A) (transitivity).

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111

mathematically precise, this is a more flexible use of the concept, enabling it to describe musical relations in more or less detail, as the context requires. Here is an example that requires consideration of the musical context. Arnold Schoenberg’s Piano Piece, Op. 33a begins with a series of six four-note chords. A classification of their PC sets according to “Tn/TnI”-type reveals a retrograde-symmetrical structure: α–β–γ–γ–β–α (ex. 4.9). This structure is reinforced by the overall pattern of rising and falling. Now, these observations are nice, but not entirely accurate. Equivalence relations between PC sets do not provide scope for a more important characteristic of these opening measures, one that entails the retrogradesymmetrical arrangement of PC set classes. The two αs not only represent PC sets of the same “Tn/TnI”-type; they are also real mirror inversions of one another. The same holds true for the two βs and the two γs. Therefore, the relations between these chords are more accurately conceived in a collection of pitch sets, a collection that we might call PSET. The pitch sets are related pairwise as follows:26 The pitch sets of the two αs:

IPI (−9, −2) ({−13, −12, −7, −2}) = {2,1, −4, −9}

The pitch sets of the two βs:

IPI (−2,3) ({−3,1,3,6}) = {4,0, −2, −5}

The pitch sets of the two γs:

IPI (3,10) ({−4,2,4,7}) = {17,11,9,6}

This representation adds a nuance that lies beyond the scope of PC sets. The pattern of rising and falling is definitely not symmetrical, since the second γ is projected higher than the first, so that the rising progression (α–β–γ–γ) is actually longer than the falling one (α–β–γ). Schoenberg’s dynamic markings stress this asymmetrical pattern of rising and falling rather than the retrograde-symmetrical structure formed by the PC set classes of these measures. The analytical exercises in Rahn’s chapter on “set types” refer to this musical fragment. However, Rahn treats the chords as realizations of PC sets, not pitch sets. He was inclined to generalize musical relationships, to strip them of all the features referring to their actual musical context. Example 4.10 shows his analysis of the beginning of the Prelude to Tristan und Isolde by Wagner. According to Rahn, the first three phrases of this prelude consist of four or five simultaneities each. If “Tn-types” are assumed, and one Greek letter is assigned to simultaneities of the same type, the resulting pattern is α–β–β–γ. However, the assumption of “Tn/TnI-types” results in a “retrograde-symmetrical structure,” namely α–β–β–α (Rahn 1980a, 78).

26. The choice of (−9, −2), (−2,3), and (3,10) as referential pitch intervals is not an arbitrary one. Each of these pitch intervals involves the PCs E♭ and B♭, which spell out Schoenberg’s favorite perfect fifths and fourths. B♭ is the first element of the prime series-form, and E♭ is its “image” (see note 23) in the T5I of the retrograde. The opening measures involve just these two series-forms.

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Mäßig 1

= 120 cantabile

α

β

2

γ

γ

β

α

Example 4.9. The first two bars of Schoenberg’s Piano Piece, Op. 33a. Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272.

Example 4.10. John Rahn’s analysis of the beginning of the Prelude to the opera Tristan und Isolde by Richard Wagner. Each type of PC set is named after its prime form, which Rahn called “representative form.” In Rahn’s notation, the representative form of a “Tn”-type is identified by parentheses. The representative form of a “Tn/TnI”-type is identified by square brackets. From Basic Atonal Theory 1st. edition by Rahn. 1980. Reprinted with permission by Wadsworth, a division of Thomson Learning: www.thomson.com. Fax 600 730–2215.

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This sort of result appealed to Rahn. Why? First of all, one of its attractions was its lack of obviousness. It showed him that analyzing music could yield profound insights. He expected his generalizing approach to reveal musical relationships that would otherwise remain concealed. Few could write about the gains of such an approach more compellingly than John Rahn: Each step up the ladder of abstraction loses particular distinctions but gains generality . . . Relations that may lie obscured in the thicket of the full particularity of things can be perceived clearly when a process of generalization has pruned away the underbrush of reality. A coherent structure of such abstract relations does exert its own perceptional pull, forming associations strongly in conjunction with and even in [spite] of the [criterion] of “same perceptual neighborhood.” (Rahn 1980a, 77)

Rahn’s analysis of the first three phrases of Wagner’s prelude shows how “a step up the ladder of abstraction”—i.e., from considering equivalence “under Tn” to considering equivalence “under Tn or TnI or both”—reveals a retrogradesymmetrical structure like that in the first measures of Schoenberg’s Opus 33a. The “underbrush” that Rahn has “pruned away” in order to arrive at this result is, first, the distinction between actual chords (like the simultaneities 4, 8, and 13 in example 4.10) and mere voice-leading events (like the simultaneities 3, 7 and 12), and, second, the difference in sound and tonal function of chords that are based on inversionally related PC sets (like the “Tristan chord” 1 and dominant-seventh chord 4). Ignoring these distinctions is certainly questionable, but Rahn did reveal a well-formed structure, one that called to mind twelve-tone serialism. Using Wagner’s prelude along with the opening measures of Schoenberg’s Opus 33a can be seen as a statement that Schoenberg’s compositional practice was built on Wagner’s. Rahn thus provided Schoenberg’s music with a historical background that added to its authority, while simultaneously substantiating his view of tonal theory as a subbranch of atonal theory (see chapter 2, p. 45).

Canonical Transformation Groups By stressing the class membership of PC sets rather than the operations connecting them, Rahn showed the equivalence of the transpositions of a PC set and its inversion in quite a satisfactory way. Moreover, he pointed out other relations among PC sets similarly yielding (smaller or larger) equivalence classes. In other words, equivalence came to be regarded as a category of different relations rather than as a relation in itself, and each of the relations of this category counted as potentially significant. The significance of a relation was actually certified by its falling in the category of equivalencies; but it was indeed potential, depending as it did on its musical function. The concept of musical equivalence

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thus became a truly “general” concept. Meanwhile, its traditional connotations were bumped down into the realm of the “particular.” One problem remains. Regarding equivalence, Rahn’s focus was not so much on the quality of a relation between two PC sets—whether it was reflexive, symmetrical, and transitive—but rather on the partitioning of PCSET. However, if it is not a single relation but a set of relations that partitions PCSET—such as the Tns and TnIs—then how do these relations cohere? Is it possible to reduce them to a single concept? David Lewin, in the introduction to an article on different ways of modeling the intervallic properties of PC sets, invoked the mathematical concept of a “transformation group” to account for the relations between operations jointly effecting a partition of PCSET. For this reason, he referred to these operations as “transformations” (Lewin 1977b, 195). The properties of a transformation group have already been touched upon at the beginning of chapter 3, where we examined the relation between the terms “operation” and “group action.” Now follows a systematic recapitulation that complies with the general definition of a mathematical group. First of all, for a set G of transformations to be called a “group,” each combination of two or more of these transformations must equal a single other transformation from the same set. In other words, if g and h are members of G, and gh = k, then k must also be a member of G. (This property is called “closure”). Secondly, combining one transformation with the result of combining a second and a third must be the same as taking the third and combining it with the result of combining the first and the second: g(hk) = (gh)k (“associativity”). The third property is the presence of an “identity”-transformation e. The combination of this and another transformation is equal in value to the latter alone: ge = eg = g. The fourth and final property—not explicitly stated but merely implied by the presentation in chapter 3—is that for each transformation g there is an inverse g−1, a transformation with which it combines to yield the value of the identity transformation: gg−1 = g−1g = e. The sets of transformations underlying Rahn’s “Tn”-, “Tn/TnI”-, and “Tn/TnI/ M5/M7” class-types all satisfy these criteria. Each member of a set-class—that is, each PC set—can be viewed as a transformation of every other member of the same set-class, and as its own T0 (the identity transformation). Therefore, all set-classes can be subsumed under the general concept of a “transformation group.” Transformations defining a set-class type in conformity with established practices are “canonical,” in Lewin’s words. The relation “is a canonical transformation of” is an equivalence among PC sets. When two of them were in that relation, Lewin called them “canonically equivalent” (Lewin 1987, 104). There are groups of 12 (T), 24 (T and TI), and 48 (T, TI, TM, and TIM) canonical transformations. However, set-classes are not always that large, since some PC sets are self-reproductive (i.e., invariant) under one or more operations (apart from T0). For example, the PC set {2,3,5,6,10} is self-reproductive

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115

Table 4.2. The numbers of set-classes generated by three canonical transformation groups

Cardinal nr.

T

T, TI

T, TI, TM, TIM

0

1

1

1

1

1

1

1

2

6

6

5

3

19

12

9

4

43

29

21

5

66

38

25

6

80

50

34

7

66

38

25

8

43

29

21

9

19

12

9

10

6

6

5

11

1

1

1

12

1

1

1

Note: See, by way of comparison, Morris 1987, 80.

under T8I; and the PC set {11,0,2,5} is self-reproductive under T0IM. The members of any set-class are self-reproductive under an equal number of operations. Table 4.2 shows the numbers of set-classes defined by each canonical transformation group. Each group can be said to “act” on the set-classes which it defines, in accordance with the definition of a group action given in chapter 3. This includes the class of PC sets of cardinal number 1, i.e., PITCHCLASS.

A Little More History: “Musical Statistics” The PC set classifications shown in table 4.2 stand in a line of development with comprehensive chord classifications from earlier periods. In chapter 2, there was already brief mention of these older classifications, but now we can study their relation to PC set theory in more depth. It should be recalled that such classifications were a reaction to the apparent harmonic freedom in contemporary music, and that they stemmed from a spirit of objectivism typical of modernist thought: as the chordal vocabulary seemed no longer subject

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to aesthetic and syntactical constraints, all a theory of harmony could do was to inventory possible chords—given some set of available pitch elements—, and perhaps to provide suggestions for their use. Or so was the opinion of the compilers of these classifications. According to Fritz Heinrich Klein (1925), a composer who had studied with Alban Berg, harmonic theory was to turn into “musical statistics” (Musikstatistik). The practitioners of this approach to harmony came up with lists of distinct chord types of each cardinality in (most often) the twelve-tone universe. Some of them only provided the numbers of chord types they had found. These numbers were significant insofar as they revealed the criteria by which one assigned chords to the same classes. It is interesting to note that, apparently, and in spite of all objectivity, these criteria could differ greatly. Table 4.3 shows the numbers pertaining to three classifications, and lists the authors who calculated or cited them.27 The following discussion will cover the notions of equivalence suggested by these classifications, while bearing in mind the contexts in which they were presented. Klein himself is one of the authors cited in table 4.3. He had presented the results of his enumerations in an article entitled “Die Grenze der Halbtonwelt,” which appeared in the German periodical Die Musik in January 1925. These results are not the main focus of the article. They merely serve as an introductory note to a discussion of compositional techniques proceeding from a twelve-tone “mother chord.” Klein’s suggestions for the derivation of harmonic combinations from this chord are important, for they reflect a mode of thought that profoundly influenced his friend and teacher Alban Berg.28 Furthermore, the “mother chord” (read from the bottom up: A0, A1, G♯2, E3, C♯4, B4, F♯5, C6, F6, G6, B♭6, D7, E♭7) is the earliest known instance of an “all-interval series,” in which each pair of adjacent PCs represents another PIC. Thus, it stands at the beginning of a long search for other such series.29 The number of chord classes that Klein distinguished corresponds to the sum total of non-equal PC sets (cf. table 2.1).30 This means that, for Klein, PC content was the distinguishing property of a chord class. From this viewpoint, all 27. This table summarizes a good deal of the information presented and discussed by Bernard (1997) and Nolan (2003). It adds the reference to Bruno Weigl, whose work is given consideration elsewhere by Menke (2005). 28. As Arved Ashby (1995) has shown, Berg’s practice of using more than one series in a composition was not so much a liberty taken with the twelve-tone technique taught by Schoenberg as the consequence of another systematic approach, for which Berg was indebted to Klein. 29. See, for example, Krenek (1937), Babbitt (1992), and Eimert (1950, 1964). This search ended with the publications of Jelinek (1961) and Bauer-Mengelberg and Ferentz (1965). With help of a computer, these authors found the total number of 46,272 allinterval series, transpositions included. 30. With the exception of the null set, which Klein did not take into account.

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equivalence

117

Table 4.3. Early comprehensive chord-class counts in the twelve-tone universe Cardinal nr.

Number of “chords”

F. H. Klein (1925)

A. Loquin (1874) E. Stein (1925) B. Weigl (1925)

A. Loquin (1895) E. Bacon (1917) J. M. Hauer (1925) W. Howard (1932)

1

12

1

1

2

66

11

6

3

220

55

19

4

495

165

43

5

792

330

66

6

924

462

80

7

792

462

66

8

495

330

43

9

220

165

19

10

66

55

6

11

12

11

1

12

1

1

1

Note: Not every author provided the numbers exactly as given in the respective column. For example, Bruno Weigl counted 454 chord classes of cardinal 6, instead of 462. Joseph Matthias Hauer, whose classification was not strictly one of chords, did not consider other cardinalities than 6.

forms of a D-major triad belong to one class, whereas a D-major triad and an Amajor triad fall into different classes. It is difficult to determine whether this was a choice of musical principle or the consequence of the chosen algorithm. In view of Klein’s pursuit of objectivity, the answer is, perhaps, both. Nevertheless, it is worth taking a closer look at his method. For reasons that will become obvious, Klein could only demonstrate its application to “dyads” (Zweiklänge) and “triads” (Dreiklänge; Klein 1925, 282). Here it is rendered so as to facilitate comparison with the methods of others. Table 4.4a shows how Klein derived his 66 dyad classes from the total chromatic. In a first cycle, he combined the lowest tone of an octave range (C) with each of the remaining eleven tones in that range (from B down to D♭). He then proceeded to the next cycle, combining C♯ with all others tones, except, of course, C. After that, there followed combinations with D (excluding those of C and C♯), D♯

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Table 4.4a. Klein’s algorithm applied to dyads B

B

B

B

B

B

B

B

B

B

B

B♭

B♭

B♭

B♭

B♭

B♭

B♭

B♭

B♭

B♭

A♯ 11th cycle (1)

A

A

A

A

A

A

A

A

A

A

A♭

A♭

A♭

A♭

A♭

A♭

A♭

A♭

G♯

10th cycle (2)

G

G

G

G

G

G

G

G

9th cycle (3)

G♭

G♭

G♭

G♭

G♭

G♭

F♯

8th cycle (4)

F

F

F

F

F

F

7th cycle (5)

E

E

E

E

E

6th cycle (6)

E♭

E♭

E♭

D♯

5th cycle (7)

D

D

D

4th cycle (8)

D♭

C♯

3rd cycle (9)

C

2nd cycle (10)

1st cycle (11)

(excluding those of C, C♯, and D), E (excluding those of C, C♯, D, and D♯), etc. Thus, Klein’s algorithm consisted of eleven successively shorter cycles. When the same method is applied to triads, it becomes more complex. As appears from table 4.4b, each cycle is then built from of a number of subcycles. All combinations from table 4.4a, except those with B, combine with the remaining tones in the same octave range to yield Klein’s 220 classes of “triads,” redundancies omitted. Klein confessed his surprise at the total number of 4,095 chord classes of cardinality 1 through 12, a number much smaller than he had expected. He probably would have been astonished to hear that Erwin Stein had found only half as many. A few months earlier, Stein, a student of Schoenberg, had contributed an essay to a special issue of the periodical Musikblätter des Anbruch, published on the occasion of his teacher’s fiftieth birthday. This essay was the first printed exposition of Schoenberg’s twelve-tone method, laying much stress on the historical conditions by which it was spurred. In the introduction, Stein welcomed the maximization of harmonic resources ensuing from the “emancipation of the dissonance” (one of Schoenberg’s tenets). With an air of offhanded authority he advised how many chords were now at the free disposal of the contemporary composer writing in a twelve-tone idiom: Thus we get 55 constitutionally different three-note chords, 165 fournote chords, 330 five-note chords, 462 six-note chords, the same number of seven-note chords, and again 330 five-note chords, 165 nine-note, 55

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B B♭ A A♭ G

D♯ 4th cycle (28)

B B♭ A A♭ G G♭

B B♭ A A♭ G G♭ F E

B B♭ A A♭ G G♭ F

B B♭ A A♭ G G♭ F E

B B B B♭ B♭ B♭ A A A A♭ A♭ A♭ G G G G♭ G♭ G♭ F F F E E E E♭ E♭ E♭ D D D♭ C 1st cycle (55)

B B♭ A A♭ G G♭

B B♭ A A♭ G

5th cycle (21)

B B♭ A A♭ G G♭

B B♭ A A♭ G G♭ F 6th

B B♭ A A♭ G

cycle (15)

B B♭ A A♭ G 8th

B B♭ A A♭ G G♭

B B♭ A A♭ G

B B♭ A 10th (1)

B B B B♭ B♭ B♭ A A A♭

B B B B B♭ B♭ B♭ B♭ A A G♯ 9th (3) cycle (6)

B B B B♭ B♭ B♭ A A A A♭ A♭ A♭ G G G G♭ G♭ G♭ F F F E E E♭ D 3rd cycle (36)

B B B B B♭ B♭ B♭ B♭ A A A A♭ A♭ G F♯ 7th cycle (10)

B B B B♭ B♭ B♭ A A A♭

B B B B♭ B♭ B♭ A A A♭

B B♭ A A♭ G G♭

B B♭ A A♭ G

B B B B B♭ B♭ B♭ B♭ A A A A A♭ A♭ A♭ A♭ G G G G G♭ G♭ G♭ G♭ F F F F E E E E♭ E♭ D C♯ 2nd cycle (45)

B B B B♭ B♭ B♭ A A A♭

B B B B♭ B♭ B♭ A A A♭

B B♭ A A♭ G G♭ F E

B B♭ A A♭ G

B B B B♭ B♭ B♭ A A A♭

B B♭ A A♭ G G♭ F

Table 4.4b. Klein’s algorithm applied to triads

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ten-note, 11 eleven-note chords, and one twelve-note chord, i.e. altogether over 2,000 chords. 31

Stein did not reveal his algorithm, if he had any. He might just have heard about these numbers. Literature on the subject was available at the time—as appears from table 4.3—but it was probably not widespread. Most classifications seem to have come about without prior knowledge of similar efforts, as Klein’s amazement at the results of his own count illustrates. Someone who certainly knew how to obtain Stein’s numbers was his contemporary Bruno Weigl. This composer and music theorist from Brünn (Brno), in the Moravian region of what is now the Czech Republic, had spelled out all the different harmonic combinations within the total chromatic, arriving at the same result reported by Stein.32 His list, which shows chords of three to five tones in staff notation while the rest are rendered in letter notation, covers almost fourteen pages of his Harmonielehre (1925).33 Although this book was written with the aim of establishing a continuity between the music of the past and that of the present—an aim which it shared with the harmonic treatises of, among others, Bernhard Ziehn (Harmonie- und Modulationslehre, 1887), Rudolf Louis and Ludwig Thuille (Harmonielehre, 1907), and Hermann Erpf (Studien zur Harmonie- und Klangtechnik der neueren Musik, 1927)—it could hardly represent the most advanced achievements of contemporary practice beyond this huge catalogue. However, it is not least because of this catalogue that it claims the interest of the historian. How did Weigl establish his chord classes? Which tone combinations did he consider equivalent? If one looks at his “table of triads” (Dreiklangstabelle; Weigl 1925, 379), he seems, at first, to proceed in the same way as Klein (see table 4.4b). But he stops after the first cycle, seeing all possible combinations of three tones represented therein. Now, since the first cycle involves all chromatic tones, and since all combinations in it are built on the same “bass” (C), we can make the following observations:

31. “So gibt es 55 ihrer Konstitution nach verschiedene dreistimmige Akkorde, 165 Vierklänge, 330 Fünfklänge, je 462 Sechs- und Siebenklänge, und wieder 330 Acht-, 165 Neun-, 55 Zehn-, 11 Elfklänge und einen Zwölfklang, ingesamt über 2000 . . . Akkorde.” (Stein 1925, 60; transl. by Hans Keller) 32. Weigl’s number of six-tone combinations (454) differs from Stein’s (462). The explanation is simple. Weigl forgot to list the following eight combinations: (C,D♭,E,A♭,B♭,B), (C,D♭,G♭,G,A,B♭), (C,D♭,G♭,G,A,B), (C,D♭,G♭,G,B♭,B), (C,D♭,G♭,A♭,B♭,B), (C,D♭,G,A♭,B♭,B}, (C,D,G,A♭,B♭,B), and (C,E,F,A♭,B♭,B) 33. A note by the author (p. IX) states that the publication of the Harmonielehre, by the Schott firm in Mainz, was originally scheduled for 1922. Due to the financial crisis in Germany, the book had to wait until 1925 to appear and was abridged to about two-thirds of its original size.

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(1) Each combination is transpositionally related to two other combinations in this cycle, except one combination that is self-reproductive under transposition (i.e., {C,E,A♭}). (2) No two transpositionally related combinations appear in the same bass position. From this it follows that each combination in the first cycle is uniquely identified with a set of pitch intervals measured from the “bass.” It represents all chords that yield the same values when these PIs are assigned to PICs. Thus, in Weigl’s view, two chords are equivalent when they are in the same bass position (or harmonic inversion). This may or may not be true for two forms of a D-major triad; and it may or may not be true for a D-major triad and an A-major triad. Weigl’s view of a chord as a set of intervals relative to a bass is reminiscent of figured-bass practice. We may ask whether this was not a mere coincidence, the outcome of a combinatorial protocol that, as Catherine Nolan points out, adds eleven elements to “a disjunct, fixed element . . . in all possible combinations” (Nolan 2003, 214). It may have been so for Anatole Loquin, whom Nolan credits as the first to apply this protocol to the elements of the chromatic scale. Loquin used the resulting partition of the collection of all possible chords (our PSET) as a springboard to a partition of this collection into T-classes. And it may also have been so for Erwin Stein, who was more than familiar with Schoenberg’s manipulation of intervals and therefore conceivably receptive to the use of T/TI-classes. For Weigl, however, the correspondence between his chord classification and one along the lines of the figured-bass tradition was not entirely coincidental. In the last section of his Harmonielehre, he retreated explicitly from the Rameauian, “root-oriented” concept of a chord, which he thought to be of no use with respect to the chromatic idiom (Weigl 1925, 394–95). Since in this idiom each tone could be combined with any other, he argued, the interpretation of a chord as “root position” or “inversion” had lost its rational basis. These forms had now become types. This explains why Weigl’s classification reflects a “bass-oriented” concept of a chord. Such a concept facilitated a comprehensive listing of chords while remaining relatively faithful to their sonic identities. Meanwhile, Weigl realized that sonic identities were also shaped by their contexts. He cautioned that his list did not account for the relations into which chords might enter. This, he wrote, would have required him to include each chord in all its enharmonic spellings (Weigl 1925, 378). Weigl was obviously closer to nineteenth-century harmonic theory than to twentieth-century PC set theory, the visionary aspects of his chord classification notwithstanding; but as to the limitations of a taxonomy of chords, he raised a point that would become a focus of criticism of PC set theory in later years. Weigl’s decision not to classify each chord of the chromatic scale with all its bass positions was certainly rational. However, it was still a choice made from

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two possible options. For why would the absence of a root—or the impossibility of assigning this role to a tone—preclude the association of chords on the basis of their type? When the different bass positions of a chord have no harmonic function to share, or no locus in a key, they still represent the same “chord,” in this sense: a PC set with such-and-such intervallic properties, which represents a harmonic entity. This definition may not capture the traditional notion of a chord in all its complexity; but at least it points to an important aspect of it—sufficiently important for writers other than Weigl to use it alone as a basis of equivalence. These others—like the composers Ernst Bacon and Josef Matthias Hauer, and the musicologist Walther Howard—classified tone combinations according to what John Rahn has called “Tn-type,” obtaining the numbers in the right-most column of table 4.3 (cf. the left-most column of table 4.2). They did not all speak of actual “chords.” Bacon’s numbers do refer to simultaneities, but the essay in which he presented them, “Our Musical Idiom,” was a speculative exercise rather than a record of compositional practice. Published in 1917 in the Monist, an American journal of philosophy, it had little impact on the musical world around him. Today, its importance rests mainly in its adumbration of PC set theory. Bacon represented simultaneities as successive-interval arrays partitioning the octave, as Richard Chrisman did with pitch sets many years later (see above, p. 100). Like Chrisman, Bacon formed classes of such arrays related by cyclic permutation. Hauer’s classification is actually one of twelve-tone series, which he reduced to complementary pairs of unordered hexachords, so-called “tropes” (Tropen). There are 44 such pairs, which Hauer tabulated in his small composition treatise Vom Melos zur Pauke (1925, 12). As eight tropes consist of transpositionally related hexachords,34 there are eighty (88 – 8) distinct hexachord types. Hauer did not enumerate tone combinations of cardinalities other than 6. The use of tropes ensured the equal distribution of PCs in a composition, in accordance with the “twelve-tone law” (Hauer 1925, 17). Within each hexachord, tones could sound simultaneously or in succession; Hauer’s hexachords functioned equally in the harmonic and melodic dimensions of music. Walther Howard saw his tone combinations in a similar way—as the raw material for harmonies as well as melodies (although he referred to them as “chords” or “simultaneities”). Howard was the author of a Wissenschaftliche Harmonielehre des Künstlers, which was published in Berlin in 1932. His aim was to present a theory not so much concerned with musical uses and styles as with the relations embodied by the pitch material itself (i.e., the twelve-tone chromatic). He expected this approach to provide a basis for any possible or desirable application, and thus 34. In six of these tropes both hexachords are inversionally symmetrical. Therefore, they relate not only as transpositions but also as transposed inversions. This relation, however, also exists between some hexachords from different tropes.

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to be most fit for pedagogical purposes (Howard 1932, 1). Counting and measuring were methods that, in his view, ensured a fresh and unbiased attitude towards music. In his book, Howard spelled out all T-classes of three-tone and four-tone combinations and provided the numbers of classes of other cardinalities (Howard 1932, 215). The concept of a T-class lacks a clear sense of verticality, such as provided by Weigl’s referential bass tone. Indeed, the members of such a class are sets rather than chords. However, we must be careful in establishing a relationship between the above-mentioned chord classifications and the concept of a PC set. The compilers of these classifications were not involved in the development of PC set theory, even though they used similar techniques of investigation. They developed their systems in relative isolation, unaware of others who embarked on similar projects and unable to create a stir. And their systems served different goals. Seen thus, it is hard to maintain that they laid “important conceptual groundwork for pc-set theory” (Bernard 1997, 12). Rather, the rise of PC set theory has given us a focus on these unconnected attempts to come to terms with the vastness of harmonic resources in the twelve-tone universe.

Sound In the 1980s, the criteria of PC set equivalence, as a topic of scholarly debate, had practically been exhausted. A general consensus of opinion had emerged regarding the relations that qualified as “equivalencies.” Moreover, people began to realize that the actual use of this designation required a good deal of common sense. This made the discussion about its mathematical underpinning more or less obsolete. In his book Composition with Pitch-Classes from 1987, Robert Morris wrote: The question of which operations to choose is guided by the attempt to provide a way of collecting “similar”-sounding sets into an equivalence class. Therefore, not every (mathematically) competent definition will do. (Morris 1987, 79)

Morris himself may have been guided by his own perception in his efforts to define useful equivalencies among pitch sets. Pitch sets are obviously not as abstract as PC sets. Pitches and pitch intervals are more specific descriptors of tone combinations than PCs and PICs. The latter help us discover certain concealed relations between tone combinations, but relations of that kind are not essential to every piece of music. As we have seen in our discussion of example 4.9 (the opening measures of Schoenberg’s Opus 33a), the use of PCs may sometimes obscure essential musical relations. John Rahn (1980a) already mentioned the possibility of defining equivalencies in PSET, but he did not go beyond that. Morris (1995) proposed three new class types. The first of these was the “pitch-

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set class” (PSC). A PSC contains pitch sets with the same intervallic spacing (i.e., the same succession of pitch intervals from low to high). Such pitch sets are, for example, {D4,F4,A♭4,C♯5}, {F3,A♭3,C♭4,E4}, and {B2,D3,F3,A♯3} (ex. 4.11a). To reduce the huge amount of possible PSCs, Morris suggested setting a limit to their vertical spans. Any two members of a PSC are related by an operation on PSET (Tn, where n is a positive or negative integer). For example: T−9({D4,F4,A♭4,C♯5}) = {F3,A♭3,C♭4,E4}

The second class type in PSET was the “pitch-class interval succession” (PCINT). A PCINT-class comprises PSCs with octave-related successive spacing intervals. The PCINT-class of which {D4,F4,A♭4,C♯5} is a member also contains the pitch sets {D3,F4,A♭5,C♯6} and {G3,B♭3,D♭5,F♯5} (ex. 4.11b). The last two pitch sets are not related to {D4,F4,A♭4,C♯5}, or to each other, by Tn. The third, and most inclusive, of Morris’s class-types was the “figured-bass” (FB-) class. This is the class-type also invoked by Weigl (1925) and, before him, Loquin (1874): an unordered set of PICs measured from a given bass (see table 4.3). In the figured-bass tradition, octave-related pitch-interval values—like a unison and an octave, a second and a ninth, or a fifth and a twelfth—were congruent mod 7. The chord symbol “6/3” indicated that each chord tone was related to the bass by an interval congruent 6 (mod 7) or an interval congruent 3 (mod 7) if it was not a “doubling” of the bass. Morris proposed a similar labeling of FBclasses partitioning PSET, which required the use of modulo-12 arithmetic. Let us continue with the earlier examples. We have seen that {D4,F4,A♭4,C♯5}, {D3,F4,A♭5,C♯6}, and {G3,B♭3,D♭5,F♯5} belong to the same PCINT-class. They also belong to the same FB-class, a class that includes pitch sets from different PCINT-classes, for example {D4,C♯5,A♭5,F6} and {G2,B♭3,D♭4,F♯4,B♭4} (ex. 4.11c). In all these sets each pitch is related to the lowest pitch (the “bass”) by an interval congruent 3 (mod 12), an interval congruent 6 (mod 12), or an interval congruent 11 (mod 12), unless they represent the same PC as the “bass.” The common designation for pitch sets with this property—that is, the “name” of their FB-class—is “3 6 11.”35 The pitch sets of one FB-class are all realizations of transpositionally related PC sets. These realizations are more likely to be heard as similar than other realizations of the same PC sets, especially when they are also members of the same PCINT-class, or the same PSC. They satisfy certain criteria of presentation that help the listener to relate them, like the spacing intervals or the pitch intervals formed with the “bass.” These are not just aural criteria; Morris’s classification of pitch sets bears a resemblance to the classification of chords in the music of the common-practice period. It thus reflects a tradition that is still represented in 35. In Morris’s notation the PIC values 10 and 11 are represented by A and B, respectively. His designation for this FB-class is “36B.”

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equivalence (a)

(b)



125

(c)

PSC PCINT FB

Example 4.11. Class types of pitch sets, after Robert Morris (1995).

concert life, in some branches of music production, and in methods of theoretical instruction. Therefore, one may feel a familiarity with Morris’s classification as a conceptual device, while still being daunted by the sonorities themselves. In the various definitions of PC set equivalence that we have discussed, “sound” has only casually been mentioned, if at all. Theorists may have taken it for granted that there is a correlation between PC set equivalence and aural similarity. However, this correlation is questionable, as the case of dyadic sets clearly shows. Various harmonic intervals are associated with a single class of such sets. Does this mean that they are similar? Does a minor seventh sound similar to a major second, or does a major tenth sound similar to a minor sixth? The relationships between these intervals may be perceived on first hearing, but this depends on the musical context, which could instead obscure such relationships. In tonal music, a major second and minor seventh both tend to resolve, and the resulting interval progressions (diatonically: 2–3 and 7–6) are related by inversion at the octave. Thus, in a tonal context we hear these intervals as if resulting from voice exchange. However, if the tonal context is abandoned, we can no longer take their similarity as obvious. In PC set theory, the concept of “interval content” has always been supposed to account for a particular sound that is intrinsic to all musical realizations of a PC set. Every other PC set with the same interval content would hence produce the same “basic” sound. This involves the transpositions of the original PC set, its inversion, and the transpositions of its inversion. It may also involve “Z-related” sets. Straus (1990a, 67) wrote, “[sets] in the Z-relation will sound similar because they have the same interval content.” Although it is hard to prove that there is always such a common “basic” sound, the concept of interval content may help one imagine it. It is true, there are some familiar cases in which a peculiar and relatively invariant sound is reflected by a highly uneven distribution of intervals. These cases include, for example, the PC sets of the “augmented triad”- type (APIC vector 000300), the “diminished seventh chord”-type (APIC vector 004002), and the “whole-tone scale”-type (APIC vector 060603). Furthermore, it is possible that conventions of tonal hearing make us particularly insensitive to the similarity of interval content between, say, the “Tristan chord” and a dominant-seventh chord

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(APIC vector 012111 in both cases). However, the sonic profile pertaining to a specific interval content is, by and large, strongly variable. This variability should not always be disregarded in the search of relations between pitch-class sets. The following will illustrate this point. In The Structure of Atonal Music Allen Forte posits a relation between the two chords in example 4.12a. The first chord, α, has been taken from Alban Berg’s song Sahst du nach dem Gewitterregen, one of the songs on picture postcard texts by Peter Altenberg, Op. 4. The second chord, β, is from Stravinsky’s Symphonies of Wind Instruments. Without suggesting a relationship between the two pieces, Forte points out that the PC sets of these two chords reduce to the same prime form (they are related as transpositions; Forte 1973, 12). Forte’s two examples strike the reader as being very randomly chosen, but on further investigation they do make an interesting comparison. First of all, both constitute strongly articulated musical events, and, what is more, both are unifying elements in their respective musical environments. Being sustained by the strings for at least an entire measure, while the voice and the other instruments remain silent, chord α effects a caesura of great power in Berg’s eleven-measure song. Its PC set is inversionally related to that of one of the song cycle’s principal melodic ideas: the line G–A♭–B♭–C♯–E (cf. the opening measures of the first and last songs). Chord β is one of the recurring elements in Stravinsky’s Symphonies of Wind Instruments, dominating the final section of this composition.36 It stands out as a musical gesture, yet as a configuration of intervals it is tightly interwoven with other elements of the Symphonies. For example, the upper section of chord β contains the PCs G, B, D and F, which outline the melody of the opening refrain. Both chords can be related to an octatonic background. If we view the octatonic scale (see ex. 1.4) as a PC set, it is self-reproductive, not only under inversion, but also under T3, T6, and T9. This means that there are only three such sets. Pieter van den Toorn (1983, 31–98) called them “Collection I”: {C♯,D,E,F, . . . A♯,B}, “II”: {D,E♭,F,F♯, . . . B.C}, and “III”: {D♯,E,F♯,G, . . . C,C♯}. The constituent five PCs of Stravinsky’s chord (β) form a part of Collection I. The chord itself results from a transposition within this collection. It is the T9 of the chord that is first presented just after the opening refrain of the Symphonies (ex. 4.12b).37 In mm. 3–5 of Berg’s song, all three octatonic collections are exposed in an intricate web of short motifs. Although this passage exploits the total chromatic, it is still possible to delineate these collections. As example 4.12c shows, they are accentuated by the instrumentation. None of the instrumental combinations (voice/harp/clarinet, horn, bassoon/viola, horns/viola) produces notes other than those belonging to one collection (either I, II, or III). 36. This final section was published separately, in a piano reduction, as a contribution to a volume of the Parisian journal La Revue Musicale dedicated to the memory of Claude Debussy (vol. 1/2, 1920, 22–23). 37. Here I follow the analysis of Taruskin (1996: 1494).

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56 Tempo I β

α

= 72

Example 4.12a. Chords from Alban Berg’s Sahst du nach dem Gewitterregen, Op. 4, no. 3, and Igor Stravinsky’s Symphonies of Wind Instruments. © 1953 by Universal Edition A.G., Wien/UE 14325. © Copyright 1926 by Hawkes & Son (London) LTD. Reprinted by kind permission of Boosey & Hawkes Music Publishers LTD.

1

56 Tempo I

= 72

T9

Octatonic collection I

Example 4.12b. Octatonic derivation of Stravinsky’s chord (after Taruskin). On the lowest staff, black note-heads represent actual chord tones.

Coll. III Al

les 4 ra

stet . . .

5

hrp + clar. 3

m.3

Coll. I

3

Coll. II

hrn.

hrns. + va.

bn. + va.

Example 4.12c. Octatonic elements in Berg’s song.

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However, in spite of their common conceptual background, we should consider what distinguishes these two chords. It is noteworthy that Berg has so clearly projected the PC set {E♭,F,G♭,A,C} as a dominant-flat-ninth chord; with the “root” (F) in the bass parts, the “fifth” (C) coming next in ascending order, and the “ninth” (G♭) in the treble parts. One may object to the designation “dominant-flat-ninth chord” because it has been adopted from the tonal idiom. Indeed, this is not the idiom of this song; but it is not far away, and Altenberg’s words provide an occasion to bring it to mind: After the summer rain did you see the forest?!?! All is glitter, quiet, and more beautiful than before.38 ....................................

We can hear Berg’s chord as illustrating these words, as a sonority that has been relieved from its traditional function of being dissonant with respect to other sonorities, and that now exists only for its own sake. In contrast to Berg, Stravinsky has laid out the elements of the transpositionally related PC set {F,G,A♭,B,D} in such a way that the listener is not so easily reminded of a dominant-flat-ninth chord. Chord β would be in an unusual position, with the fifth (D) in both bass and treble. Although Stravinsky’s chord is not entirely devoid of tonal elements either (NB the major triad in the upper voices), it is not as suggestive of functionally directed harmony as Berg’s. It is by the relative autonomy of sound vis-à-vis PC set that these “equivalent” chords are yet so different—as different as, for example, statues carved from the same stone. The concept of PC set equivalence embodies a view of musical coherence as independent from the limits of aural perception. However, we have seen that it has its own limits: it is rather indifferent to distinctions of sound. This indifference is inherited from music theory rather than from mathematics. The search described in this chapter was not a search for musical relations that satisfy a mathematical definition. It was primarily a search for mathematical formulations. These should underpin already well-established notions of musical relatedness, which could also help extend their range of application. Most of these notions involve operations that preserve what are, in specific contexts, essential characteristics of a tone combination—operations like the registral redistribution of PCs (context: figured-bass practice, theory of harmony), permutation/ harmonic inversion (context: double counterpoint, theory of harmony), transposition, mirror inversion (context: imitative counterpoint), and interval expansion (context: melodic variation).

38. “Sahst du nach dem Gewitterregen den Wald?!?! / Alles rastet, blinkt, und ist schöner als zuvor.” (transl. by A. Kitchin)

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PC set equivalence assumes the existence of a musical identity that remains the same under these operations. Such a persistent identity can only be very abstract; it has proven difficult to locate even in the PC sets themselves. One possible locus of identity, the total interval content, was abandoned as a criterion for the classification of PC sets. What united such sets was to be found between rather than in them. However, there remained an interest in relations between PC sets on the basis of their intrinsic properties. “Similarity” was a concept by which one hoped to approximate these relations.

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Chapter Five

Similarity The discussion of “similarity” has been one of the major threads running through PC set theory. It revolves around the question of whether PC sets, when they are not connected by a more or less obvious operation, or do not share a more or less obvious property, can still be “related.” The aim has been to model such relatedness, and thus render it tangible. There is no apparent link with the familiar geometrical concept of similarity, which refers to the equal proportions of different-sized objects. Nor is there a link with the set-theoretic relation of the same name, which involves the order relations in two sets. The most likely source of the concept is information science, especially insofar as it deals with uncertainty. Similarity relations are defined there as “fuzzy” relations. They involve a degree of imprecision, and thus better approximate everyday human thought and speech than “crisp” equivalence relations (Klir and Folger 1988, 83–85.). In PC set theory, “similarity” is an umbrella term covering concepts with different origins, ranging from geometry to statistics. In this chapter, these concepts will be analyzed with respect to their underlying questions, intuitions, and assumptions. None of them has become the “most common” concept of PC set similarity; nor do they all refer to the same properties. PC sets can be similar in different ways; generally speaking, however, similarity is a matter of degree. In that respect, similarity relations are different from equivalence relations. PC sets are either in an equivalence relation or not, but they can be more or less similar. The study of PC similarity has been motivated by the prospect of a subtle and flexible analytical framework. The terms “equivalent” and “similar” may both be appropriate to a single pair of PC sets. In other words, they are not mutually exclusive; they reflect different ways of thinking about relations. However, similarity relations involve a much greater number of PC sets: any pair of PC sets will be more or less similar on some scale. This inclusiveness has been another reason to search for reliable measures of PC set similarity.

Concept and Referent What does it mean when two PC sets are called “similar”? How do these PC sets resemble one another? And in what regard can PC set similarity be meaningful? A PC set can be realized in a multitude of ways, causing different musical sensations.

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131

Although in ordinary language the word “similarity” denotes a resemblance, the similarity of two PC sets may not be apparent to a listener at all. It is, in other words, an abstract relation, regardless of how one defines it. This is one of the puzzling aspects of the debate on PC set similarity. A tool to describe relations beyond techniques of direct musical derivation, the similarity concept might account for the intuitive perception of a unified harmonic-melodic vocabulary. But how can it do that if such perception is dependent upon the variables of musical reality; upon the registral and temporal distribution of PCs; upon articulation and syntax? And if PC set similarity does not tell us something significant about our perception of a work, what can we learn from it? The earliest texts dealing with PC set similarity did not raise this question. The concept was brought into play without a proper introduction. Forte started his discussion of it as follows: To define an appropriate, musically meaningful similarity relation for any two sets of different cardinality is a task beyond the scope of this paper . . . It is possible, however, to define a simple yet effective criterion for similarity in the case of two sets of the same cardinality. (Forte 1964, 149–50)

Forte did not state what he meant by “musically meaningful” and “effective.” From his application of the concept—in an analysis of No. 4 of the Five Movements for String Quartet, Op. 5 by Anton Webern—one can infer that he wanted to come to grips with the relations between different equivalence classes of PC sets. It is not difficult to see the reason for this. If equivalence (as defined on the basis of transposition, transposition and/or inversion, or the equal-vector relation) were the sole criterion of PC set-relatedness, it would be hard to prove that PC sets are significant as units of structure in post-tonal music. Often, there are relatively few equivalent pairs among the PC sets that can be extracted from a musical composition (considering the most straightforward grouping criteria). Therefore, if PC sets are important, it is logical to look for more possible connections between them than pure equivalence. The above suggests that measures of similarity supported the concept of PC sets as real musical objects, while providing tools for their study. It suggests, in other words, that analytical concepts may not only clarify musical structures, but also enable, or uphold, an analytical practice, more or less for its own sake. Indeed, this is a possibility we cannot ignore, particularly since other authors working in the field of PC set theory have been just as reticent as Allen Forte concerning the musical insights gained by the use of a similarity measure. In his search for “generally applicable structural criteria in the atonal works of Schoenberg, Webern, and Berg,” Richard Teitelbaum investigated several relations between the “harmonic collections” (i.e., the PC sets) in these works, including “the degree of similarity (or dissimilarity) of intervallic distribution” (Teitelbaum

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1965, 74–75, 88). However, he was equally concerned with the use of computers in music analysis. As Teitelbaum wrote, the application of the computer was appropriate in the laborious process of determining and comparing the distribution of intervals in a large number of PC sets (Teitelbaum 1965, 76). It is possible that such an analytical endeavor was not only enabled, but also inspired by the availability of the computer. A computer can only deal with questions that lend themselves to algorithmic formulation. If analysts want to use it, they must raise such questions. The question of which interval classes are associated with a particular type of PC set belongs to this category. It translates into the following algorithm: •

List all unordered pairs of the PC set’s elements.

•

Determine the interval class represented by each of these pairs.

•

Compute the number of occurrences of each class.

•

Order these numbers by class, from APIC 1 to APIC 6.1

The result of these consecutive operations is the familiar APIC vector (see chapter 2, p. 48). Teitelbaum also used a protocol returning a value for the relation between two different APIC vectors, a value called “similarity index.” This will concern us later. The main point here is that he made no attempt to argue the need for such a protocol; he just claimed his similarity index to be “a significant indicator of the similarity or dissimilarity of interval content between two harmonic collections” (Teitelbaum 1965, 88). Its significance, however, may have had a lot to do with the state—and the status—of computer research (see further chapter 8A). We must realize that at the time of Forte’s and Teitelbaum’s early explorations in the field of PC set similarity little was known about the early atonal repertoire that was their main research object. They were searching for adequate means of description. Their similarity measures served as heuristic devices, scaffolding for yet unknown determinants of musical coherence. Actually, this holds true for PC set theory at large. A PC set analysis may reveal the consistent use of certain intervallic configurations in a musical composition, but it may also reveal the opposite. In both cases, the concept of a PC set has proven its worth, not as the signifier of actual units of structure, but as an analytical tool. This would mean that similarity measures should be seen—and judged—as refinements of what is basically a heuristic apparatus. However, the heuristic tools soon attained the status of descriptive models. Writing in 1964, Allen Forte could not restrain himself from blurring the fundamental distinction between these two categories: The theory of set-complexes has special significance for the study of syntactic structure in the case of atonal music, and work is now in progress to refine and 1.

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Cf. Teitelbaum’s description of this process (1965, 78).

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test related research tools, with the expectation that useful general analytic methods and descriptive techniques will result. (Forte 1964, 178)

The relationship between “analytic methods” and “descriptive techniques” is unclear. It is possible that the former term refers to the heuristic dimension of PC set theory; but Forte may have just wanted to draw a distinction between, on the one hand, analyzing music and, on the other, communicating the results of an analysis. In the preface to The Structure of Atonal Music, the heuristic function of PC set theory is not even mentioned: [I]t is the intention of the present work to provide a general theoretical framework, with reference to which the processes underlying atonal music may be systematically described. (Forte 1973, ix)

Forte assumes a correspondence between the theoretical framework and the musical processes or structures. In other words, the theoretical framework is supposed to be fully descriptive. But to which kind of process or structure do similarity relations correspond? The PC set is itself a problematic concept, as we have seen in previous chapters, but at least it is tied to music history, including the history of music theory. It can be shown to derive from serial composition, and although it is not actually an outgrowth of the chord or the scale, it has much in common with them. Furthermore, when we identify PC sets and trace transpositional, inversional, complementary, and inclusional relations between them, we are in the analytical tradition that has developed at professional schools of music, and that has had a formative influence on composers, performers, scholars, teachers, and editors alike. Whatever the geographical variety, this tradition has always focused strongly on relations involving themes, motifs, harmonies, and keys (correlative to its focusing on the repertoire of the common-practice period). Notwithstanding its initial alliance with contemporary composition, PC set analysis has established itself as an extension of this analytical tradition, and it carries a good deal of weight derived from it. This is not the case with similarity relations. PC set similarity was originally a concept without a clearly defined function, and without a history that would render it authoritative. This situation has never really changed. New similarity measures have been introduced because they were more discriminating than previous ones, or involved a wider range of PC set pairs.2 The relevance of PC set similarity itself was taken for granted. Robert Morris created a similarity index (labeled SIM) with the explicit aim of approximating the judgment of the ear. According to Morris—who 2. See, for example, Isaacson (1990, 2), who had formulated three criteria for the measurement of intervallic similarity: it should “1) provide a distinct value for every pair of sets; 2) be useful . . . for sets of any size; 3) provide a wide range of discrete values.” See further below.

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had taken an interest in PC set theory primarily as a composer—SIM would “provide a rationale for the selection of sets that insure predictable degrees of aural similitude” (Morris 1980, 446).3 Here he is on thin ice. Aural similitude always goes hand in hand with aural difference. It is the analyst’s decision to stress the similitude. This decision is necessarily based on a particular conception of what is being heard. For example, it is possible to stress the sameness of the chords C–E–G and E– G–C by reference to their constituent “pitch-classes” (as Jean-Philippe Rameau did); but it is just as possible to stress their distinctness, by reference to their intervallic structures (as Gioseffo Zarlino did). Neither view is the negation of the other. Rather, each rests on a different valuation of a single set of properties.4 In this case, these valuations correspond to certain structural or syntactic functions. In other words, they concern the “nuts and bolts” of past compositional and improvisational practices. PC set similarity, however, has always lacked such a clear musical referent. Therefore, equating PC set similarity with aural likeness is bound to remain a matter of faith. Similarity measures have attained a high degree of sophistication. In combination with their lack of an obvious source of validation, this renders them unfit for pedagogical purposes. Similarity is thus also an area of PC set theory falling outside the traditional scope of music analysis (which has always had a generally pedagogical orientation). In many texts dealing with this topic, the questions of analytical procedure have been taken for granted as much as the questions of musical relevance. In The Structure of Atonal Music, Allen Forte—in many ways still a traditional music theory teacher—offered a set of musical examples showing instances of his R0-, R1-, R2-, and Rp-measures of PC set similarity (ex. 5.1; Forte 1973, 59; see further below). These examples were probably not meant as evidence of the frequency of these relations. At the time, Forte used to apply not empirical but logical criteria of significance. He considered a similarity relation worthier of note the less likely it was to occur between any two PC sets. The question of what it meant to encounter it in a work seems to have been of minor importance to him. As to example 5.1, he made no effort to clarify the method of his analysis, or to state its purpose. This musical example, like many others in his book, served merely as illustration.

3. In a footnote added to his article—which responded to the criticism passed on an earlier, spoken version of it—Morris confided: “I have been and still am concerned with attempting to provide a similarity relation which corresponds in some way to what I—at least—hear when two sets are presented in some similar musical setting.” (Morris 1980, 458) His “similarity index” was tested empirically by Cheryl Bruner (1984), who judged it too narrowly defined to reflect a listener’s aural experience. 4. See Carl Dahlhaus’s discussion of this topic with reference to the theories of Zarlino and Rameau. (Dahlhaus 1968, 211–14.)

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similarity 61. Scriabin, Seventh Piano Sonata Op. 64



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5

(a)

6-Z49 [4,6,9,10,0,1]

5-16 6-Z19 [1,2,5,6,8,9]

333

(b)

6-Z13 [9,10,0,1,3,4]

6-Z24 6-34 [9,10,0,1,3,5] [3,4,6,8,10,0]

62. Similarity relations between parts a and b in example 61 R0, Rp 6-Z49

6-Z19

R1, Rp 6-Z24

6-Z13

6-34 R0, Rp R0, Rp

Example 5.1. Similarity relations among PC sets in the Seventh Piano Sonata, Op. 64 by Alexander Scriabin, as pointed out by Allen Forte in The Structure of Atonal Music. Copyright © 1973 by Yale University.

Contrary to Forte, Richard Teitelbaum took a truly heuristic approach, using his similarity index (which he labeled “s.i.”), and other concepts, to characterize a musical work’s harmonic idiom. The similarity indices resulting from Teitelbaum’s analyses were not simply documented as analytical results, but as raw observations from which certain conclusions could be drawn. For example, Teitelbaum noticed that PC sets in Schoenberg’s early atonal works tended to be similar to a much lesser degree than the PC sets he had extracted from Webern’s Five Movements for String Quartet, Op. 5. He presented this as an indication of a difference in “harmonic practice” between the two composers (Teitelbaum 1965, 108). Another author who provided an analytical context for his discussion of PC set similarity was Charles Lord. Lord’s “similarity function” (sf) was almost identical to Morris’s similarity index, but he emphatically presented it as a corrective to Forte’s

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R0-, R1-, R2-, and Rp-measures, claiming that it would give “a more complete and specific picture of intervallic similarity” (Lord 1981, 109). This claim was justified, for the sf denoted relative PC set similarity, and thus involved any possible pair of PC sets. However, with the single musical example he presented—an excerpt from the Ninth Piano Sonata, Op. 68 by Alexander Scriabin—Lord did not show the musical functionality of PC set similarity any more than Forte did, particularly since he had copied this example from The Structure of Atonal Music, complete with Forte’s segmentation (cf. Forte 1973, 117). Still other authors, including Regener (1974), Rahn (1980b), Morris (1980), and Isaacson (1990), presented similarity measures without referring to musical examples at all. Nevertheless, they were confident about their practical value: All of the various similarity relations . . . can be combined into one monstrous graph . . . through which wander all the paths of all the similarity relations. And since these paths cross—that is, share element-nodes—the adventurous composer can, with her or his moving finger, trace new similarity paths created by the concatenations of segments from the crossing paths of the similarity relations that created the graph. (Rahn 1980b, 497) IcVSIM, as an intuitively satisfying measure of intervallic similarity in the abstract, should prove beneficial for concrete applications in the analysis of atonal music. (Isaacson 1990, 25)

It is not necessary to downplay the speculative origin of the similarity concept; nor is speculation objectionable. It is perfectly possible for a composer to conceive a musical process along John Rahn’s “similarity paths.”5 In that case, a listener’s ignorance of the process does not detract from the functionality of the similarity concept. In music analysis, it is true, this functionality is hard to verify. However, this in itself does not disqualify PC set similarity as an analytical concept. Even though it is not grounded in pre-existing compositional or analytical practices, it can still be viewed as a mental image of musical relatedness. And it is impossible to ban such images from the history of music, since all musical relations—including the ones that do play a verifiable role in musical practice—rest on conceptualization.6 5. Rahn’s article inspired a composition by James M. Bennighof. See Bennighof 1984. 6. The relationship between conceptual and perceptual structure is a recurring topic in discussions about music analysis. Nobody wants to deny the important role of the human ear when it comes to verifying the results of an analysis. On the other hand, it cannot play this role without conceptual guidance; and sometimes it cannot play this role at all. Research conducted by Nicholas Cook in the 1980s revealed that a fundamental analytical concept like “tonal closure” exerts a relatively weak influence on music perception, and only within short spans of time (Cook 1987b; for a critical comment on Cook’s experiment, see Gjerdingen 1999). Of course, it can be useful as an analytical concept, but it may not tell us anything about perception.

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What we need to know is what brings about these “images of musical relatedness.” Besides a proximity between elements, they involve a purpose, and the competence to make this proximity serve that purpose. A proximity of elements may not be discerned if it does not reflect the light that purpose and competence cast over the music. The origination of Richard Teitelbaum’s similarity index from an investigation into analytical applications of the computer is a case in point. Going further back in time, we may think of Schoenberg’s preoccupation with the principle of “developing variation” in the music of, among others, Beethoven and Brahms. This principle reflected his own compositional concerns. Joseph Straus (1990b) has depicted Schoenberg as a composer who tried to bring an omnipresent musical past into line with his own artistic mission. Carl Dahlhaus did not make such a strong historical claim, but he, too, pointed to the idiosyncrasy of Schoenberg’s analyses: The abstraction with which Schoenberg the analyst operated consisted as a rule in the fact that intervals or complexes of intervals appeared as the true substance of music, whereas the other features of the composition . . . were treated as the mere “surface,” more a matter of “presentation” than of “idea.”7

Schoenberg’s strong inclination to close reading reflects his devotion to the musical work of art as embodied in notation. This attitude was not idiosyncratic. It correlated with the work-concept originating in the Romantic aesthetic theory of fine art. Lydia Goehr (1992) has pointed out the developments in the spheres of music notation, performance practice, concert behavior, music criticism, and copyright law by which this work-concept gained footholds in the nineteenth century. For another example of a pragmatically conditioned analytical perspective, we may think of Rameau’s basse fondamentale, whose complex and dynamic historical background has been unraveled by Thomas Christensen (1993a). Elsewhere, Christensen writes: To understand Rameau’s conception of the fundamental bass in the Traité de l’harmonie means reconstructing the musical conditions of his time that impelled his discoveries, reconstituting the varied pedagogical traditions to which Rameau was heir . . ., and identifying the intellectual ideas with which he came into contact. (Christensen 1993b, 17)

Bearing in mind the richness and complexity of conceptualization, we will now examine a number of similarity measures from the viewpoint of their underlying intuitions, and the origins of these intuitions. 7. “Die Abstraktion, mit der Schönberg als Analytiker operierte, bestand in der Regel darin, daß Intervalle oder Intervall-Komplexe als die eigentliche Substanz der Musik erscheinen, während die übrigen Tonsatzmerkmale . . . als bloße ‘außenschicht’—mehr ‘Darstellung’ als ‘Gedanke’—behandelt werden.” (Dahlhaus 1986, 282; transl. Derrick Puffett and Alfred Clayton).

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1. Triads consisting of three consonances:

2. Triads consisting of two consonances and one mild dissonance:

3. Triads consisting of one consonance and two mild dissonances:

4. Triads consisting of two consonances and a sharp dissonance:

5. Triads consisting of one consonance, one mild dissonance, and a sharp one:

6. Triads consisting of one mild dissonance, and two sharp ones:

Example 5.2. Krenek’s chord classification (Studies in Counterpoint, 1940).

Gradations of Harmonic Tension When Forte started discussing PC set similarity in 1964, the topic had already been raised twice, though in a different connection. Paul Hindemith, in his Unterweisung im Tonsatz (1937), had proposed a comprehensive classification of chords employing the twelve notes in the chromatic octave. And Ernst Krenek, in his Studies in Counterpoint (1940), had classified all possible triads. Strictly speaking, when chords are assigned to classes, we are dealing with equivalence rather than similarity. The classifications of Hindemith and Krenek invite a comparison with PC set similarity because both composers plotted their classes on scales of increasing harmonic tension. This means that they brought all chords into a relationship of greater or lesser similarity.

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Hindemith and Krenek gave different explanations for the experience of harmonic tension. They both pointed to the qualities of the intervals contained in a chord. Krenek distinguished consonances (PICs 3, 4, 7, 8, and 9), mild dissonances (PICs 2 and 10), sharp dissonances (PICs 1 and 11), an interval that could be perceived as consonant or dissonant, depending on the musical context (PIC 5), and a “neutral” interval (PIC 6). He must have struggled with these categories, for his argumentation was loose and contradictory on this point. On the one hand he treated them as objective gradations of harmonic tension; on the other, he pointed to the aesthetic, and hence arbitrary, basis of attributes like “consonant” and “dissonant.” The degree of the tension may be explained by vibration-ratios, combinationtones, or other acoustical phenomena; yet, the decision of what shall be considered a dissonance, and how it should be handled is an arbitrary assumption inherent in a particular musical style, for it depends exclusively on aesthetic concepts. (Krenek 1940, 7)

In any case, Krenek correlated the tension of a triad with its constituent intervals. For example, he classified a triad consisting of two consonances and a sharp dissonance (for example, C4–D♭4–A4) as tenser than a triad consisting of a consonance and two mild dissonances (for example, C4–D4–B♭4; ex. 5.2; Krenek 1940, 19–20). Hindemith, who had grounded his theory of harmony physically—although he sacrificed physics for tradition when the two were in conflict (see note 9 below)—arranged the intervals by the extent to which they were “burdened” (belastet) by their combination tones, specifically their difference tones. When the best-perceivable difference tones (designated D11 and D21)8 just amplified the members of the interval, the latter sounded clear and stable. The most strongly amplified member was designated the “root” of the interval. Thus, C4 was the root of the perfect fifth C4–G4 (D11 = C3 and D21 = C3), a very stable interval. The perfect fourth G4–C5 (D11 = C3; D21 = C4)—which had its root on top—was somewhat less stable. However, Hindemith attached a greater harmonic value to this interval than to those producing non-amplifying difference tones, like the minor sixth E4–C5 (D11 = G3). Hindemith’s arrangement of intervals—his “series II”—was the following: (1) octave; (2) perfect fifth and (3) perfect fourth; (4) major third and (5) minor sixth; (6) minor third and (7) major

8. This is a common German notation, originally proposed by Heinrich Husmann. D11 and D21 are called difference tones of the first and second order, respectively. When f1 and f2 are the frequencies of two pitches forming an interval, and f2 is higher than f1, then the frequency of D11 equals f2 − f1, and the frequency of D21 equals 2f1 − f2 (Husmann 1953, 12–13). Husmann’s proposal to let the order indications “first,” “second” etc. begin with the two primary pitches has not found general acceptance.

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© By kind permission of the Schott Music GmbH & Co. KG, Mainz, Germany.

Example 5.3. Hindemith’s chord classification (Unterweisung im Tonsatz, 1937).

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sixth; (8) major second and (9) minor seventh; (10) minor second and (11) major seventh; (12) tritone.9 Like Krenek, Hindemith determined the value of a chord by considering the values of the intervals of which it was formed. However, they valued certain intervals differently, like the perfect fourth (a fairly stable interval according to Hindemith, but one that could also sound dissonant, according to Krenek) and the tritone (a highly unstable interval according to Hindemith, but a “neutral” one according to Krenek). Moreover, Hindemith also considered the mutual positions of the intervals in a chord. In his view, a chord was maximally stable when its bass note was also the root, that is, the root of its “best” interval (in terms of Series II). Hindemith distinguished six principal classes of chords, which he identified by Roman numerals. A lower numeral indicated a greater harmonic value (ex. 5.3; Hindemith 1937, 116–21). The classifications of Hindemith and Krenek are, in fact, classifications of intervallic properties, or combinations of such properties. Hindemith’s label “IIb2” signifies the combined properties “containing a tritone, a major second and/or minor seventh (i.e., not a minor second or a major seventh), and a bass note that is not the root.” Having one such set of properties in common is an equivalence relation in the collection of chords encompassed by the twelve tones of the chromatic octave. Hindemith’s classification of chords differs from the PC set classifications of, say, Forte and Rahn in that there is no rule of transformation—or a coherent set of such rules—connecting all members of one class, like transposition, transposition and/or inversion, etc. (The same goes for Krenek’s classification.) His classes even consist of chords of different cardinalities. Moreover, since the root concept serves as a criterion for the stability of a chord, two realizations of one PC set may end up in different subclasses of his system. (Krenek—who did not use the root concept—would not see them as distinct.) For example, Hindemith assigned the chords C4–G4–B♭4 and B♭3–C4–G4 to the classes III1 and III2, respectively. There is something else worth mentioning. Unlike the common PC set classifications, Hindemith’s system does not seem to constitute a “phenomenology of all chords” (Hindemith 1937, 120). It has obviously been devised for chords of six tones or less, since it is unable to distinguish between larger ones. Each such chord, of seven or more tones, contains representatives of all PICs. As a consequence, all these chords would belong to just one of Hindemith’s classes, namely IV. (Of course, Krenek’s classification of triads did not have this problem.) Rather than simply viewing this imbalance as a shortcoming of Hindemith’s system, we should consider its underlying pragmatic considerations: chords of more than 9. Series II is not entirely consistent with Hindemith’s basic principles. For example, the members of a minor third are not amplified by either of this interval’s two prominent difference tones. This means that the minor third would have to come after the major second, the lower member of which is amplified by D11. In an apparent attempt to satisfy the musical intuition of his readers, Hindemith presented the minor third as a “shading” (Trübung) of the major third (Hindemith 1937, 89).

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Example 5.4. Hindemith’s way of showing the increase and decrease of harmonic tension (Harmonisches Gefälle) in his Unterweisung im Tonsatz. © By kind permission of the Schott Music GmbH & Co. KG, Mainz, Germany.

six tones contradicted the principles of textural clarity and harmonic functionality that guided the change in his style in the 1930s. Which problems did Hindemith and Krenek actually want to solve? Their classifications were obviously a response to the expansion of harmonic vocabulary in the early twentieth century. In that respect, they fall in line with the chord-type inventories discussed in chapter 4. However, Hindemith and Krenek did not content themselves with the mere identification and tabulation of chords. They wanted to provide new general criteria for their use, though for very different reasons. Hindemith aimed to keep harmonic theory as the intermediary between art and nature at a time when it obviously had been shaken to its foundations.10 His “updating” of it would not be complete without new instructions for the succession of chords. Krenek—who, in the 1930s, acted as a spokesman for the musical modernism which Hindemith opposed11—addressed his instructions for the succession of triads to students of serial composition. Harmony was, in a way, a blind spot of early twelve-tone serialism. Apart from the combinatoriality principle—which helped Schoenberg to avoid octave doublings (see chapter 4, p. 94)—there were no general rules for the treatment of chords. It was not Krenek’s aim to establish such a rule. Rather, he wanted to bring the issue to the notice of his readers, for he felt that no composer could manage without a keen sense of harmonic progression (Krenek 1940, 19). In the opinion of these two men, one of the criteria for a good harmonic progression was the ebb and flow of harmonic tension—Hindemith’s harmonisches Gefälle—which should be planned in order to make sense. Krenek showed how the more dissonant simultaneities might prepare and underline the focal point of a melody, and how the use of milder ones might induce a relaxation (Krenek 1940, 21–23). Similarly, Hindemith showed how a conscious choice of chords from the various classes of his system might outline different tension curves (for example, 10. This was also the time that brought forth some electro-acoustical facilities with which one could gain firm control of tone production. Hindemith’s decision to base his new theory of harmony on the phenomenon of the combination tone—a phenomenon commonly known for almost two centuries—was inspired by his experiments with the trautonium, an electronic musical instrument invented in the late 1920s (Hindemith 1937, 78). It is ironic that an electronic instrument helped Hindemith to take a conservative stance in view of musical modernism. 11. Especially in his six lectures on “new music” in 1936 (Krenek 1937).

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curves rising and falling very rapidly, or curves rising and falling more slowly). Or he did the reverse, drawing such tension curves from harmonic progressions (ex. 5.4; Hindemith 1937, 138). Hindemith also invoked other criteria for harmonic progressions—criteria involving voice leading and root progression—but it is this one that is of special interest in relation to PC set similarity. Harmonic tension, based on intervallic structure, provided an index of relatedness which might be useful for composers. In PC set theory, however, it was the intervallic structure itself that mattered—not a slippery attribute like “tension.” Such an attribute implied causal relationships—tensions and resolutions—contradicting the sense of musical logic that PC set theory had inherited from twelve-tone serialism.

Distributive Patterns In “A Theory of Set Complexes” Allen Forte attempted to define meaningful similarity relations given two PC sets with equal cardinal numbers. Since these relations were based on the APIC vectors, they also involved the transpositions, inversions, and Z-correspondents of these PC sets. According to Forte, two PC sets were maximally similar if four interval classes (i.e., APICs) were represented by the same number of unordered PC pairs in each. In other words, four entries in the APIC vectors of these two PC sets should be identical (Forte 1964, 150). (Four is the maximum number of entries, for APIC 1–6, that two non-equal APIC vectors can have in common.) Eric Regener stated this relationship a little more formally. According to Regener, two PC sets A and B, with a and b occurrences of APIC i respectively, were maximally similar if: ai = bi for four of the i [i ≠ 0]. (Regener 1974, 205)

The following example has been drawn from Forte’s analysis of Webern’s Five Movements for String Quartet, Op. 5, no. 4, the analysis concluding his 1964 article. The violin tremolos in mm. 1–2 (ex. 5.5) realize the PC sets α = {B,C,E,F} (that is, {11,0,4,5}; Forte name “4–8”) and β = {B,C,F,F♯} (that is, {11,0,5,6}; Forte name “4–9”). A comparison of their APIC vectors (minus the entries for APIC 0) reveals why Forte called them “maximally similar”: α:

200121

β:

200022

Four entries correspond; two are different. (To facilitate comparison, different entries between two APIC vectors appear in bold typeface in this and the following examples.) In PC set α there is one instance of APIC 4. This class is not represented in PC set β. On the other hand, the single instance of APIC 6 in PC set α is outweighed by the

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similarity Sehr langsam



145

= ca. 58

mit Dämpfer

am Steg . . . . . . . . . . . . . . . . . . mit Dämpfer

am Steg . . . . . . . . . . . . . . . . . . α β

Example 5.5. Maximal similarity between the PC sets {11,0,4,5} (α) and {11,0,5,6} (β) at the beginning of Webern’s Opus 5, no. 4. An example mentioned by Allen Forte in “A Theory of Set-Complexes for Music” (1964). © 1922 by Universal Edition A.G., Wien/UE 5888, UE 5889.

two instances of this interval class in PC set β. If we take these numbers into account—and if we assume a similarity relation to involve two PC sets with nonequal APIC vectors—the predicate “maximally similar” is fully appropriate. Yet a further differentiation is possible. Some PC sets may be more “maximally similar” than others. For example, Forte distinguished between “first-order” and “second-order” maximal similarity, using common mathematical terminology to denote levels of quantity.12 APIC vectors in a first-order maximal similarity relation consist of the same digits, whereas APIC vectors in a second-order maximal similarity relation—like those of the PC sets α and β above—do not. In Regener’s formulation, a first-order maximal similarity relation meant that: ai = bi for four of the i and in addition aj = bk (which implies ak = bj) for the other two interval classes j and k (Regener 1974, 205)

In his analysis of Webern’s Opus 5, no. 4, Forte also mentioned an instance of this relation, involving the PC sets {0,4,5} and {11,0,4} on the one hand (γ; Forte name “3-4”), and {11,0,5} and {11,4,5} on the other (δ; Forte name “3-5”). These PC sets have the following APIC vectors: γ:

100110

δ:

100011

APIC 4 is represented in γ by the same number of unordered PC pairs as APIC 6 in δ. The same is true, of course, of APIC 6 in γ and of APIC 4 in δ. 12. Babbitt had used this expression with reference to combinatorial hexachords. According to Babbitt, “first-order” hexachords created combinatorial relationships at only one transpositional level, “second-order” hexachords at two, etc. (Babbitt 1955, 58).

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Table 5.1. Distributive patterns in the relation 5:1 (Forte: second-order maximal similarity) PC set α = {B,C,E,F}, or {11,0,4,5} (Forte name “4-8”) {4,5} {11,0} APIC 1

2

{0,4} 4

3

{0,5} {11,4} 5

{11,5} 6

{0,5} {11,6} 5

{0,6} {11,5} 6

PC set β = {B,C,F,F♯}, or {11,0,5,6}(Forte name “4-9”) {5,6} {11,0} APIC 1

2

3

4

Note: The columns represent the successive APICs from 1 through 6. Five out of six PC pairs from each set are distributed to the same classes. The black bars mark the common number of pairs per class.

The maximal similarity relation appears to be even more flexible when the sizes of the vector entries are taken into account. Each APIC vector represents a distributive pattern: the unordered pairs formed by the elements of a PC set are distributed to a maximum of six interval classes. When we compare the APIC vectors of two PC sets of the same cardinality, we should know how many pairs from either set belong to the same classes, and how many belong to different classes. The ratio of these numbers can serve as an expression of the relation between these PC sets. In the case of the PC sets α and β above, this ratio is 5:1, as table 5.1 shows. Both sets have two PC pairs in APIC 1, two in APIC 5, and one in APIC 6. The single remaining pairs in α and β are assigned to APIC 4 and 6, respectively.13 Now let us compare the APIC vectors of α and a new PC set: ε = {4,5,6,11} (Forte name “4-6”): α:

200121

ε:

210021

The APIC vectors of the PC sets α and ε represent distributive patterns in the same relation as those of α and β, that is, 5:1 (table 5.2). Unlike β, however, ε is in a firstorder maximal similarity relation to α. Compared to the single APIC-4 representative {0,4} in PC set α, it accommodates an APIC-2 representative—{4,6}—which does not have a “class-mate” either. In PC set β, however, the counterpart of {0,4} is an APIC-6 representative, of which there was already one in PC set α. 13. The relation between the vectors of two PC sets α and β is symmetrical only if α and β have the same cardinal number. In that case, there is one proportion representing α’s relation to β as well as β’s relation to α.

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Table 5.2. Distributive patterns in the relation 5:1 (Forte: first-order maximal similarity) PC set α = {11,0,4,5} (Forte name “4-8”) {11,0} {4,5} APIC 1

2

3

{0,4} 4

{0,5} {11,4} 5

{11,5} 6

4

{11,6} {11,4} 5

{11,5} 6

PC set ε = {4,5,6,11} (Forte name “4-6”) {5,6} {4,5} APIC 1

{4,6} 2

3

Table 5.3. Distributive patterns in the relation 4:2 (Forte: first-order maximal similarity) PC set ζ = {11,0,4,6} (Forte name “4-16”)

{11,0} APIC 1

{4,6} 2

3

{0,4} 4

{11,4} {11,6} 5

{0,6} 6

{0,4} 4

5

{0,6} 6

PC set η = {0,3,4,6} (Forte name “4-12”)

{3,4} APIC 1

{4,6} 2

{0,3} {3,6} 3

Two new PC sets—ζ = {11,0,4,6} (Forte name “4-16”) and η = {0,3,4,6} (Forte name “4-12”)—are in a first-order maximal similarity relation as well: ζ:

110121

η:

112101

However, from table 5.3 it appears that their distributive patterns are in the relation 4:2. Seen thus, they are more different than α and ε are, and even more different than α and β, which are in a second-order maximal similarity relation according to Forte. Apparently, the notion of “maximal similarity” may include smaller or larger differences between the interval contents of two PC sets. The same is true for another similarity measure that Forte had defined in 1964: “minimal similarity.” In two PC sets that are minimally similar, each interval class

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is represented by a different number of PC pairs (Forte 1964, 150). The APIC vectors of the PC sets θ = {10,0,2,4,6} (Forte name “5-33”) and ι = {11,0,2,4,6} (Forte name “5-24”) provide an instance of this relation: θ: 040402 ι: 131221 Table 5.4 shows that these vectors represent distributive patterns in the relation 6:4. Now, θ is minimally similar to other sets as well, for example κ = {11,0,1,2,4} (Forte name “5-2”): θ: 040402 κ: 332110 In this case, however, the relation is 4:6 (table 5.5). Apparently, there are various degrees of “minimal similarity.” This may seem contradictory, but it only shows that PC set similarity can be defined in more than one way. In The Structure of Atonal Music, Forte defined his similarity measures again, now labeling them “R0” (minimal similarity), “R1” (first-order maximal similarity), and “R2” (second-order maximal similarity).14 He also added a new measure: “maximal similarity with respect to pitch class,” designated “Rp.” Two PC sets of cardinal number n are in this relation if they contain a common subset of cardinal number n −1 under transposition and/or inversion (Forte 1973, 47, 50). Forte combined this measure of inclusion with each of the intervallic similarity measures so as to reduce the number of PC sets satisfying either measure. As said before, Forte regarded a relation as more significant when it involves a smaller number of PC sets: [T]he pc similarity relation Rp is not especially significant taken alone, since by that measure a given set may be similar to many others. When Rp is combined with the [interval-class] similarity relations, however, a considerable reduction is effected. (Forte 1973, 49)

Forte thus spoke of musical relations as independent from musical context. He measured their significance against the criteria of a self-contained system of which the musical work was only an example. Forte’s approach to PC set similarity was later criticized for failing to provide comprehensive coverage of that system (Lord 1981; Isaacson 1990). Indeed, Forte considered a few special cases 14. Regener rightfully criticized these labels as being meaningless. The indices 0, 1, and 2 follow the order: least similarity—great similarity—not as great similarity (Regener 1974, 205).

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149

Table 5.4. Distributive patterns in the relation 6:4 (Forte: minimal similarity) PC set θ = {10,0,2,4,6} (Forte name “5-33”)

APIC 1

{10,0} {4,6} {2,4} {0,2} 2

3

{6,10} {10,2} {2,6} {0,4} 4

5

{10,4} {0,6} 6

{2,6} {0,4} 4

{11,6} {11,4} 5

{0,6} 6

PC set ι = {11,0,2,4,6} (Forte name “5-24”)

{11,0} APIC 1

{4,6} {2,4} {0,2} 2

{11 {11,2},2} 3

Table 5.5. Distributive patterns in the relation 4:6 (Forte: minimal similarity) PC set θ = {10,0,2,4,6} (Forte name “5-33”)

APIC 1

{4,6} {10,0} {2,4} {0,2} 2

3

{2,6} {10,2} {6,10} {0,4} 4

5

{10,4} {0,6} 6

{0,4} 4

{11,4} 5

6

PC set κ = {11,0,1,2,4} (Forte name “5-2”) {1,2} {0,1} {11,0} APIC 1

{11,1} {2,4} {0,2} 2

{1,4} {11,2} 3

of PC set similarity only, identifying them by distinct labels. He did not provide a full scale of relative values. For this, he may have had his reasons: any two PC sets are “similar” to some extent, but this does not mean that any similarity relation is worth using. Forte’s measures are thus a selection from all the possible similarity relations. On the basis of the vector entries, he could, for example, have specified other degrees of similarity besides the maximal and minimal—APIC vectors with one, two, or three corresponding entries—but he did not. Which musical intuition underlies Forte’s intervallic similarity measures? It is important to recall that they only apply to PC sets with the same number of elements. What does this restriction tell us? We already know that Forte, when he

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presented the intervallic measures in 1964, referred to the possibility of comparing other sets as well, saying that such an exploration would exceed the limits of his paper. In The Structure of Atonal Music, however, he seems to have dropped the matter. There is no mention of other similarities than those between PC sets of equal size: [I]t might be useful to define relations for sets of the same cardinal number so that given two such sets known to be non-equivalent one could determine the degree of similarity between them. (Forte 1973, 45)

Do Forte’s similarity relations themselves exclude comparisons between PC sets of different sizes? Indeed, there can be no first-order maximal similarity between sets of different cardinalities. Second-order maximal similarity is possible if one set contains one more element than the other. The relation between the PC sets ζ (see above) and λ = {0,1,4,6,9} (Forte name “5-32”) is a case in point (table 5.6): ζ: 110121 λ: 113221 Minimal intervallic similarity as defined by Forte becomes a more trivial relation between PC sets as the difference between their cardinal numbers increases. Therefore, it is best to use this relation with pairs of equal-sized PC sets. But then there are conflicting reasons for defining this as the domain of both minimal and maximal similarity. The significance of minimal similarity is based on its relative infrequency among equal-sized PC sets. Maximal similarity, however, can hardly be found in any other domain. It would be logical if maximal similarity were defined for pairs of PC sets with different numbers of elements, like ζ and λ above. For Forte, however, comparing such pairs fell within the scope of PC set inclusion (see chapter 6), rather than similarity relations. Forte’s intuition of intervallic similarity can be summarized in a number of statements: Statement 1: PC pairs (intervals, subsets of cardinal number 2) are the key attributes of a PC set.

Each pair of PCs receives a value (normally the APIC value). Similarity is founded on the correspondence of these values. Statement 1 assumes interval content as a distinctive property of a PC set.

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Table 5.6. Distributive patterns of PC sets with different numbers of elements PC set ζ = {11,0,4,6} (Forte name “4-16”)

{11,0} APIC 1

{4,6} 2

3

{0,4} 4

{11,4} {11,6} 5

{0,6} 6

{9,1} {0,4} 4

{1,6} {4,9} 5

{0,6} 6

PC set λ = {0,1,4,6,9} (Forte name “5-32”)

{0,1} APIC 1

{4,6} 2

{1,4} {6,9} {9,0} 3

Note: The PC sets satisfy Forte’s definition of second-order maximal similarity.

The next two statements are numbered 2a and 2b, since they both deal with the same question: is a similarity relation exclusive to certain pairs of PC sets? Statement 2a: Similarity is a relation between equal-sized PC sets.

The APIC vectors of these PC sets constitute partitions of the same number (3 in the case of PC sets with three elements, 6 in the case of PC sets with four elements, 10 in the case of PC sets with five elements, etc.). This is considered the basic condition of intervallic similarity between two PC sets. It is actually an equivalence relation, but PC sets in this relation may resemble one another to a greater or lesser degree. The equal number of elements provides a common basis enabling a relatively simple comparison between them. As an aside, it should be noted that measuring the “size” of a PC set in distinct elements is not only a mathematical convention. It also reflects a musical convention: the way in which tonal chords are identified as triads or seventh chords without regard to any doublings. Statement 2b: Similar PC sets do not have the same interval content.

This statement is in keeping with the common use of the word “similar,” a word that signifies a resemblance, but not an identity. When two APIC vectors form partitions of the same number, they are supposed to be similar—unless they are identical. Statement 3: The degree of similarity depends on the number of interval classes (APICs) that are equally represented in both PC sets.

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= IVEC (L) = differences = IVEC (I)

[3

2

1

0

0

0]

(4-1)

[2

2

1

1

0

0]

(4-2)

[1

0

0 -1

0

0]

1 2 +0 2 +0 2 +1 2 +0 2 +0 2

2

=

s.i.

Example 5.6. Richard Teitelbaum’s computation of the “similarity index” (s.i.) of two PC sets with Forte names “4-1” (for example {0,1,2,3}) and “4-2” (for example {0,1,2,4}), respectively. “IVEC” is Teitelbaum’s abbreviation of “interval vector”—that is, our APIC vector. Richard Teitelbaum, “Intervallic Relations in Atonal Music,” Journal of Music Theory 9/1 (1965): 88. Used by permission of Duke University Press and Yale University.

The fact that this number ranges between 0 (minimal similarity) and 4 (maximal similarity) reveals how rough Forte’s measures actually are. Five possible degrees of similarity seem inadequate in view of the twelve set-classes of cardinal number 3 that Forte distinguished, not to mention his thirty-eight set-classes of cardinal 5, or his fifty set-classes of cardinal number 6.15 Forte’s measures can tell us whether an interval class is equally represented in two PC sets. However, they do not register the extent to which it is represented in either set, that is, its share in the total number of PC pairs. As a consequence, we do not know in what proportion similarity stands to difference. When the difference depends on two interval classes only, as in the case of maximal similarity, it can still be smaller or greater. Table 5.2 and table 5.3 illustrate this. In brief, any measure of similarity should include a measure of difference.

Squaring the Difference In 1965, shortly after Forte had presented his similarity measures, Richard Teitelbaum came up with an alternative approach. Teitelbaum had just graduated from Yale University, and would make a name for himself as a composer and live-electronics musician in the following years. His article “Intervallic Relations in Atonal Music” appeared in Journal of Music Theory, like Forte’s “A Theory of Set-Complexes.” They seem to have been prepared simultaneously. Like Forte, Teitelbaum claimed to fill the need of a systematic approach to the atonal works of Schoenberg, Webern, and Berg; and he made no secret of his indebtedness to Forte, much of whose terminology he had adopted. But he did not refer to Forte’s similarity measures. He defined a new measure—the similarity index (s.i. for short)—which responded to the problem that Forte had left unsolved: the proportion of similarity to difference of two PC sets. 15. If the four statements above are taken for granted, the total number of similarity relations is 66 for set-classes of cardinal number 3, and 595 for the set-classes of cardinal numbers 5 and 6.

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To compute the s.i. for two PC sets, Teitelbaum, like Forte, compared their APIC vectors. Given two vectors, Teitelbaum proposed the following protocol: [T]ake the difference between the corresponding entries, square each difference, sum the squares and take the square root of the sum.(Teitelbaum 1965, 88)

He also provided an example, shown here as example 5.6. Teitelbaum’s protocol was actually based on a geometrical interpretation of the APIC vector. In such an interpretation, the entries of a row vector represent the coordinates marking a certain point in an x-dimensional space. The vector is an arrow connecting the origin of the coordinate system with this point (ex. 5.7a). Its value is equal to its length, which can be computed by extracting the root of the sum of the squared coordinates. (A simple case of this is Pythagoras’s theorem, which applies to a two-dimensional space: a = a x 2 + a y 2, where a is the vector, and where ax and ay are positions on the two axes of the coordinate system.) Seen thus, the APIC vector is a vector in a six-dimensional space. It can be notated (a1, a2, a3, a4, a5, a6), and its value is: a1 + a 2 + a 3 + a 4 + a 5 + a6 2

2

2

2

2

2

Teitelbaum’s s.i. is the difference of two such vectors. To put it in a slightly different way, it is the value of the difference vector. The latter results from the subtraction of the entries of one vector from those of the other. Given two vectors a = (a1, a2, a3, a4, a5, a6) and b = (b1, b2, b3, b4, b5, b6), the value of the difference vector a − b is: (a1 - b1 ) 2 + (a 2 - b2 ) 2 + (a 3 - b3 ) 2 + (a 4 - b4 ) 2 + (a 5 - b5 ) 2 + (a6 - b6 ) 2

Geometrically, the subtraction of one vector from another can be seen as a specific way of joining them together. Example 5.7b illustrates this, employing the two-dimensional space of example 5.7a. Teitelbaum’s s.i. offered more different values than the hidden scale against which Forte had plotted his degrees of intervallic similarity. Also, it sometimes contradicted Forte’s notion of PC set similarity. A comparison of the values that Forte and Teitelbaum would assign to the same pairs of PC sets reveals their varying degree of correspondence. Earlier on, the PC sets α = {11,0,4,5} and β = {11,0,5,6} were shown to have APIC vectors representing distributive patterns in the relation 5:1 (see table 5.1). Another pair of PC sets, α and ε = {4,5,6,11} – yielded the same result (see table 5.2). The s.i. would be 1.41 (√2)16 in both 16. The values of the s.i. are usually irrational. One can write them as the square root of an integer, as Teitelbaum himself did (provoking the criticism of Charles Lord; see Lord 1981, 111, note 7), or one can provide decimal numbers as approximations of these values. Both notations are used in the present discussion.

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ay

a

ax

Example 5.7a. The geometrical representation of a vector called a.

ay -b a (a y - b y ) (a - b) b ax (a x - b x )

Example 5.7b. The subtraction of a vector called b from vector a. An arrow as long as b, but pointing in the reverse direction (−b), departs from the far end of a, marking a new point with the coordinates (ax− bx) and (ay − by). The difference of a and b is a vector connecting the origin with this point.

cases, indicating a small difference of interval content. When seen from Forte’s perspective, these pairs are maximally similar, but they fall into the “second-” and “first-order” categories, respectively. The s.i. of the PC sets ζ = {11,0,4,6} and η = {0,3,4,6}—with APIC vectors representing distributive patterns in the relation 4:2 – (see table 5.3)—would be 2.82 (√8). In other words, they would be more different than α and β or α and ε. According to Forte’s classification, however, ζ and η are in a closer relation than α and β: a relation of first-order maximal similarity. What is the cause of these discrepancies? We have seen that Forte considered it important when two PC sets contained the same numbers of one interval class

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(Statement 3). We also have seen that his measures of intervallic similarity are insensitive to the actual sizes of these numbers. Now, the above observations suggest that the s.i. of two PC sets increases with the difference between their distributive patterns. The explanation for this is simple. In two sets, the distribution of PC pairs to the six interval classes is the same up to a specific number of these pairs (cf. the black bars in Table 5.1 to 5.8). That leaves a number of pairs of which the distribution is not the same. The s.i. registers this difference, the blind spot of Forte’s similarity measures. It does even more than that, as the following examples will show. First, let us compare the APIC vectors of the PC sets µ = {2,3,4,5} (Forte name “4-1”) and v = {2,3,4,8} (Forte name “4-5”): μ: 321000 v: 210111 The distributive patterns of µ and v are in the relation 3:3 (table 5.7), which suggests that these PC sets are less similar than, say, ζ and η. The distributive patterns of the last mentioned pair were shown to be in the relation 4:2 (see table – 5.3). However, the s.i. of µ and v—2.44 (√6)—is smaller than the s.i. of ζ and η— – 2.82 (√8)—thus indicating a greater similarity of interval content. This discrepancy can be explained as follows. The difference between ζ and η involves two interval classes only (APIC 3 and APIC 5). In the case of µ and v, however, the difference is spread over all six interval classes. By squaring the differences between the corresponding vector entries, Teitelbaum’s s.i. suggests that two PC sets provide a larger contrast as their difference is more concentrated in one interval class. When the relation of the distributive patterns does not change, the s.i. may still indicate a greater or lesser difference between PC sets. The comparison of µ’s vector with that of a new PC set, ξ = {2,4,7,9} (Forte name “4-23”), illustrates this: μ: 321000 ξ: 021030 These vectors represent distributive patterns that are in the same relation as – those of µ and v—that is, 3:3 (table 5.8). The s.i., however, is not 2.44 (√6), but –– 4.24 (√18). It strongly emphasizes the difference, as this involves the minimal number of interval classes (two: APIC 1 and APIC 5). Interestingly, Forte would assign a high degree of similarity to µ and ξ for the same reason. In his view, these PC sets would be in a first-order maximal similarity relation. Teitelbaum’s conception of intervallic similarity can be summarized as follows: Statement 1: PC pairs (intervals, subsets of cardinal number 2) are the key attributes of a PC set.

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Table 5.7. Distributive patterns in the relation 3:3 PC set μ = {2,3,4,5} (Forte name “4-1”) {4,5} {3,4} {2,3} APIC 1

{3,5} {2,4} 2

{2,5} 3

4

5

6

{4,8} 4

{3,8} 5

{2,8} 6

4

5

6

4

{2,9} {4,9} {2,7} 5

6

PC set v = {2,3,4,8} (Forte name “4-5”) {3,4} {2,3} APIC 1

{2,4} 2

3

Note: The difference is spread over six interval classes.

Table 5.8. Distributive patterns in the relation 3:3 PC set μ = {2,3,4,5} (Forte name “4-1”) {4,5} {3,4} {2,3} APIC 1

{3,5} {2,4} 2

{2,5} 3

PC set ξ = {2,4,7,9} (Forte name “4-23”)

APIC 1

{7,9} {2,4} 2

{4,7} 3

Note: The difference only involves two interval classes.

Statement 1 was also made in connection with the similarity measures of Forte. Statement 2: No pair of PC sets is excluded from the similarity relation.

The s.i. can be computed for each pair of PC sets, regardless of their cardinal numbers, and regardless of other relations they may have (such as an identical interval vector). In other words, all PC sets are considered similar to a greater or lesser degree. The fact that Teitelbaum did not apply his s.i. to PC

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sets with different cardinal numbers does not change this condition. Nor is it essential that the s.i. lose its effectiveness as this difference increases, as Eric Isaacson (1990, 19) observed. This may only be a reason to limit the domain of investigation. Statement 3: The similarity of two PC sets depends on the amounts by which the frequencies of the interval classes in one set exceed those in the other.

The smaller these amounts, the greater the similarity. Teitelbaum’s s.i. is actually a measure of dissimilarity.17 Statement 4: The difference between two PC sets is greater as it involves a smaller number of interval classes.

This becomes obvious when we consider two PC sets characterized by an uneven distribution of intervals, one stressing an interval class that is entirely lacking from the other. A PC set of the “diminished seventh-chord” type—stressing APIC 3 (four PC pairs)—and a PC set of the “whole-tone tetrachord” type—stressing APICs 2 and 4 (three and two PC pairs, respectively) will together yield a high s.i.

Common Subsets Most of the similarity measures discussed so far involve abstract intervals—that is, intervals dissociated from their constituent elements. (The dissociation was expressed in Statement 1 above.) Alternatively, two PC sets may be similar because they share one or more subsets. We can say that set-classes of cardinalities 3 and higher are more likely to be heard as common to a pair of PC sets than interval classes. In other words, subset content may provide a more reliable measure of similarity between PC sets than interval content does. However, the definitions of PC set similarity that take subset content as their starting point are no less abstract than those based on interval content. The meaning of the term “subset” has been broadened by theorists: it may involve common subsets under transposition and/or inversion, operations preserving the subset’s interval content, though some theorists have excluded inversion from consideration. Others, however, did not even specify operations, thus leaving open possibilities such as a similarity based on common subsets under multiplication. Forte had already brought up subset content as a measure of PC set similarity in The Structure of Atonal Music: his Rp-relation was based on it. But the issue received a more general treatment from Eric Regener (1974), who was a critical 17. This is true for other measures as well. See Rahn 1980b, 489, and Isaacson 1990, 27, note 7.

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reviewer of Forte’s book. The general tenor of Regener’s comments—which were published in Perspectives of New Music—was Forte’s lack of mathematical rigor, but he also cast doubt upon the musical value of some of Forte’s relational concepts. Regener proposed the partition function as a tool that would measure the similarity of two PC sets more accurately than Forte’s Rp, R0, R1, and R2. The partition function is the number of “configurations” (that is, transpositions) of one PC set having n elements in common with another. The partition functions for the successive ns—from 0 up to the cardinal number of the smallest set—form the partition vector,18 which can detect structural similarities between the two sets. To illustrate how Regener’s concept works, let us return to the PC sets ζ = {11,0,4,6} and λ = {0,1,4,6,9}. From table 5.6 we can see that the interval content of the larger set λ includes that of ζ. Does this result bear a relation to the subset content of ζ and λ? Table 5.9a shows the number of elements (“notes”) each transposition of ζ has in common with λ. When ordered by interval of transposition, these numbers form the common-note vector of ζ and λ.19 Table 5.9b shows how the partition vector can be derived from the common-note vector by counting the numbers of the equal entries in the latter. There are no 0s and 4s in the common-note vector. However, it contains four 1s, seven 2s and one 3. This results in the partition vector 13710, ||13710|| in Regener’s notation. One thing that can be read from the partition vector is the degree of inclusion. This is the cardinal number of the largest common subset(s) under transposition. It is represented by the right-most non-zero entry of the partition vector. In the case of ζ and λ, the degree of inclusion is 3, and is represented by the entry 1 (which means: “ζ has three elements in common with λ at only one transpositional level”). This degree of inclusion tells us that no transposition of ζ—a set of four discrete elements—is entirely included by λ, despite the total inclusion of ζ’s interval content. We may conclude that the number of instances of an interval class held in common between two PC sets is not indicative of the degree of subset inclusion. However, Regener’s measure fails to provide sufficient evidence of this, since it does not take into account every subset with the same interval content as ζ. The partition function should be extended to include the inversion of ζ and its transpositions.20 Table 5.10 does this, and shows that the degree of inclusion of I(ζ) and λ is the same, namely 3. Therefore, no straightforward relation exists between common interval content and common subset content. The degree of inclusion should, of course, be considered relative to the cardinal numbers of these sets. But even then it lacks discriminating power, as the following examples illustrate. It would seem that the number 3 represents a 18. This row vector is a partition of the number of transpositions, i.e., 12. 19. This row vector is a partition of the product of the cardinal numbers (20 in this case; each element of a set of cardinal number 5, like λ, must appear four times in a transposition of a set of cardinal number 4, like ζ (see table 5.9a). 20. David Lewin (1977b) proposed such an extension. See further below.

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Table 5.9. Partition function of the PC sets ζ = {11,0,4,6} and λ = {0,1,4,6,9} (a)

Tn(ζ): 0 1 2 3 4 5 6 7 8 9 10 11

(b)

= = = = = = = = = = = =

{ 11, 0, 4, 6 { 0, 1, 5, 7 { 1, 2, 6, 8 { 2, 3, 7, 9 { 3, 4, 8, 10 { 4, 5, 9, 11 { 5, 6, 10, 0 { 6, 7, 11, 1 { 7, 8, 0, 2 { 8, 9, 1, 3 { 9, 10, 2, 4 { 10, 11, 3, 5

} } } } } } } } } } } }

Number of elements in common with λ: 3 2 2 1 1 2 2 2 1 2 2 “Common-note vector”: 0 322112221220

Number of elements in common with λ: Instances of Tn(ζ) (“partition vector”)

0 1

1 3

2 7

3 1

4 0

Table 5.10. Partition function of the PC sets I(ζ) = {1,0,8,6} and λ = {0,1,4,6,9} (a)

Tn I(ζ): 0 1 2 3 4 5 6 7 8 9 10 11

(b)

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= = = = = = = = = = = =

{ 1, 0, 8, 6 { 2, 1, 9, 7 { 3, 2, 10, 8 { 4, 3, 11, 9 { 5, 4, 0, 10 { 6, 5, 1, 11 { 7, 6, 2, 0 { 8, 7, 3, 1 { 9, 8, 4, 2 { 10, 9, 5, 3 { 11, 10, 6, 4 { 0, 11, 7, 5

} } } } } } } } } } } }

Number of elements in common with λ: 3 2 0 2 2 2 2 1 2 1 2 “Common-note vector”: 1 320222212121

Number of elements in common with λ: Instances of Tn I(ζ) (“partition vector”)

0 1

1 3

2 7

3 1

4 0

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significant degree of inclusion for two PC sets containing four elements each. Yet, the PC sets µ = {2,3,4,5} and ξ = {2,4,7,9} are both included by this degree in no less than eighteen T-classes of cardinal number 4. The same degree results from their comparison to fifty-seven T-classes of cardinal number 6. And they are included by a degree of 4—a degree as large as their cardinal number—in twenty-one of these classes. Now, let us compare PC sets of cardinal numbers 7 and 8. Three T-classes of cardinal number 8 include the set {0,1,2,4,7,8,9} (Forte name “7–20”) with a degree of 5; but 36 include it with a degree of 6, and four include it with a degree of 7. Deriving a degree of inclusion from the partition vector thus leads to a very uneven classification of similarity relations between PC sets. The partition vector could also be used for an alternative measurement of PC set similarity. Regener proposed comparing the partition functions of two pairs of PC sets (Regener 1974, 208). What kind of result did he expect? Having determined the partition vectors of these pairs, he could perhaps tell how similar their relations were. This is not the same as telling how similar just two PC sets are. However, both pairs may consist of two identical sets. For example, one pair may consist of two µs, and the other of two ξs. In that case, µ and ξ are compared with respect to their “self-partition functions.” We count the number of elements each PC set has in common with each transposition of itself. This is practically equivalent to applying Forte’s intervallic similarity measures. Regener rightly noted that two PC sets in Forte’s R1-relation often have the same self-partition functions (Regener 1974, 207). More specifically, they have the same selfpartition function unless they have different vector entries for APIC 6.21 Since self-partition functions are very similar to intervallic similarity as conceived by Allen Forte, they involve the same problems. When it appears from the self-partition vector that one PC set has, say, three elements in common with two of its transpositions, and the same is true for the other set, the transpositional levels concerned may or may not be the same for both sets. If they are not the same, this is an indication of different interval contents, which the self-partition vectors fail to recognize as much as Forte’s R1-relation. For example, the 21. The APIC vector of a PC set reveals the numbers of PCs shared by the set and its transpositions. Given a PC set A, a vector entry an (“a instances of APIC n”) most often tells us that the number of elements in common between A and Tn(A) is a. If so, this is also the number of elements in common between A and T−n mod 12 (A). Only when n = 6 will A and Tn(A) share 2a elements. In a previous section (on Forte’s similarity measures) we have seen that the vectors of R1 -related sets consist of the same digits, and that four of their entries correspond. This means that for most T(A)s sharing a elements with A, there is a T(B) sharing a elements with the R1 -related set B. Consequently, A and B are likely to have the same self-partition vector. This is not the case, however, when the entry for APIC 6 in the vector of A corresponds to the entry of another APIC in the vector of B—that is, when a6 in the vector of A, and an in the vector of B (n ≠ 6). Then, B shares a elements with Tn(B), whereas A shares 2a elements with T6 (A).

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self-partition vectors of the PC sets µ and ξ (see table 5.8) are both ||52221||. However, the transpositional levels at which µ retains three of its elements are 1 and 11, whereas ξ retains three elements when it is transposed by 5 or 7. This is caused by the difference—so evident from table 5.8—between the interval contents of µ and ξ. Compared to the three pairs of PCs in µ that belong to APIC 1, ξ contains three pairs belonging to APIC 5. Regener also considered the partition functions of two heterogeneous pairs of PC sets. As said before, he thus applied the concept of similarity to relations between PC sets. He did this because he had discovered that “[the] number of different partition functions [was] relatively restricted.” Speaking of hexachords, he pointed out that: [T]o the 3240 (that is, 80·81/2) different pairs of hexachords correspond only 70 different partition functions, of which 14 cover the combinations of the 80 hexachords with themselves.

(Regener considered the collection of hexachords as partitioned by Tn alone. This explains why he identified a larger number of different hexachords than Forte—that is, 80 instead of 50.) The number of pairs sharing the same function . . . ranges from 1 to 490 (for the class ||0044400||); fifty-one of these classes comprise 20 pairs or less. (Regener 1974, 208)

These numbers, though interesting in themselves, are not particularly informative with respect to PC set similarity. What does it mean when two pairs of PC sets have the same partition function? It means that the PC sets in question are of the same sizes, and that the pairs have the same number of common subsets under transposition. Moreover, the sizes of these common subsets correspond. Thus, it can be said that the two pairs are similar to the same degree; but that degree can still be high or low. Although Regener’s alternatives to Forte’s similarity measures were by no means a great success, David Lewin (1977b) took up this idea of comparing PC set relations in terms of partition functions. Unlike Regener, Lewin considered common elements under any group of canonical transformations (see chapter 4). The partition function of two PC sets A and B might not only involve the twelve transpositions of A, but also the twelve transpositions of A’s inversion. When the partition functions were the same, Lewin argued, two pairs of PC sets could be similar to various degrees. They were “weakly similar” if the similarity ended there. They were “superstrongly similar” if they also yielded the same common-note vector. And they were only “strongly similar” if one pair yielded the same common-note vector as did a canonical transformation of the other. In Lewin’s view, then, the common-note function (the number of elements each canonical transformation of one PC set has in common with the other) was

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more significant than the partition function (the number of canonical transformations of one PC set sharing a specific number of elements with the other). Lewin was right that the common-note function is a more selective measure. This can be ascertained from the reflexive forms of the partition and common-note functions. We have seen that the identical self-partition functions of the PC sets µ and ξ conceal the difference between their interval contents. The common-note vectors of µ and ξ and their duplicates, however, are most revealing in this regard:22 (µ,µ): 4 3 210 0 0 0 012 3 (ξ, ξ): 4 0 210 3 0 3 012 0 The PC sets µ and ξ are therefore only “weakly similar.” The conditions under which any two PC sets A and B are strongly similar depend on the canonical transformation group. Given the group of twelve transpositions, (A,A) and (B,B) are superstrongly similar if A and B have the same interval content. Given the group of transpositions and inversions, however, that condition would not even ensure a strong similarity between the two. Lewin discovered that some of the As and Bs with equal interval contents “fell by the wayside” when compared with respect to their common-note functions. In other words, they were “weakly similar.” He found these As and Bs among the “Z-related” pairs (Lewin 1977b, 227). Which statements could embody the ideas underlying the measures of Regener and Lewin? First of all, what was their conception of a PC set? Statement 1: A PC set consists of discrete elements.

In other words, the PCs in a PC set are related only insofar as they belong to the same set. Forte and Teitelbaum, when measuring the similarity of PC sets, considered the intervals formed by the PCs. In other words, they saw each element as a member of several pairs. Statement 2 (“No pair of PC sets is excluded from the similarity relation.”) is not different from the corresponding statement applying to Teitelbaum’s s.i. In the next statement, however, we see how Regener and Lewin conceived of PC set similarity: Statement 3: Two PC sets are similar when they are partly the same.

Basically, Regener and Lewin measured the similarity of PC sets by enumerating the elements they shared. It follows from Statements 2 and 3 that the “part that is the same” may be one of the PC sets itself, or both of them (in which case the PC sets are identical). 22. Apart from its first entry, the common-note vector of a PC set and its duplicate is identical to the PIC vector of that PC set.

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The next and final statement is “a step up the ladder of abstraction” (to quote John Rahn): Statement 4: What is considered “the same thing” (PC set) may involve a transformation.

The elements may be replaced by others, as long as this is the consequence of a canonical transformation. Regener was much more restrictive than Lewin in regarding transformations as “canonical.” In the context presented here he worked with transpositions only.

“Absolute” Measures The “Set Theory” session of the second National Conference of the Society for Music Theory, which took place in New York in 1979, contained two papers that addressed the similarity issue. One was read by Robert Morris, the other was a response to it by John Rahn. Both papers were published, along with the other papers of that session, and with an additional contribution by David Lewin, in the journal Perspectives of New Music. In its published form, this set of papers received considerable attention. It set the agenda for future work in the field of PC set theory, as one can see from later publications like Morris (1987), Isaacson (1990 and 1996), and Rogers (1999). In these years, PC set similarity was a very promising area of research. Morris started off by proposing a new similarity measure: the similarity index (SIM). At first sight, it looked like a rudimentary version of Teitelbaum’s measure of the same name (which Morris did not mention). Like Teitelbaum, Morris compared the APIC vectors of two sets (“V,” in his notation), taking the differences of the corresponding entries. However, instead of squaring these differences, and taking the root of the sum of the squares, Morris simply added up the differences. Here is his definition:23 If S and R are two sets: V(S) = [a1,a2,a3,a4,a5,a6] V(R) = [b1,b2,b3,b4,b5,b6] . . . the similarity relation is written . . . SIM(S,R) . . . and is called the similarity index between (or of) R and S. It is evaluated . . . 6

SIM(S,R)=

Σ |an-bn|

n=1

(Morris 1980, 446)

23. In this definition, the numbers of instances per interval class (n, if not specified) should actually have been given in subscript.

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What is the consequence of a procedure like this? R and S both have n elements. These elements, in turn, form ½(n2 − n) pairs in each set,24 which are distributed over six interval classes. As we have already observed, there is a number of pairs in S and in R for which the distribution corresponds. Morris called this number “k” (Morris 1980, 448). A number of pairs may then remain. Since R and S are of the same size, this number is equally divided between them. The pairs that remain in S are assigned to classes other than the remaining pairs in R. Like Teitelbaum’s s.i., Morris’s similarity index (SIM) deals with these remaining pairs. Both indices are actually measures of the difference between two PC sets. However, Morris’s SIM does not measure the “spread” of the difference, that is, the number of interval classes involved. Therefore, it is a less sensitive tool than s.i. For example, Morris would not see a distinction between the relation of µ and v (s.i. = 2.44) and that of µ and ξ (s.i. = 4.24; see table 5.7 and 5.8). His SIM would point at 6 in both cases. As PC set similarity rose higher up the agenda of music theory at the beginning of the 1980s, a special problem related to it became more obvious. Post-Fortean similarity measures were supposed to return values for relations between any two PC sets; as noted at the beginning of this chapter, this was one of their main attractions. But these values could not always be compared, as they were relative to the cardinal numbers of the PC sets involved. Morris was aware of this problem, which also affected his own SIM: Obviously, a similarity index of 2 between two hexachords is not directly comparable to the same index between two trichords. As the total number of [interval classes] available becomes larger, a given index becomes a correspondingly finer indication of similarity. (Morris 1980, 450)

Therefore, he provided an additional measure: an “absolute similarity index” (ASIM). Morris’s idea was to divide the similarity index of two PC sets R and S by the number of PC pairs in R plus the number of pairs in S.25 In his notation: ASIM(R,S) = (SIM(R,S)) / (#V(R) + #V(S))

When the similarity index is equal to the total number of PC pairs in R and S—indicating the maximum difference between any two PC sets (no PC pair in S is distributed to the same interval class as any pair in R)—Morris’s ASIM returns the value 1. As a consequence of Morris’s definition of PC set similarity, 24. The unisons are excluded; see chapter 2, p. 48. 25. Morris’s SIM is the sum of (a) the number of PC pairs by which S differs from R, and (b) the number of PC pairs by which R differs from S. (These numbers are equal when R and S have the same size.) It is thus a “two-way” relational concept, as is Teitelbaum’s s.i. For this reason, the similarity index of R and S must be divided by the total number of pairs in both sets.

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only PC sets below cardinal number 5 can be maximally different.26 At the other extreme, when two PC sets have the same APIC vectors, his ASIM returns the value 0. What does this mean for the PC set relations discussed so far? The PC sets γ = {0,4,5) and δ = {11,0,5} yield the same similarity index as α = {11,0,4,5} and ε = {4,5,6,11}—that is, 2. We have seen that these PC sets differ by one pair only. Their distributive patterns are in the relations 2:1 and 5:1, respectively. Now, Morris’s ASIM suggests that this difference loses significance when the total number of PC pairs is larger. In the case of γ and δ (two times three PC pairs), its value is 0.33. Therefore, γ and δ are less similar than α and ε (two times six PC pairs), the absolute similarity index of which is 0.16. This line of reasoning is clear. However, it proceeds from the APIC vector as an abstract statistical format, that is, as a display of the frequency distribution of interval classes in a PC set. Morris’s interpretation of it does not seem to consider the nature and behavior of these “data.” Teitelbaum’s use of the APIC vector showed a stronger sense of history in this regard. As we have seen, his s.i. responds strongly and specifically to an unequal distribution of intervals, which reflects a musical intuition buttressed by compositional practices (e.g., the use of equal divisions of the octave). Morris’s contribution to PC set similarity expanded the area of the thinkable. Others followed his lead, allowing the APIC vector to influence their musical imagination and theoretical writings more and more. John Rahn’s paper “Relating Sets” was more than just a response to Morris. Apart from discussing and refining the tools the latter had offered—for example, Rahn preferred Morris’s k to his SIM, and he proposed an “absolute k” as the counterpart of Morris’s ASIM27—it also evaluated the similarity measures provided by Regener (1974) and Lewin (1977b). And it contained a new set of proposals, starting with a concept measuring the “mutual embedding in sets A and B of sets X of size n,” designated “MEMBn(X,A,B)” (Rahn 1980b, 492). Rahn’s concept was based on Lewin’s embedding number (EMB): “the number of distinct forms of [a PC set] X . . . which are subcollections of [another PC set] Y” (Lewin 1977b, 197; the word “form” refers to a canonical transformation). Stressing subset content as a basis of PC set similarity, MEMBn was also reminiscent of Regener’s partition function. It is one thing to invent a concept like the partition function or MEMBn; to specify a value for the similarity of two PC sets is quite another. We have seen that 26. For two PC sets to be maximally different, the interval classes represented in one should not be represented in the other, and is thus dependent on a sufficient number of zero entries in both APIC vectors. The smaller the set, the larger this number. In this regard, Morris’s ASIM reflects the intuition that chords consisting of just a few tones provide larger contrasts than chords containing many different tones. 27. To obtain this measure, 2k was divided by the number of PC pairs in R plus the number of pairs in S. In Rahn’s notation: ak(A,B) = 2·k(A,B)/#V(A) + #V(B) (Rahn 1980b, 489)

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Regener had difficulties in deriving a scale of values from the partition function. Rahn decided to count the number of all mutually embedded (common) subsets of more than one element. Thus, he obtained a value for the “total mutual embedding in A and B of subsets of all sizes greater than one” (Rahn 1980b, 493). The following formula tells us that TMEMB results from a summation of the values of MEMBn for all n between 2 and 12 inclusive: 12

TMEMB(A,B) =

Σ MEMBn(X,A,B) (ibid., 492) n=1

It should be noted that Rahn did not just count the common subsets under transposition, like Regener. MEMBn was generalized to any group of canonical transformations. And Rahn focused on the number of common subsets, whereas Regener had been more interested in their sizes. For example, Regener would have found that the PC set ζ = {11,0,4,6} had a subset of three elements in common with λ = {0,1,4,6,9}. According to Rahn, they would share four distinct subsets: {0,4,6}, {0,4}, {0,6}, and {4,6}. In his view, each common subset consisting of three elements would entail three common subsets of two elements. Similarly, each common subset of four elements would entail six common subsets of two, and four common subsets of three elements, etc. Thus, the value of TMEMB grew exponentially with the number of elements in common between two PC sets. This was an acceptable idea in itself—much more acceptable than a concept of PC set similarity fluctuating in exact proportion to the number of common elements. What was problematical, however, was Rahn’s counting every instance of a common subset: X must be embedded at least once in both sets A and B to be counted; then all instances of X in either set are counted. (Rahn 1980b, 492)

As we have seen, the PC set ζ = {11,0,4,6} has {0,4,6} in common with λ = {0,1,4,6,9}. This common subset would thus count for two on Rahn’s scale. Of course, the same goes for the smaller subset {0,6}. The common subset {0,4}, however, also matches another subset of λ, namely {1,9}. Therefore, it would count for three. The general idea underlying Rahn’s way of measuring PC set similarity can be rendered as follows: imagine a thing (A) consisting of several components; next, imagine a second, similar, thing (B). Suppose there is one component (X) of A that is also a component of B; this is a similarity between A and B. Under Rahn’s protocol, A and B are considered more similar as the number of X in B is greater. (Of course, the similarity increases further when more X is also added to A.) For the moment, it may be helpful to interpret A and B as two different dishes both of which contain, say, ginger. What would happen if more ginger were added to B? Although we can expect the flavor of ginger to become stronger in the latter, the similarity between A and B need not increase as a result. Dish A may not even taste of ginger much at all. This example may seem out of place, but it

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is not clear why a similarity concept like Rahn’s would match the world of musical tones any better than the world of dishes and their ingredients. In any case, when a common subset was represented by different numbers in two PC sets, Rahn did not see this as a dissimilarity. Like Morris, Rahn devised an “absolute” similarity measure for PC sets, using TMEMB as a basis. To normalize the values of TMEMB between 0 and 1—representing minimal and maximal similarity, respectively—he divided TMEMB(A,B) by the total number of relevant subsets in A and B. The result was “ATMEMB.” PC sets that were related by a canonical transformation always yielded the value 1, which meant that each subset of A matched at least one in B. PC sets that were not so related, but still had the same interval content, did not come out as maximally similar (as they do under Morris’s SIM). The statistical approach to the similarity issue made itself felt in Rahn’s paper as much as in Morris’s. Both authors defined PC sets in terms of loose components: PC pairs (Morris) and subsets of any size between 2 and 12 inclusive (Rahn). A component of one PC set might or might not match with one of another. The number of matching components—or, in Morris’s case, the number of non-matching components—was set against the sum of all components in both PC sets. Thus, similarities between PC sets were represented in proportion to their cardinal number. David Lewin’s “A Response to a Response” was the shortest, but by far the most complex of the three papers on PC set similarity that were sent in for publication in the 1980 Spring-Summer issue of Perspectives of New Music. It provided a measure, “REL,” that was meant as an alternative to Rahn’s ATMEMB. REL now exists in various formulations. Lewin’s own was not incorrect, as he himself suggested to Eric Isaacson (who reported this without further comment; Isaacson 1990, 27, note 9), but the readers of his paper may have found it a hard nut to crack: REL(A,B) = (1/SQR(TOTAL(A)TOTAL(B))) times ΣSQR(EMB(/X/,A)EMB(/X/,B)) (Lewin 1980, 500) In his book Composition with Pitch-Classes, Robert Morris provided a more compact version of the same definition, replacing some of Lewin’s labels (“SQR” for “square root,” for example) by mathematical symbols: REL(A,B) =

1 ·V W

V= W= and the sums are taken for all SCs [set-classes] X.

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In his definition of W, however, Morris had wrongly substituted the plus sign for the multiplication sign, so that in his version the square root was taken of the sum of the numbers of X in A and B instead of their product (cf. Morris 1987, 107). Although REL was a measure for the total common subset content of two PC sets, Isaacson (1990) and Rogers (1999) applied it to subsets of a single cardinal number: 2. Hence the term “REL2” in the following formula, which Lewin handed down to them as an alternative for the one he had presented in 1980. This new formula involved the cardinal numbers (#) of the two PC sets (X and Y): REL2(X,Y) = (Isaacson 1990, 11; Rogers 1999, 79)

All these formulas amounted to the same mathematical notion. When a is the sum of n integers ai (ai = a1, a2, a3 . . . an), and b is the sum of n integers bi (bi = b1, b2, b3 . . . bn)—in brief, when a = Σai and b = Σbi—the following equation holds:

if and only if

bi ai

is a constant

The final clause stipulates a very important condition: for the equation to be valid, ai and bi should be directly proportional. In Lewin’s interpretation, ai and bi represented the numbers of instances of a set-class i in the PC sets A and B. A valid equation meant that A and B were maximally similar. Maximal similarity was thus dependent upon a constant ratio between the numbers by which equivalent subsets were contained in A and B. REL converted the above equation to a rational form: or, alternatively, (cf. Lewin 1980 and Morris 1987)

so that a value between 0 (minimal similarity) and 1 (maximal similarity) could be obtained for any pair of PC sets. In the formula of REL2, the numerator was multiplied by two so as to bring it into correspondence with the new denominator. The values of the terms #A(#A −1) and #B(#B −1) in this denominator were twice the values of a and b, for which they had been substituted.28

28. REL2 can be thought of as measuring the similarity of interval-class vectors. In the above definitions, a is the sum of the APIC vector entries of a PC set A; #A(#A −1) is the sum of the PIC vector entries of that set (see chapter 2). It follows that #A(#A − 1) = 2a.

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or, alternatively, (cf. Isaacson 1990 and Rogers 1999)

Although Lewin’s measure was no doubt mechanically complex, the underlying idea was straightforward; much more so, in any case, than the idea basic to Rahn’s ATMEMB. Lewin’s measure also provided greater discrimination. Rahn added the numbers of subsets of one equivalence class in A and B together. As a consequence, he made no difference between a set-class represented by one against four instances in A and B and a set-class represented by two against three. The set-class in question counted for 5 in both cases. By taking the products of these numbers, Lewin arrived at two different values, i.e., 4 and 6. As it was more sensitive to differences such as these, REL could be seen as a more reliable measure of the delicacy of human perception than ATMEMB. Lewin’s 1980 definition of REL, the one he would later reject, did not allow PC sets to be maximally similar unless they were related by a canonical transformation (transposition and/or inversion). In that case, the constant ratio between the numbers of subsets X in A and B was 1:1. It would, however, have been interesting to find another constant ratio as well. But such a thing could only obtain for all X of a single cardinal number. This has possibly been the reason for Lewin’s proposal of a specialized form of REL, namely REL2. David Rogers, who provided a geometrical interpretation of the new measure (along lines similar to our interpretation of Teitelbaum’s s.i.; cf. example 5.7), specified six pairs of different-sized PC sets for which it returned the maximum value of 1 (Rogers 1999, 79). In both PC sets of each pair the same set-classes of cardinal number 2—in other words, the same interval classes—were represented in exactly the same proportion. This can be seen from their APIC vectors: {0,1,4,5,8} (Forte name “5-21”): 202420 {0,1,4,5,8,9} (Forte name “6-20”): 303630 In this example the constant ratio is 2:3. In every PC set pair singled out as “maximally similar” by Rogers, there is a constant ratio for the classes of subsets of each cardinal number. These ratios are all different, as table 5.11 shows. The similarity of the PC sets presented in this table can easily be verified. One member of each pair includes the other, which is not smaller in size by more than one element. The larger member is transpositionally and inversionally symmetrical: it “closes the circuit” that is left open by the smaller one (ex. 5.8). The PC sets {0,1,6}, {0,3,6}, and {0,2,4,6,8} relate to their counterparts—{0,1,6,7}, {0,3,6,9}, and {0,2,4,6,8,10}—as nearly complete cycles (modulo 12). All PC sets involve a relatively small number of interval classes. The larger member of a pair does not add new interval classes to those contained in the smaller.

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Table 5.11. The constant ratios between the subset classes of all cardinal numbers (#) in Roger’s six maximally similar pairs

Schuijer.indd Sec1:170

3–5

4–9

5–21

6–20

#1

3

4

#1

5

6

#2

1

2

#2

2

3

#3

1

4

#3

1

2

#4

1

3

#5

1

6

{0,1,6} (Forte name “3-5”)

{0,1,4,5,8} (Forte name “5-21”)

{0,1,6,7} (Forte name “4-9”)

{0,1,4,5,8,9} (Forte name “6-20”)

3–8

4–25

5–33

6–35

#1

3

4

#1

5

6

#2

1

2

#2

2

3

#3

1

4

#3

1

2

#4

1

3

#5

1

6

{0,2,6} (Forte name “3-8”)

{0,2,4,6,8} (Forte name “5-33”)

{0,2,6,8} (Forte name “4-25”)

{0,2,4,6,8,10} (Forte name “6-35”)

3–5

4–9

7–31

8–28

#1

3

4

#1

7

8

#2

1

2

#2

3

4

#3

1

4

#3

5

8

#4

1

2

#5

3

8

#6

1

4

#7

1

8

{0,3,6} (Forte name “3-10”)

{0.1.3,4,6,7,9} (Forte name “7-31”)

{0,3,6,9} (Forte name “4-28”)

{0.1.3,4,6,7,9,10} (Forte name “8-28”)

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5

1

3

1

1

1

1

6

3

3

5

1

4

3

4

4

2

2

2

6

2

2

4

2

2

{ 0, 2, 4, 6, 8 } , { 0, 2, 4, 6, 8, 10 }

{ 0, 2, 6 } , { 0, 2, 6, 8 }

2

2

2

2

3

3

2

4

2

{ 0, 3, 6 } , { 0, 3, 6, 9 }

3

3

3

6

3

2

2

1

1

2

2

{ 0, 1, 3, 4, 6, 7, 9 } , { 0, 1, 3, 4, 6, 7, 9, 10 }

1

1

Example 5.8. Maximal similarity in terms of REL2. Six pairs of PC sets singled out by David Rogers (1999).

{ 0, 1, 4, 5, 8 } , { 0, 1, 4, 5, 8, 9 }

3

1

{ 0, 1, 6 } , { 0, 1, 6, 7 }

5

1

1

1

2

2

1

3

2

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similarity

When measuring the similarity of two PC sets A and B by means of REL, Lewin did not count any set-class that was absent from either A or B.29 In that case, the product of the share this set-class had in A (that is, ai) and its share in B (that is, bi) is 0. Consequently, when the subset contents of A and B did not intersect at all—as in the case of a “diminished-seventh chord” ({0,3,6,9}; Forte name “4-28”) and a “whole-tone trichord” ({0,2,4}; Forte name “3-6”)—the value returned by REL is 0. A and B are thus “minimally similar.” David Lewin intended his REL to determine the degree of proportionality between the subset contents of two PC sets. (To apply the “ginger” analogy again: the taste of ginger in a dish does not change if one adapts the amount of ginger to the amounts of the other ingredients.) Measuring PC set similarity against size, he continued the line of investigation that had also yielded Morris’s ASIM and Rahn’s ATMEMB. All these measures embody the following basic ideas: Statement 1: A PC set is equal to the sum of its subsets.

But not all subsets were counted; not even all the proper ones. A count either included all proper subsets consisting of two or more elements (such a count was fundamental to Rahn’s ATMEMB or Lewin’s REL), or it included only those consisting of two elements (as with Morris’s ASIM or Lewin’s REL2; from the perspective of Statement 1, the latter are consistent with Forte’s R0, R1, and R2, and with Teitelbaum’s s.i.). Statement 2 (“No pair of PC sets is excluded from the similarity relation”) was already fundamental to the measures of Teitelbaum and Regener. The following two statements, however, concern the specifics of the similarity measures discussed in the present section: Statement 3: Two PC sets are more similar as they contain more subsets that correspond with respect to class.

Morris used the concept of an interval class (APIC). In this regard he followed Teitelbaum. Morris’s values, too, increase as the similarity decreases. Rahn and Lewin considered classes of subsets as being defined by a group of canonical transformations, but they did not specify this group in their formulae. Statement 4: The correspondence takes on a higher value when the total number of subsets in both sets is smaller. 29. Nor did Rahn. In the latter’s MEMBn and ATMEMB, however, the exclusion was not the automatic result of the calculation. Rahn needed to add a clause to this effect to his formula, which Lewin rejected as an “arithmetic awkwardness” (Lewin 1980, 501).

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173

Every single subset is supposed to contribute to the identity of a PC set, just as every instance of an interval class is supposed to contribute to the identity of a chord. But when a PC set is compared to other PC sets, the absence of particular classes of subsets is equally important. (For example, when a major triad is compared to a chord with some dissonant intervals, the absence of such intervals in the former acquires incidental significance.) “Being similar to A” not only means “having a considerable number of things in common with A,” but also “not having too many things that are absent from A,” or otherwise “not lacking too many of the things that are found in A.”

An Almost Unreasonable Wish In 1990, Eric Isaacson judged most similarity measures that had been developed so far “inadequate.” Isaacson, who had chosen PC set similarity as a topic for his dissertation at the University of Indiana, and who would reanimate the discussion about it in the 1990s, found that no existing measure succeeded in ruling out completely the influence of the cardinal number. Few things are more natural and predictable than a range of similarity degrees varying with the sizes of the PC sets in question. REL2 covers the whole range of values from 0.0000 to 1.0000 when a trichord is compared to other trichords, whereas it covers only a small range (from 0.5690 to 0.6395) when this trichord is compared to nonachords. This makes perfect sense. Even though there are as many set-classes of cardinal number 9 as set-classes of cardinal number 3, the former simply cannot differ as much (cf. note 26). However, Isaacson was concerned by this “inconsistency” (Isaacson 1990, 13). He presented a measure ensuring the widest possible range of values for any pair of cardinalities, thus fulfilling the almost unreasonable wish that the similarity debate had been pursuing from the beginning; as Allen Forte had written in 1964: “[to] define an appropriate, musically meaningful similarity relation for any two sets of different cardinality. . .” Isaacson’s IcVSIM (“Interval-class Vector Similarity”) was an application of a statistical tool: the “standard deviation function.” Given a series of numerical values representing the frequencies of items in a sample, the standard deviation function measures the spread of these values about their mean. In the case of IcVSIM these numerical values were taken from the APIC difference vector (the “Interval-difference Vector” in Isaacson’s words,”IdV” for short) of two PC sets. The difference vector had played a role in Teitelbaum’s concept of PC set similarity, too. We have seen that his s.i. assigned a value to the difference vector. This value increased with the differences between the corresponding terms of the two APIC vectors. Since these differences tended to be greater in the case of PC sets with different cardinal numbers, Teitelbaum’s s.i. failed as a measure

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of the similarity between such sets. It was unable to return the lower values that indicated greater similarity, except with pairs of equal-sized PC sets. As said before, this was not a disadvantage in itself. However, by relating the differences to their mean, one could undo the dramatic effect of a difference in size on the s.i. of two PC sets. Here is how Isaacson defined his IcVSIM, which he called “a scaled version of Teitelbaum’s s.i.”: IcVSIM(X,Y) = σ(IdV), where IdV = [(y1−x1) (y2 −x2 ). . . (y6 −x6 )] Here, X and Y are IcVs [“interval-class vectors,” that is, APIC vectors] and σ represents the standard deviation function. The standard deviation function, defined in terms of the Interval-difference Vector, is:

∑(IdVi − IdV) 2 6 where: IdVi = the ith term of the Interval-difference Vector, and IdV = the average (mean) of the terms in the IdV (Isaacson 1990, 18) σ=

What does IcVSIM actually measure, if it is not just the difference in representation of the interval classes? It measures how evenly (or unevenly) this difference is distributed over the various interval classes. A constant difference between the entries of two APIC vectors, however big it might be, yields the value 0, indicating maximal similarity. When the PC sets X and Y are of the same size, this constant difference can only be 0. This means that X and Y are maximally similar if they have equal APIC vectors—in other words, if they are related by transposition or by inversion-plus-transposition, or if they are Z-related. When they do not have the same number of elements, it is still possible that they are maximally similar if they meet the following condition: the surplus of unordered PC pairs in the largest PC set should amount to an integral multiple of six, so that it can be distributed evenly over six interval classes.30 Isaacson provided the following example, involving the PC sets {0,3,6} (Forte name “3-10”) and {0,1,3,6,7,9} (Forte name “6-30”): {0,3,6}: 002001 {0,1,3,6,7,9}: 224223

30. Isaacson stated this in more general terms: if the above-mentioned condition were met, X might share an IcVSIM value relative to Y with a set of another cardinal number (Isaacson 1990, 22). Thus, IcVSIM(X,Y) might equal IcVSIM(Y,Y) (maximal similarity).

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175

Since a PC set of cardinal number 6 contains twelve more unordered PC pairs than a PC set of cardinal number 3 (fifteen versus three), the only possible constant difference between the APIC vector entries of these sets is 2. Isaacson found eleven instances of maximal similarity between PC sets of different sizes (Isaacson 1990, 19–22 and 27–28, note 16). The cardinal numbers of these PC sets were 3 and 6 (three versus fifteen unordered PC pairs: a surplus of twelve in the largest set; constant difference: 2), 5 and 8 (ten versus twenty-eight PC pairs: a surplus of eighteen in the largest set; constant difference: 3), or 6 and 7 (fifteen versus twenty-one PC pairs: a surplus of six in the largest set; constant difference: 1). There are two more categories of PC set relations satisfying the abovementioned condition for maximal similarity in terms of IcVSIM. In PC sets of cardinal number 7 there are eighteen more unordered PC pairs than in PC sets of cardinal number 3 (twenty-one versus three). This surplus can be divided by six, which is also true of the surplus in PC sets of cardinal number 9 with respect to PC sets of cardinal number 4 (thirty-six versus six, a surplus of thirty PC pairs). However, these combinations of cardinal numbers do not produce a single instance of maximal similarity. One would expect members of complement-related set-classes to be maximally similar in terms of IcVSIM. The complement theorem asserts that the difference between the frequencies of an interval class in two complementary PC sets is always equal to the difference between their cardinal numbers (see chapter 4, p. 147). Unfortunately, this is not true of APICs. The complement theorem was conceived with reference to PICs (Lewin 1960). And the PIC vector of a PC set differs from the APIC vector in that the frequency of PIC 6 is twice the frequency of APIC 6 (see chapter 2, tables 2.2 and 2.3). When we apply IcVSIM to PIC vectors instead of APIC vectors, all complementary PC sets will turn out to be maximally similar; but then {0,3,6} and {0,1,3,6,7,9} no longer deserve the designation they received from Isaacson: {0,3,6}: 00200200200 {0,1,3,6,7,9}: 22422622422 There is no constant difference between the corresponding terms of these PIC vectors. When IcVSIM were modified to deal with PIC vectors (a modification that would entail the addition of five more terms to the “Interval-difference vector,” and the substitution of 11 for 6 as a divisor in Isaacson’s formula), this would not be a case of maximal similarity. Of all the similarity measures discussed in this chapter, only Teitelbaum’s s.i.

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and Isaacson’s IcVSIM are profoundly affected by the use of PIC vectors instead of APIC vectors. 31 We can start a summary of Isaacson’s ideas on PC set similarity with the first two statements related to Teitelbaum’s s.i: “PC pairs (intervals, subsets of cardinal number 2) are the key attributes of a PC set” (Statement 1); and “No pair of PC sets is excluded from the similarity relation” (Statement 2). The third statement, however, cannot be maintained. In Teitelbaum’s view, PC set similarity was dependent on the differences between the frequencies per interval class. For Isaacson’s IcVSIM, the following statement holds: Statement 3: The similarity (or dissimilarity) of two PC sets depends on the variation in difference between the frequencies of the successive interval classes.

The less marked the variation, the greater the similarity. The differences themselves are of lesser importance. This means that PC sets of different cardinal numbers can be as similar as PC sets of the same cardinal number. IcVSIM was not the last similarity measure to be developed. The debate on PC set similarity has been ongoing, and has yielded several new measures, of which Marcus Castrén’s recursive relation (RECREL; Castrén 1994) is the most prominent. Castrén not only compares two PC sets, but also all subsets of these PC sets in terms of their subset contents, using a vector that represents the percentual shares of all subset classes of a given cardinality in a PC set. The result is a conceptually complex, but highly nuanced, measure of PC set similarity. Another noteworthy contribution is Robert Morris’s pitch measure (PM), a similarity measure for pitch sets (instead of PC sets; Morris 1995, 226). It registers the number of pitches and the number of (absolute) pitch intervals shared by two such sets. Its value lies not so much in its subtlety or universal applicability as in its relative simplicity. It is easy to integrate in everyday analytical practice, and capable of yielding sensible results, especially when applied to open textures. The increasing number of similarity measures called for surveys to evaluate them, which were provided by authors such as Richard Hermann (1993), Marcus Castrén (1994), Michael Buchler (1997), and Ian Quinn (2001). It also elicited critical comments. Certain assumptions that had been taken for granted began to meet with serious doubt. Isaacson (1996), for example, questioned the commonly accepted idea that each interval class is equally different from every other one. While acknowledging the merits of the previous, mostly PC-based, similarity measures (including his own), he advocated taking into account more dimensions than pitch alone.

31. This is a consequence of the fact that difference values—one of which may change when the PIC vector replaces the APIC vector—are squared.

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As stated before, there seems to be no similarity measure whose utility has gained a general consensus. The second edition of Joseph Straus’s Introduction to Post-Tonal Theory (2000)—a book intended to represent PC set theory to an audience of undergraduate music majors (p. vii)—does not mention the topic at all. Yet, it is this topic that has been stimulating years of intensive labor, reflected in papers that have won acclaim for their subtlety and penetration. Apparently, it is so engaging intellectually as to survive the lack of lasting success. How are we to assess all the work that has been done in the field of PC set similarity? Similarity relations provide an open concept of musical relatedness, a concept yielding what John Rahn has called “one monstrous graph,” in which each PC set is connected to all the other ones. Such a concept enables the composer to trace—and the analyst to retrace—“similarity paths” (Rahn) much in the same way that Hindemith created his “tension paths” (his harmonisches Gefälle). Like the latter, similarity paths can be more or less curved, depending on the similarity values of the PC sets underlying successive sonorities. The similarity concept enables PC set theory to address music as a temporal phenomenon. This is much more difficult to accomplish with the common forms of PC set equivalence. Similarity relations supplement equivalence relations in another way as well. When we check a sample of sets for equivalent pairs, we show them to satisfy (or not) an extrinsic criterion of relatedness. The term “extrinsic” means that this criterion is not necessarily a property of the two sets. For example, when we say, with Allen Forte (1973), that equivalent PC sets should reduce to the same prime form, we do not require them to be in prime form. When we say, with John Rahn (1980a), that they should be members of the same equivalence class, we do not demand that they appear with all other members of that class. Actually, we do not even compare these PC sets themselves, but just the results they yield after going through a special kind of identification test. For the investigation of similarity relations we define a property that each PC set exhibits in a specific way, or in some measure. Our comparison now concerns these ways and measures, and is thus more direct. It is for this reason that similarity relations are supposed to accord with an intuitive understanding of music better than equivalence relations do. They bespeak a concern with musical experience, and with musical freedom, which many found lacking in PC set theory originally. However, the purpose of measuring these relations has been strangely ambiguous. For all their openness, similarity measures have also been created to enhance our conceptual control over music. They require intellectual skill; the freedom to follow whichever similarity path can only be enjoyed at the highest level of abstraction. What we see here is the interdependence of purpose and competence, the self-perpetuating power of a technical vocabulary. Although it may seem that

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similarity measures helped PC set theory deal with issues beyond its original purview, these issues actually came up in response to the challenges implicit in its own apparatus. The set and the row vector facilitated a comparative study of relations among tone combinations, while perhaps seriously limiting its scope in some ways.

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Chapter Six

Inclusion Much of what we call “pitch-class set theory” today originally served as the introduction to a theory. An inventory of classes of musical objects, and of relations between these classes, its function was to prepare the ground for a model of large-scale pitch organization in music of the twentieth century: the pitch-class set complex. This model was the main focus of Allen Forte’s seminal article “A Theory of Set-Complexes for Music” from 1964 and of his book The Structure of Atonal Music from 1973. Set-complex theory deals with the analysis of entire compositions, or movements and sections of compositions. More specifically, it deals with the question of how these can be unified in terms of PC sets. This requires a PC set relation that links several PC sets across boundaries of cardinality (unlike Tn/TnI equivalence), but also sets a clear limit to their number (unlike the various “absolute” similarity measures, like ASIM, ATMEMB, and REL). In other words, it should delineate a cross-cardinality “family” of PC sets. Such a relation is the inclusion relation. “Inclusion” means that one set is contained in another; the two sets are related as subset and superset. It is on this relation that the PC set complex is primarily based. Over the years, the set complex faded into the background somewhat. Ironically, it never had the appeal of the ideas that once built up to it; even a topic so notoriously abstruse as PC set similarity aroused more interest and discussion. Although set-complex theory failed to make a lasting impact on the study of post-tonal music, Forte remained dedicated to the aim of fitting PC set relations into a larger pattern. In the late 1980s, he embraced the PC set genus as a source of overarching coherence in music (Forte 1988a). This, too, is a “family”-like concept based on PC-set inclusion. However, it is both more restrictive and more universal in scope than the set complex. It is more restrictive, because one usually distinguishes a relatively small number of distinct PC-set genera;1 and it is more universal in scope, because genera “exist” independently from individual musical works. The set complex is defined contextually (like the subject of a fugue, the theme of a variation cycle, or a twelve-tone series), while the genus is defined communally (like “sonata form,” the Dorian mode, or C major). The present chapter relates such concepts to tonal theory. It concentrates mostly on the set complex, because of its role in the emergence of PC set theory. 1. Forte (1988a) lists twelve PC-set genera, Parks (1989) only five, compared to 114 set complexes.

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Considering that PC set theory was predicated on the structural autonomy of “atonal” music, it is an interesting idea that it originally paralleled older schemata of music explanation. The elaboration of this idea in subsequent pages will necessarily involve some speculation. Forte was reluctant to interpret twentieth-century works with reference to historical models. For Forte, the music of Schoenberg or Stravinsky was interesting not for what it retained from the past, but for the new knowledge he thought it embodied. And he believed that this knowledge was only accessible through systematic observation. Hence Forte’s unceremonious introduction of the set complex in The Structure of Atonal Music. Where it came from mattered less than what it could tell about pitch relations in the classical works of the twentieth century: [The set complex] provides a comprehensive model of relations among pc sets in general and establishes a framework for the description, interpretation, and explanation of any atonal composition. (Forte 1973, 93)

Forte’s cursory introduction of the set complex recalls the taken-for-grantedness of his ideas on PC-set similarity. However, the previous chapters have shown that such ideas did not always evolve uniquely in response to new musical developments; they may already have their locus in the world. This chapter proposes an ontology of PC set inclusion that shows the tonal descent of the set complex; and, as a corollary, it offers a view of the tonal system through the lens of setcomplex theory.

An Analytical Exploration Inclusion is the correlate of segmentation. Since music is organized in segments of various kinds and magnitudes,2 there are many musical relations to which the term “inclusion” is applicable. It may refer to relations between motifs and phrases, intervals and chords, chords and polychords, or linear progressions and themes. 2. This seems a point of consensus between creative, analytical, and cognitive perspectives on music; witness the following three quotes: “It will be useful to start by building musical blocks and connecting them intelligently. These musical blocks (phrases, motives, etc.) will provide the material for building larger units of various kinds, according to the requirements of structure” (Schoenberg 1967, 2); “While the phrase may be thought of as a segment of some independence, it is usually associated with one or more other phrases as an integral part of a larger structure” (Leon Stein 1979, 24); “If confronted with a series of elements or a sequence of events, a person spontaneously segments or “chunks” the elements or events into groups of some kind” (Lerdahl and Jackendoff 1983, 13). In some publications the term “segment” functions at a higher level of abstraction than it does here. Morris (1987) uses it as the designation of an ordered set, reserving the term “set” itself for unordered sets.

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181

A closer examination of such relations reveals that the superset/subset definition of that term is not always adequate or appropriate. Firstly, segmentations can be ambiguous; and, secondly, the structural identity of each individual segment may not reside in the sum of its elements and their associated intervals. Consider, for example, Gabriel Fauré’s Élégie for Cello and Piano, Op. 24 (1880). On the face of it, the main melody in C minor (mm. 2–5) is divided into four segments of one measure each by sequential repetition (ex. 6.1a). Harmony, however, decides otherwise. Note the crucial difference between the E♭4 in measure 2 and its melodic correspondent B♭3 in measure 3. The former is a relatively stable tone: the third of the tonic chord; the latter is the very first accented dissonance. As such, it is the peak of an initial tension curve. This means that B♭3 is still part of the first segment, which ends more convincingly with its resolution to A♭3 than with the metrically weak G3 in measure 2. The actual process of repetition, then, starts on the third beat of measure 3, and another segmentation of the melody suggests itself (ex. 6.1b). Although this segmentation is the most preferable from the viewpoint of musical phrasing, it does not discard the previous one. The melodic parallelism between the successive measures is too obvious to be missed; and such alternative, if subordinate, segmentations can gain in meaning as the piece progresses. Other considerations may then come into play. For example, it may at first not seem right to detach the eighth notes in measure 2 from the subsequent appoggiatura. Yet, it is this eighth-note figure that launches the middle section in A♭ major (mm. 23ff.) and serves as an accompaniment to its ornate theme. This is a reason to rate it as an important segment of the main melody (ex. 6.1c). Segmentations, then, apart from possibly being ambiguous in themselves, may rest on criteria of totally different orders. Now, if the eighth-note figure represents an important segment of the Élégie’s main melody, how should we characterize it? The first two forms shown in example 6.1c (mm. 2 and 23) have the same succession of intervals, and each of them occupies the same position relative to the local tonic (C and A♭, respectively)—an interesting combination of properties in view of the modal change from minor to major. However, the figure retains neither these intervals nor this harmonic position over time. Apart from its rhythm, its most permanent characteristic is the succession of an accented higher chord tone and an unaccented lower one, the former being embellished by its upper neighbor. It is hard to express such a characteristic set-theoretically. As will be recalled from chapter 2, a PC (or pitch) set offers no differentiation between chord tones and embellishments, or between accented and unaccented tones. Nor does it differentiate between chord tones that are part of a melodic progression (such as C4 in ex. 6.1a, m. 2) and chord tones that are not (such as G3 in the same measure). Seen thus, a PC set is a highly indeterminate concept. And the general tendency of PC set theory has been toward even greater indeterminacy—perhaps to expand its range of application, or to reflect the uncertainty of

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2

5 sempre

Example 6.1a. The main melody of Gabriel Fauré’s Élégie for Cello and Piano, Op. 24.

2

5 sempre

Example 6.1b. A segmentation of Fauré’s melody, based on considerations of harmony and phrasing. 2

23

sempre molto adagio

26

sempre

Example 6.1c. A small segment and its transformations.

musical perception. The shift of focus from equivalence to similarity is a case in point. Another typical example has been the development of a “theory of contour” within the purview of set theory by Michael Friedmann (1985) and Robert Morris (1987, 1993). A contour is an ordered set of pitch elements that are not specified beyond their relative heights. This concept can help us cope with the flexible identities of themes and motifs in tonal music. In example 6.1c, most forms belong to the same contour class3—that is, they are contour-equivalent— because their first tone is lower than the second, higher than the fourth, and 3. As defined by Friedmann (1985, 227). Morris adopted the term “contour class” in 1987, but replaced it with “cseg” (“contour segment”) in 1993.

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inclusion (a)

(b)

A (e)

(c)

E Mm7 (f)

E



183

(d)

A

B Mm7

(g)

A

E

Example 6.2. Members of a contour class (cf. ex. 6.1c).

equal to the third. However, other members of this class may not satisfy our earlier description of these forms (ex. 6.2, the members labeled f and g). Therefore, contour alone is not a distinguishing feature of Fauré’s eighth-note figure. An adequate description of it ought to take into account the functional relations between its elements. At this point, the reader may be reminded of Fred Lerdahl’s doubts about the musical relevance of the set concept (see chapter 1, pp. 23–25). His observation that “some pitches are heard as more or less structural than other pitches” certainly holds true for all forms of the eighth-note figure in Fauré’s Élégie. And it may also apply to tone combinations in the post-tonal repertoire that have been treated in the literature as reified PC sets. Is the opening melodic line of Schoenberg’s Piano Piece, Op. 11, no. 1 (ex. 6.3a, mm. 1–3) an occurrence of “Set [-Class] 6-Z10,” as Forte had it in his analysis of this piece (“The Magical Kaleidoscope,” 1981, 136)? Such a statement implies that each pitch has the same importance. It thus sanctions a comparison of this line with any other six-tone combination. For example, Forte compared it to the left-hand part of measures 4–5, which represents “6-Z39.” The PC sets in this class are the complements of those in “6-Z10,” which means that they have the same interval vectors (333321). And he regarded the two accompanying chords in measures 2–3 as another set of six PCs, as did Gary Wittlich (1974) before him. These chords jointly represent the Fortean set-class “6-16,” which is related to “6-Z10” by the common inclusion of, among others, “3-3”—the set-class of both the head motif (B4,G♯4,G4) and the chord {B♭2,A3,D♭4}(ex. 6.3b).4 4. The inclusion of members of the same set-classes in two PC sets can be taken as a measure of the similarity between these PC sets. Rahn’s MEMB-number, discussed in chapter 5, is such a measure. A PC set of class “6-Z10” includes four members of “3-3,” whereas a PC set of class “6-16” includes two. Hence, MEMB3(“3-3,” “6-Z10,” “6-6”) = 6 (cf. Rahn 1980b, 492) Of course, this number is only meaningful in comparison to similar numbers obtained from other pairs of PC sets, or as part of a summation of all set-classes represented in the two mentioned sets (Rahn’s TMEMB).

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Mäßige

5

langsamer

10

rit.

Example 6.3a. Arnold Schoenberg, Piano Piece, Op. 11, no.1, mm. 1–11. Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272. ''6-Z10''

2

3

''3-3'' ''6-16''

2

3

''3-3''

Example 6.3b. A PC-set analysis of the first three measures of Schoenberg’s Piano Piece, reflecting Forte’s emphasis on the two six-tone combinations in the right and left hands. 2

3

Example 6.3c. George Perle’s analysis of the same three measures. George Perle, Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern. Copyright © 1991, the University of California Press.

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The head motif is obviously part of a larger whole, and the melodic line that includes it is itself a recurring element. But how do we know that the second chord forms one structural unit with the first? Don’t we hear these chords as two syntactically related events rather than as a single event taking two measures (including a rest) to unfold? It is true, PC sets need not coincide with units of musical syntax such as motifs or chords. But segmentation of a musical surface into PC sets should nevertheless depend on a judgment of plausibility. In this case, the segmentation is plausible. Whereas in this piece “6-Z10” is primarily associated with restatements of the opening melody (mm. 34–36, 46– 47, and 53–55), “6-16” presents itself in different ways, as shown in example 6.4. The six segments (a–f) in this example are all based on Tn-related forms of it. We can view the succession of the two chords in measures 2–3 (a) as a partial ordering on the PC set {5,6,9,10,11,1}, in the sense that each element of the subset {5,6,11} precedes any element of {9,10,1}.5 The Tn-forms in the same column of example 6.4 (b–d) preserve this ordering. In other words, the order relations between the elements of {5,6,9,10,11,1} also hold between their correspondents under Tn. Consider the relation between example 6.4a and 6.4b: T3 ({5,6,11},{9,10,1}) = ({8,9,2},{0,1,4})

This partial ordering, together with the pointed beginning of the piece, emphasizes the role of “3-3” as a class of subsets held in common by the PC sets of “6Z10” and those of “6-16.” In the first column of example 6.4, “3-3” is represented by the last tones of each segment. In two segments (b and d) these are also the highest tones. The second column shows two occurrences of {1,2,5,6,7,9}—a set we already know (see ex. 2.8), and another member of “6-16.” Here, “3-3” is even more prominently set off as a subset. Each segment incorporates two statements of the head motif, transposed and with octave displacements: (F4,D4,C♯5) and (A4,F♯4,F5).6 The relations shown in example 6.4 assume the arbitrariness of the musical surface. They exist independently of melodic shape and harmonic function. Under the same assumption, George Perle (1962) interpreted the first measures of Opus 11, no. 1 as a dense sequence of operations on the head motif (ex. 6.3c). Clearly, Perle’s segmentation is committed to the consistency of the intervallic structure rather than to the integrity of the melodic line, which appears 5. The discussion of partially ordered sets was prompted by the “license of simultaneous statement of pitch-classes” in serial music (Babbitt 1962, 60). The concept of a partial ordering enabled the study of relations between twelve-tone series beyond the 48 canonical transformations. Lewin (1976) explored this territory. Morris (1987, 198–202) discussed the concept as part of a wider range of pitch-resources. 6. Recall that any PC set of class “6-16” includes two members of set-class “3-3” (see note 4 above). In the last two segments of example 6.4 (e and f), this property is exploited to the maximum.

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(a) mm. 2−3

T8 (b) m. 12

(e) m. 13

[] T3

(c) m. 29 T3

(d) m. 39

(f) mm. 49−50 T8

T11

T0

Example 6.4. The representation of Fortean set-class “6-16” in Schoenberg’s Opus 11, no. 1. Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272.

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as a composite of the higher tones of successive segments. But in view of the year of composition (1909), and in the absence of any detailed information on the actual conception of this piece,7 melodic and harmonic patterning cannot simply be set aside as a function of what Forte has called “external form” (Forte 1981, 130). Now that we have examined the relation of the opening melody to its accompaniment on PC-set basis, let us return to the question concerning this melody: is it the realization of a PC set, or does it just represent itself? As mentioned, “6-Z10’s” association with the opening melody of Schoenberg’s piano piece is quite exclusive. Other occurrences of this set-class form a somewhat heterogeneous group. Forte found eight of them, but not all of these are as conspicuous as the occurrences of “6-16” (ex. 6.5, cf. ex. 6.4).8 This suggests that mere content is not the essence of this melody. Rather, its arrangement is what matters. The essence is a linear mold that can also accommodate PC sets of other classes, like “6-21” (mm. 9–11) or “5-Z17” (mm. 50– 52). This mold imposes a hierarchy on the elements of these sets. It gives us a sense of tonal rhythm, which invites us to hear A4 in measure 2 as an incomplete neighbor of G4, and F4 in measure 3 as a retardation of E4.9 Such hierarchical distinctions challenge a set-theoretical interpretation of Schoenberg’s piece as much as Schoenberg’s choice of harmony challenges a tonal interpretation. It would be wrong to assign structural significance to the PC set of the opening melody, if for no other reason than that the accompaniment represents a set of such a significant class (i.e., “6-16”). We simply cannot offer a judgment of plausibility as to the claim that this melody represents a PC set. It definitely is a source of smaller sets, such as those of class “3-3,” but should these be considered subsets? Let us draw a few lines: •

A first condition for the musical pertinence of the set-theoretic inclusion relation is that musical segments can be plausibly defined on the basis of their abstract content.

7. There are no extant sketches for the Three Piano Pieces, Op. 11, and the only theoretical text by Schoenberg that might shed a contemporary light on these pieces is his Theory of Harmony of 1911. In this text, one searches in vain for references to procedures so obviously foreshadowing the serial techniques of later years. 8. When hearing example 6.5e (mm. 45–46), the listener may recognize the succession of a minor third, i.e., {E4,C♯4}, and a major third one step lower, i.e., {D4,B♭4}. This succession is also a characteristic of the opening melody, in which {A4,F4} follows {B4,G♯4}. In measures 45–46, however, A3 and C3 should be added to form a set of class “6-Z10.” A3 connects to B♭4 as E4 did to F4 in the original melody, but the role of C3—a mere grace note to the following C♯4—is in no way comparable to that of its correspondent G4 in measure 2. And if it is counted as a member of this segment, why exclude D♯3 in measure 44? Although in this excerpt the reference to the opening melody is obvious, it does not obviously involve the melody’s PC set. 9. Lerdahl, in his analyses of this Piano Piece (1989, 82–84 and 2001, 353–66), also sees the E as more important, though on the basis of a psychologically founded system of preference rules.

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inclusion

(a) mm. 1−3

(b) mm. 9−11

(e) mm. 45−46 T0

T7

(c) m. 34

(f) m. 51

T3 I T11 I

(d) m. 39

(g) m. 61

T9

T2

Example 6.5. The representation of Fortean set-class “6-Z10” in the same piece. Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272.

This condition is met by the accompaniment of the opening melody of Schoenberg’s Opus 11, no.1, but not by the melody itself, though it is met by the threetone head motif of this melody. The main melody of Fauré’s Élégie, however, also fails to meet the condition, and so does the eighth-note figure that becomes so important in the middle section of that piece. •

A second condition for the musical pertinence of the set-theoretic inclusion relation is a salient representation of the set-classes of both the subset and its superset.

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This is especially important in view of the role of this relation in the delineation of set-families (see further below). But what is “a salient representation”? “Salience” is the generic term that Fred Lerdahl (1989) has used to denote a number of properties by which a “pitch event”—that is, for Lerdahl, a tone or a chord—can stand out as important in a sequence of such events. These properties range from the more sensory (for example, duration, loudness, or registral position) to the more abstract (for example, structural prominence). In Lerdahl’s wake—but not quite in his spirit!—John Doerksen (1994, 1998) has developed salience criteria for set-classes, to determine which classes represented in a composition have “the highest degree of involvement at the musical surface” (Doerksen 1998, 196). Doerksen checks a composition for relevant segments, thereby exclusively following boundaries created at the musical surface (voices, simultaneities, rests). He then identifies these segments by their set-class names. The degree of involvement of a set-class depends on the musical functions of the segments that represent it (for example, the articulation of a formal division), and on the number and nature of its relations to other set-classes in the composition (equivalence, complementarity, inclusion). We need not fix the number of salience categories, nor restrict ourselves to set-classes found on the musical surface. However, for the set-theoretic inclusion relation to reflect a particular type of pitch organization, members of both setclasses should indeed be woven into the musical fabric. To put this more pointedly, for a relation to exist, its terms must exist. And the terms of an inclusion relation typically do not exist through this relation, as one might claim for the terms of an equivalence relation. This idea merits some elaboration:

With the Help of the Family It is true, as we have said, that the single things are nothing as determined by their relations which are the negation of their singleness, but they do not, therefore, cease to be single things . . . On the contrary, if they did not survive in their singleness, there could be no relation between them—nothing but a blank featureless identity. —Thomas Hill Green, Prolegomena to Ethics

In a world of objects, one object can gain identity from a property it has in common with other objects. It is then no longer an anonymous individual object— one out of many—but represents, or instantiates, this particular property. In turn, the property concerned takes on general value. It is no longer restricted to only one object. Consider the now familiar example of a tone that instantiates a PC, for example E♭: this tone can be accentuated by other tones with which it shares the E♭ property. But the property does not exist without multiple tones sharing it. It is actually an equivalence relation that calls “E♭” to life.

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In so speaking, we involve ourselves in the metaphysical debate on “universals”: things that can be said of multiple concrete objects; things like “red,” “two meters tall,” “parent,” “E♭.” Metaphysicians often distinguish between two kinds of such things, namely, properties and relations. However, the example of a PC shows that there can be much reciprocity between them. The term “property” is used here as including relational as well as non-relational attributes. What is it about universals that has triggered a debate? Philosophers struggled with the question “how there can be oneness in the multiplicity” (RodriguezPereyra 2000, 256), or “how numerically different particulars can nevertheless be identical in nature” (Armstrong 1978, 41). Universals have served as an answer to these questions, but their ontological status has always been a matter of controversy. Do they exist independently from the individual objects—the “particulars”—to which they apply? Do they exist in these objects? Or do we only use them as convenient labels for these particulars, thereby imposing distinctions of language on a world that is otherwise unfathomable?10 PC set theory seems founded on the conviction that it makes sense to speak of universals as real things, that what matters about an object is what can be abstracted from that object. We have seen that it defines pitch events in terms of certain properties, like PCs, interval classes, PC sets, interval vectors, etc. It examines the relations between these properties, and it tacitly assumes that these relations also hold between their bearers. This assumption has drawn criticism. In the words of James A. Davis: Once pitch-classes become the elements of an analysis, one is no longer dealing with the pitches found within a composition. Instead, the analyst is manipulating classes which are by no means the same entities as the pitches which constitute their membership. The problem is intensified when these pitch-classes are gathered to form pitch-class sets, and these sets are then subjected to various manipulations. At this point we have set-theoretical functions applied to sets whose members are classes. Unfortunately, the original pitches found within the composition are no longer a part of the ontological domain in which the theorist is working. (Davis 1995, 513–14)

Is Davis right to have concerns about the level of abstraction on which PC set theory operates? Not if his concern is that such a level is unprecedented in the history of Western music theory, or that it is a priori musically irrelevant. When PC set theory began, the notion of octave equivalence was already firmly established in musical thought, and the concept of a PC set had its referents in compositional practice. Were this not the case, the theory might not have achieved its institu10. Two text collections provide an overview of the debate: Properties, ed. by D. H. Mellor and Alex Oliver (Oxford 1997), begins with contributions of Gottlob Frege and Bertrand Russell; The Problem of Universals, ed. by Andrew B. Schoedinger (Atlantic Highlands 1992), also covers ancient Greek and Medieval philosophy (Plato, Aristotle, Abelard, Aquinas).

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tional status. What Davis deplores in PC set theory (its abstraction) is actually continuous with the Western classical music-theoretical tradition. His observation of an ontological ambiguity is well taken, but such ambiguity does not disqualify the theory as a member of this tradition. A PC set—or a PC set-class, for that matter—exists in the same way that “E♭” exists: in a give-and-take with multiple concrete objects. It exists insofar as it is instantiated by different tone combinations.11 In turn, it supplies these tone combinations with a common structural identity. The important point here is, however, that it contributes nothing to the reinforcement of an included smaller combination or an including larger one. An inclusion relation is not based on what its terms have in common. With respect to a set of sets, SET for short, the proposition “A ⊆ X” (“A is a subset of X”) rather gives us a property (“⊆ X”) that A might share with other sets ( A ⊆ X, B ⊆ X, . . ., X ⊆ X). Together, the sets with this property form a subcollection of SET that we shall call the family of subsets of X.12 In example 6.6a, its graphic image is a triangle with one upper vertex (associated with X, the family’s “progenitor”) and two lower ones.13 Set A may not only have one such property. It may also be a subset of sets other than X (A ⊆ Y, A ⊆ Z), and this holds true for any set with which it shares the property “⊆ X.” Each of these inclusion relations (. . ., “⊆ X,” “⊆ Y,” “⊆ Z”) generates a different, but not entirely distinct family (ex. 6.6b). The term “family” is appropriate, because families, in the ordinary sense, have members in common, members that are related by marriage: brothers-, sisters-, fathers-, and mothers-in-law. And the term is applicable to various musical domains, not only that of PC sets. Medieval hexachords, tonal keys, and twelve-tone series can be seen as families (of diatonic pitches, of pitches and harmonies, and of seriesforms) loosely or tightly connected by the bonds of common membership. The property “⊇ A” (“is a superset of A”) is the basis of yet another family. If A is a member of the family of subsets of X, the family of supersets of A includes, of course, X. In example 6.7a, the image of this family is a triangle with one lower vertex (associated with the progenitor A) and two upper ones. The family of subsets of X is still visible as a dotted triangle. Obviously, a family of n subsets involves a further n families of supersets (ex. 6.7b). 11. For an illustration, see example 2.9. 12. The reader may wonder whether “the power set of X” is not a better term. However, I intend the term “family” to have a broader meaning. In the following, it will be used for sets of sets that are related by inclusion. This not only involves the subsets, but also the supersets of a given set. 13. In his treatise on PC-set genera, Allen Forte (1988a) also uses the term “progenitor.” In his view, the only PC sets that qualify as progenitors of a family (more specifically, a genus) are those of cardinal number 3—first, because 3 is the smallest cardinal number yielding more than one representative of an interval class; and second, because trichords represent (uniquely or pairwise) all possible combinations of two APICs from 1 through 6. Each such combination forms the “DNA” of another genus.

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(a)

(b) Y

X

B

A

C

X

Z

A

Example 6.6. (a) A graphic representation of the family of subsets of X. (b) Set A is a subset of a number of families. (a)

(b) X

X

A

B

A

C

Example 6.7. (a) A graphic representation of the family of supersets of A. (b) Set X is a superset of a number of families. (a)

(b) X

X

A

A

Example 6.8. (a) Inclusion of the family of subsets of A. (b) Inclusion of the family of supersets of A.

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All these families are unique.14 However, they are not easily delimited. They cut across each other, and larger families contain smaller ones. Each family of subsets that includes A also includes the family of subsets of A (ex. 6.8a). And the family of supersets of A is included by the family of supersets of each subset of A (ex. 6.8b).15 As a consequence of their criss-crossing boundaries, families of subsets and supersets can be united to form larger entities; or they can be divided into smaller ones. This suggests another reason why the term “family” serves us well here. The size of a human family is equally indeterminate. The number of its members varies with the “width of the frame.” We may think of a family as a household of two parents and their children, or a group that additionally includes grandparents, uncles, aunts, and cousins. We also use the term “family” to refer to an entire genealogical tree. Furthermore, each member can be chosen as a point of perspective on the family. From this perspective, certain members are near while others are more remote. “Nearness” and “remoteness,” too, are concepts that have a history in music discourse, like the concept of “having a common family member”; but this history will concern us later. To recapitulate, we are investigating the question of whether and when PC set inclusion can rightfully be considered important. The answer may not lie in the inclusion relation itself, but in the nature, power, and/or frequency of its

14. Theoretically speaking, the only two exceptions are the families of the universal set and the null set. By their definitions (see chapter 2, p. 45), these sets have equal families. 15. For an accurate sense of the numerical constraints on the dimensions of such a family network, we can take our perspective from one set, and calculate the number of families of which it is a member. Let SET be the power set of ELEMENT. Further, let the cardinal number of ELEMENT be n. Then, each set A in SET of cardinal number m is a member of 2m families of supersets and 2n−m families of subsets (including such oddities as families with progenitors consisting of zero, single, or all elements). If we set n to 12—equating ELEMENT with PITCHCLASS—and m to 3, A is a member of eight families of supersets and 512 families of subsets. If subsequently we set m to 7, A is a member of 128 families of supersets and 32 families of subsets. These numbers are considerably smaller when we investigate the family affiliations of not one, but two or more distinct sets. Let us take from SET an arbitrary number of sets, A, B, C, . . ., regardless of cardinality. The number of families of supersets of which all of A, B, C, . . . are members is dependent on the size of their intersection. If i is the cardinal number of A ∩ (B ∩ (C ∩ . . .)), there are 2i families of supersets that contain A, B, C, . . . The same sets can be found together in a number of families of subsets. This number, however, depends on the size of their union. If u is the cardinal number of A ∪ (B ∪ (C ∪ . . .)), there are 2n−u families of subsets that contain A, B, C, . . . Thus, any two or more sets belong to fewer families as their intersection is smaller, or, equivalently, as their union is larger. In other words, the number of families involved decreases with the number of common elements between these sets. For example, if we keep n = 12 as the cardinal number of ELEMENT, two sets of three elements with zero intersection are contained in 64 families of subsets, 448 less than the number containing each of them.

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instantiation. “Including” a PC set, or “being included” in it, is a relation that can become a property (like “E♭,” or like each PC set itself). It should then be had in common with other PC sets, so as to transcend the multiplicity of relations—not least inclusion relations—in the twelve-tone universe. Hence the importance of families of subsets and supersets. But the abundance of such families poses a serious problem. How can PC set A instantiate the property “⊆ X” if it is also included in Y, Z, and/or other sets? What makes X prevail as a superset of A? And, conversely, how can X instantiate the property “⊇ A” if it also includes B, C, and/or other sets? What makes A prevail as a subset of X? In short, how, and to what extent, do these two PC sets exist? Let us return once more to the opening melody of Arnold Schoenberg’s Opus 11, no. 1 and its accompaniment. PC sets of highly involved set-classes are those of the melody’s head motif, the chord succession in measures 2–3, and the second chord of this succession (m. 3). We have not taken special note of the first chord, {G♭2,F3,B3} (m. 2), which is a representative of class “3-5.” This set-class is not as frequently and saliently represented in the piano piece as “3-3,” the set-class of the head motif and the second chord. It does not coincide with a foreground motif of the opening melody, nor does it notably occur independently from “6-16.” And since, as we have seen, the opening melody cannot be put on a par with its PC set, the only PC sets related as subset and superset are those of the second chord and the chord succession as a whole. This inclusion relation deserves to be noted, but it is not actually the focus of instantiation. The subset and its superset exist by the properties that they represent for themselves, that is, by their recurrent set-classes. In other words, the musical pertinence of PC set inclusion here actually rests on other relations (specifically Tn/TnI equivalence). Is it conceivable, though, that musically pertinent relations rest on PC set inclusion? The answer is yes, with the help of the family. Consider the family of subsets of X. Recall that it is unique, in the sense that it is the only family containing all these and no other sets. Although each member may also belong to other families, X can be outlined by a number of them collectively. We have already seen how in Stravinsky’s Agon, measures 414–15, the octatonic scale emerges from a chain of interlocking three-tone combinations (see chapter 1, ex. 1.3c). Each combination can be said to instantiate a portion of that scale by virtue of appearing with the others. Thus, if A and X are PC sets represented in a musical composition, and A ⊆ X, A is a stronger instance of the property “⊆ X ” as the number of PC sets with which it shares this property (B, C, . . . and X itself) is larger. This number should perhaps not be the only measure of the “subsetness” of A, but for the moment we shall assume that under its increase “⊆ X” can become “a way of being A.” Similarly, the family of supersets of A can come to the aid of X, and provide it with “a way of being X.” All chords in the “march” section of Claude Debussy’s Fêtes, the second of the orchestral Nocturnes, are members of the

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family of supersets of {8}, a one-element set. This is obvious, as the section is built on a pedal A♭/G♯. But there is more. Debussy’s march theme has been characterized as a “sui generis example” of a theme based on “non-tonally-functional chromatic progressions” (Pomeroy 2003, 161). When viewed from our “family” perspective, these progressions interact in an interesting way with the structure of the theme, and with its instrumentation. The march section begins in measure 116 (“Modéré mais toujours très rythmé”). It is based on a repeated pair of phrases, α and β, each eight measures long. Muted trumpets introduce this pair in measures 124–39 over an accompaniment of harps and pizzicato lower strings. (The trumpet parts are shown in ex. 6.9a.) Woodwinds take over in measure 140 with T3(α), and then play T6(β) in alternation with the horns. Finally, from measure 156, trumpets and trombones play both phrases full force at their original pitch amid a whirl of orchestral sound. Example 6.9b shows the harmonic basis of the march section of Fêtes. All chords are represented on the upper system. Below that we see a harmonic reduction that gives visual prominence to the common tones between the structural chords. The parsimonious voice leading16 conceals the relation of model and sequence between the progressions supporting phrase α and the whole of β and T3(α). This relation is rendered distinct on the lowest staff in example 6.9b. The next phrase, T6(β), could mark the beginning of yet another sequence. However, the harmony loops back into the restatement of α. As said before, the single unifying element of the entire chord succession is the bass pedal A♭/G♯. But some other pitches are held for a considerable time, too. Together with A♭/G♯, they define shorter spans of chordal accompaniment. We hear Fs throughout the first statement of phrase α. All chords of phrase β, and a number of those supporting the subsequent T3(α), include B3 and D4. The remainder of T3(α) sees the return of F (m. 144), which does not give way until the last statement of β (m. 164). From these observations we can infer two things. First, apart from the family of supersets of {8}, two smaller families can be distinguished: the family of supersets of {5,8} and the family of supersets of {2,8,11}. Second, the alternation of these two smaller families stands in a contrapuntal relationship with the phrase structure of the march. Example 6.9c shows where the families take over from each other: at the beginning of the first statement of β, halfway through T3(α), and at the beginning of the final statement of β. There is no such change at the beginning of T3(α), nor at that of T6(β) or the final statement of α; but these moments are marked by a major change of instrumentation, as if for want of harmonic momentum. 16. Studies by Cohn (1996, 1997), Kopp (2002), and others provide support for the idea that parsimonious voice leading—the transformation of one chord into another by a minimum of melodic motion—was a principle of harmonic progression through a large part of the nineteenth century.

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124

3

α

3

3

129

β

3

3

3 3

134

3

3

Example 6.9a. Claude Debussy, the march theme from Fêtes, mm. 124–39. α

124

128

Model:

β

132

136

Sequence:

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T 3 (α)

T 6 (β)

140

144

148

152

Sequence: (interrupted)

Example 6.9b. The harmonic progressions underlying Debussy’s march.

Trp

Wdw

α

F A {5,8}

β

D B A {2,8,11}

T3 (α)

F A {5,8}

Hrn/Wdw

T6 (β)

Trp/Trb

α

β

D B A {2,8,11}

Example 6.9c. Changes of “family” and instrumentation plotted against the phrase structure of the march.

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What do these families tell us about Debussy’s march? Perhaps this: certain chord changes are “stronger,”17 more disruptive, than others, even when they affect the same number of pitches or PCs, or when they are equivalent in some way. For example, measure 144 may give us a stronger sense of harmonic change than measure 128, in spite of the correspondence between these measures. The progression from measure 143 to measure 144 eliminates a PC set that has remained invariant under a number of chord changes: the set consisting of G♯, B, and D. The progression from measure 127 to 128, on the other hand, is the very first in the march section. Until that moment, we have heard only one chord, albeit for eight measures. Another reason to classify the chord change in measure 144 as “relatively strong” is that it introduces—or actually reintroduces—a PC set that is held through a considerable number of subsequent changes: the set consisting of A♭ and F. This assumes that the impact of a chord change increases with the persistence of not only the eliminated PCs but also the new ones. (The degree of that impact is thus established retrospectively.18) In this respect, too, the change in measure 128 is weaker: a PC set of relatively great persistence is introduced four measures later.19 By the same token, the chord change in measure 148 is of lesser impact than the corresponding one in measure 132. Thus, with the help of the family, we have discerned a misalignment between phrase rhythm and harmonic accent in the march section of Fêtes. This can be explained as a way to achieve gradual motion, as a remedy against sudden breaks and abrupt shifts in the musical process. But it is also in keeping with the formal function of this section. The pedal is traditionally a signal of anticipation, most typically in the middle section of a small ternary form, or to the end of a sonata 17. I use the term “strong” in roughly the same sense as Arnold Schoenberg in his Structural Functions of Harmony. According to Schoenberg, root progressions a fourth up (e.g., V–I, I–IV, etc.) are strong, “because great changes in the constitution of the chord are produced.” This is even more true of progressions like IV–V or V–VI. Hence, these progressions are “superstrong” (Schoenberg 1969, 6ff). 18. From the phenomenological perspective presented by David Lewin (1986), the event at the first beat of measure 144 is the object of various interrelated perceptions. Each of these perceptions involves a specific context of which the event is part, for example, measures 124–44 (perception: “sequential repetition”) or measures 132–44 (perception: “{G♯,B,D} eliminated”). And each perception provides itself a context for new events to come. In these contexts, the event in measure 144 may be perceived in yet another way. The chord change at the beginning of measure 148 denies the sequential repetition noted there, insofar as this were to involve all of measures 124–39, but it places greater value on the subset of PCs sustained from measure 144, i.e., {A♭,F}. 19. I mean {G♯,B,D}, but I realize that this is a questionable point. The chord change in measure 128 does bring G♯ (alias A♭) and D together. Why should we not consider this smaller PC set to be the progenitor of a family, since it is extended over more chord changes than its superset {G♯,B,D}? My intuition is to give priority to the superset, unless the number of chords that contain only the subset (i.e., {G♯,D}) is larger.

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development section. By slackening the pace of harmonic change, it delays the moment of arrival, which consequently receives greater emphasis. A misalignment of structural layers, such as described above, can reinforce this effect. Metaphorically speaking, it can divide the music into different “streams,” which create a strong sense of arrival when they finally converge. The hemiolas that typically precede important cadences in eighteenth-century triple-meter dances serve the same goal. All of the above proceeds from the intuition that, if one, two, or more PCs are shared by a number of chords in succession, each chord relates itself to them. These PCs are not so much the center of gravity as the focus of contemplation, more so as the number of chords increases. Such a relation can serve as an instance of PC set inclusion. It is important to note that our approach involves a reinterpretation of the concept of a subset. It now refers, not to a distinct entity A within X, but to a component of X. This is not a mere shift of emphasis. For example, we can hear a tritone as included in a dominant seventh chord. However, this is not a subset in the same sense as some of the three-tone configurations in Schoenberg’s Piano Piece, Op. 11, no. 1. It receives no reinforcement whatsoever from every, or any, other tritone. For convenience, we shall call this type of subset “settled.” Subsets as they occur in Opus 11, no. 1, are then “nomadic,” because they are defined in terms of properties that transcend the boundaries of any particular superset. Now the question is, with respect to subsets of the “settled” type: how can they meet the requirement of a salient representation when their role is to make up particular supersets, as in Stravinsky’s Agon? Since they can play this role only collectively, as a family of subsets, what enables individual members of this family to stand out from the rest? An important condition, as we shall see shortly, is their membership in different families of subsets, or, equivalently, their being progenitors of a family of supersets.

Plural Identities It is not peculiar to PC set theory to define musical entities on the basis of their constituent pitch elements and (or) abstract intervals; there are scalar or modal theories that do the same. A scale may not represent a relational system, but serve as a stock of tones (assuming octave equivalence) from which various combinations can be formed. Olivier Messiaen expressed this view when he wrote about his “modes of limited transposition”: [They] can be used melodically—and above all harmonically, melodies and harmonies never going beyond the notes of the mode . . . They exist within the spheres of influence of several keys at once, without polytonality—the

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composer being free to let one of the keys prevail, or to leave the tonal impression undetermined.20

Example 6.10 is an excerpt from Le Banquet Céleste (1928), which Messiaen has analyzed himself in Technique de mon langage musical (Messiaen 1944, 96). The first chord {C♯,E♯,A♯,B} is a subset of the second transposition of his octatonic Mode 2 (Pieter van den Toorn’s “Collection I”). At the same time, this chord is highly suggestive of the key of F♯ major, in which it would serve as a dominant thirteenth.21 Messiaen corroborates this suggestion by the appearance of an actual F♯-major chord (with a non-modal E♯) on the downbeat of the next measure, a chord marking the modulation from the second to the first transposition of Mode 2 (“Collection III”). Inclusion is a crucial concept here. Messiaen’s chords are subsets of his modal scales, but they can form tonal patterns with other chords from the same scale, or with chords from other scales. By their involvement in such patterns they achieve a certain prominence. Their tonal affiliations cause them to stand out from their precompositionally defined supersets. The octatonic scale is noted for its capacity to accommodate various elements of tonal harmony: triads (major, minor, and diminished), seventh chords (minor, dominant, and half-diminished), and French augmented sixth chords. There are also more complex formations, such as the dominant thirteenth. Most relations between these harmonic elements are not reminiscent of traditional tonality. For example, the octatonic scale does not allow a dominant seventh chord to resolve to its tonic. On the other hand, this chord can move to a dominant seventh at another pitch-level. It is a consequence of the overall symmetry of the octatonic scale that it contains a number of each available chord type. We saw that symmetry at work earlier, in Stravinsky’s Symphonies of Wind Instruments (chapter 4). The origin of the scale lies in sequential procedures of nineteenthcentury composers, particularly those procedures that employ one sequential interval (PI 3, or −3) and preserve the interval content of the sequence model. Taruskin’s account of the emergence of the octatonic scale stands as a classic of musical scholarship (Taruskin 1985, 1996, 255–306). Stravinsky’s “Petrushka chord” (ex. 6.11) is a subset of Collection III. It contains two major triads with no common tones, or, looked at another way, two dominant seventh chords with two common tones, their roots a tritone apart. Before Arthur Berger’s discovery of the octatonic scale as a regulative force in 20. “[Ils] peuvent être utilisés mélodiquement—et surtout harmoniquement, mélodies et harmonies ne sortant jamais des notes du mode . . . Ils sont dans l’atmosphère de plusieurs tonalités à la fois, sans polytonalité—le compositeur étant libre de donner la prédominance à l’une des tonalités, ou de laisser l’impression tonale flottante.” (Messiaen 1944, 85; my translation) 21. A chord owing its status of a “type” to the music of, among others, Schumann and Chopin.

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(R:+ bourdon 16) (chantant)

(clair)

(staccato bref, à la goutte d’eau)

Example 6.10. A tonal progression in Messiaen’s Mode 2, the octatonic. An example from Le Banquet Céleste (1928) used by Messiaen in his Technique de mon langage musical (1944). By kind permission of Alphonse Leduc, owner and publisher for the worldwide.

the music of Stravinsky (Berger 1963), many took a chord like this as “polytonal.” Today, this is a controversial designation, although the voicing in example 6.11a does indeed articulate the two major triads, and although Stravinsky himself confided that he “had conceived of the music [i.e., the music of the second tableau of Petrushka] in two keys” (Stravinsky and Craft 1962, 156). Berger was right to observe that an incomplete octatonic scale does not allow either key to present itself in more than a fragmentary way. As he wrote: [A] “polytonal” interpretation, insofar as it may have any validity at all, is even more problematic than the determination of a single priority. For the “gapped” scale affords far too little information for the delineation of “keys” of any kind. (Berger 1963, 26)

However, such a “polytonal” interpretation can be meaningful in another respect. The C-major and F♯-major triads in the Petrushka chord owe their salience, not only to a particular musical articulation, but also, and maybe more so, to their referentiality. “These triads—whose identity reflects past conventions and traditions—are at one and the same time octatonic subsets and residual emblematic reflections of common-practice tonality,” writes Kenneth Gloag (2003, 83). The latter role is reinforced by their relationship: C major and F♯ major are maximally apart in the circle of fifths. It seems to be a statement on the condition of tonal music to blend them together as Stravinsky does. Notwithstanding the inclusion of the Petrushka chord in an octatonic scale, its interpretation as a synthesis of two opposing poles is entirely rational, given the weight and the influence of the tonal tradition at the time of Petrushka’s composition. In turn, this interpretation has added hugely to its salience as an octatonic subset. Berger’s

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 a)

inclusion 3

3

3

b) 3

Example 6.11. (a) The “Petrushka chord.” Igor Stravinsky, Petrushka (1911), from the second tableau. (b) Octatonic derivation of the “Petrushka chord.” © Copyright 1912 by Hawkes & Son (London) LTD. Revised version © Copyright 1948 by Hawkes & Son (London) LTD. US Copyright renewed.

proposal to hear it as a “unitary sonic event” (Berger 1963, 22) would have been pointless if it had not previously been categorized as a polychord. What, we may ask, is the Petrushka chord? Or where is the Petrushka chord? Can it be separated from the commentary it has elicited? This question invites a general reflection on music theory and its relation to music practice, which will be given space in chapter 7. For the moment, it suffices to say that an inclusion relation involves musical objects with plural identities. A subset of the octatonic scale may also function as a tonal chord (Messiaen, Stravinsky); a segment of a melody may reappear in the accompanying figuration of another melody (Fauré). A plural identity comes about by an object’s occurrence in different environments, most typically when this object can be assimilated as an integral part of these environments—that is, when the environment comprises similar objects. Two classic cases of a plural identity are pivot chords and temporary tonics. Consider a family of chords that are diatonic to a given key. It is of no immediate concern here how many chords are in this family, or whether these chords are frequent or rare; nor do we, at this point, distinguish between structural and embellishing chords. What matters, for the present discussion, is that a chord can gain salience in one key from its invoking another. However, it cannot invoke a second key without the support of chords that are foreign to the first. In example 6.12, the opening phrases of Bach’s chorale Nun lob, mein Seel, den Herren, BWV 17/7, the first F♯-minor chord is a fully integrated member of the family of chords in A major. The beginning of the second phrase brings to light its affiliation to the F♯-minor family. This involves another chord, with which it shares the property of being a member of this family, but which does not belong to the family of A major (C♯ major, m. 5). It helps the F♯-minor chord stand out from A major, almost like a “family member by marriage.”22

22. When it comes to instantiation of the property “Member of the F♯-minor Family of Chords,” the help is mutual of course. However, the C♯-major chord does not have a plural identity in this example.

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           

F#m

               

      

        

  C#M



203

    

              F#m

Example 6.12. Johann Sebastian Bach, Nun lob, mein Seel, den Herren, BWV 17/7. The F♯-minor chord as a member of two families.

Bach’s chorale provides an example of what is generally known as “tonicization.” Here, we describe it as an instance of family intersection, not because this accounts better for tonal practice, but because an aspect of this practice can be viewed through a set-theoretic lens—more specifically, through the lens of PC set inclusion. Tonal chords are “subsets” of a key, that is, if we define a key in terms of its constituent PCs. This definition is one-sided to be sure, but it has been invoked in harmonic theories from early on, especially in discussions of key relations. According to Abbé Georg Johann Vogler (Tonwissenschaft und Tonsetzkunst, 1776), closely related keys were those whose key signatures differed by no more than one sharp or flat. Vogler thus looked at the size of their intersection. Anton Reicha (Cours de Composition Musicale, 1816) observed that the tonic of a key serves as a diatonic triad in its closely related keys. This statement reduces a chord to a set of pitches (or PCs) that is included in various supersets. It is a set, not a chord, that is invariant under a change of key. Gottfried Weber (Versuch einer geordneten Theorie der Tonsetzkunst, Vol. 1, 1817) actually used the word “family” to denote the sum total of diatonic triads and seventh chords uniquely pertaining to a key.23 In his view, this family existed prior to the scale, which he presented as the linear succession of the tones from which the key’s triads and seventh chords were built (Weber 1817, 223). Like Vogler and Reicha, Weber considered two keys more closely related as their scales had more tones in common. However, relations between keys a fifth apart were more important than others. Furthermore, Weber made an exception for relations between parallel keys (e.g., C major and C minor), in that he added them to the “first-degree” relations in spite of their relatively small intersection. On his famous chart of key relationships (example 6.13; Weber 1917, 301) one could measure the distance between any pair of keys. We will return to that chart in the following section.

23. “Und so sind jeder Tonart nur gewissen Harmonieen angehörig, und diese, die Familie der einer Tonart eigenthümlich angehörenden Harmonieen, nennen wir die eigenthümlichen Harmonieen der Tonart.” (Weber 1817, 218; my italics)

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Example 6.13. Gottfried Weber’s chart of key relationships (1817). Nederlands Muziekinstituut, Den Haag. NMI 2 O 12.

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A later theorist for whom proximity was, among other things, a matter of shared content was Arnold Schoenberg. In his Structural Functions of Harmony (posthumously published 1954), Schoenberg presented a theory of “monotonality.” Instead of a system of keys, this theory posited one tonality with different “regions.” These regions were associated with consonant harmonies at a greater or lesser distance from the tonic. One of the measures of this distance was the number of tones (PCs) a region had in common with the tonic region. For example, regions in a “direct and close” relationship had five or six tones in common, while an “indirect but close” relationship was indicated by three or four tones being in common. Schoenberg’s “classification of relationship” (Schoenberg 1969, 68) is not consistent, due to his using a handful of criteria for the relative proximity between regions (apart from a minimal intersection of three tones, he included a fifth relation, a common dominant, the close proximity to a closely proximate region, or a structural analogy); and this handful does not even include the reason why the “Dorian” region (D minor with respect to C major) is ranked as more distant from the tonic than the “minor mediant major” (E♭ major).24 However, common pitch content was important enough for Schoenberg to mention as one basis of tonal relationships. We have seen how family intersection can create the conditions for inclusion relations to become musically manifest. To summarize: a member A of Family X may appear to be also a member of Family Y when it is joined by a non-member of X with which it shares the property “member of Y.” The inclusion relation between A and Y may lead to a heightened salience of A in Family X. Such A are the chord {C♯,E♯,A♯,B} in Messiaen’s Le Banquet Céleste (see ex. 6.10) and the second F♯-minor chord in Bach’s setting of Nun lob, mein Seel, den Herren (see ex. 6.12). Now, the following is important to note. If we conceive of these chords as plural identities, we abstract certain contextual properties away from them. In that moment of abstract reflection, they become PC sets: blank, featureless identities, uninterpreted entities, members of no family. In reality, however, their surroundings “tug and pull” at them, in Lerdahl’s words. It is this tug and pull that can bring them into prominence. In serial twelve-tone music, unordered PC sets emerge in a similar way: as a neutral body of objects that are made prominent as they are mapped onto different series. In chapter 4, such relations were explained under the principle of combinatoriality, a principle that ensured the conditions for a twelve-tone series to retain particular PC sets under particular transformations. 24. Carl Schachter (1987, 316, note 22) already made this observation. Schoenberg’s reason may have been that the fundamental-bass progression from C to D was “superstrong” and hence disruptive (see note 17). The defectiveness of such progressions—or, equivalently, the need to normalize them by an intermediate fundamental—was a tenet of Viennese fundamental-bass theory, of which Schoenberg the harmony teacher was a descendant. For an authoritative account of this tradition, see Wason 1985.

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The way in which Allen Forte introduces the inclusion relation in The Structure of Atonal Music reminds one of the studies of combinatoriality by Milton Babbitt, David Lewin, and others. When confronting the question of how to assess the relative importance of all the subsets of a PC set, he points to “the notion of invariance.” Forte defines this as the intersection of transformationally related PC sets: Given equivalent pc sets A and B, a third set C is determined by the elements that are both in A and B. This set is called the intersection of A and B. . . . The elements of such a set will be called invariant pitch-classes. (Forte 1973, 30)

Recall that, in Forte’s view, equivalent PC sets were related as transpositions or as transposed inversions. “Under these transformations some pc sets may hold certain pitch-classes fixed, while other pcs will change,” he writes elsewhere (Forte 1978, 9). Invariance provided him with a rationale for harmonic progression, just as it provided Babbitt with a rationale for the succession of twelve-tone series: The concept of invariance is intimately bound up with the intuitive musical notions of development, change, continuity, and discontinuity. (Forte 1973, 31)

Interestingly, these “intuitive musical notions” do not require PC sets with invariants to be related transpositionally and/or inversionally. This arbitrary restriction betrays the influence of twelve-tone serialism. It enables the “prediction” of the number of PCs held over from one set to the other.25 As a corollary, it helped Forte determine the conditions for “minimum,” “maximum,” and even “complete invariance.” Thus, in the first chapters of The Structure of Atonal Music his focus is on settled subsets. He considered these of structural importance when he could find a larger representation of their supersets in the music; the conjunction of these supersets was their primary means of instantiation.26 One could say that Forte conceived of the property “⊇ A” as defining a family of PC sets, and of A itself as the progenitor of this family. However, the notion of

25. See chapter 3, note 21. For any element a in a PC set A to be also included in Tn(A), A should include b = (a − n) mod 12. Generally speaking, then, the intersection between A and Tn(A) is as large as the frequency of the PIC n in A—which can be read from the PIC vector. For any a in a PC set A to be also included in TnI(A), it must sum to n with itself, or with another element of A. Therefore, the intersection between A and TnI(A) is as large as the number of distinct PC values in either set that sum to n with another PC value in the same set. These are but two of a number of “common-tone theorems.” For a listing of these, see, for example, Rahn 1980a, 97–123. 26. Forte had not addressed the problem of a subset’s instantiation in “A Theory of Set-Complexes for Music.” This paper covered inclusion relations only insofar as they were relevant for an understanding of the PC set complex.

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207

“family” is latent in this introduction to PC set inclusion. It enters the discussion later, in the second part of Forte’s book, where it effects a major change in the notion of “subset.”

Study in K(h) As noted before, about one half of The Structure of Atonal Music is devoted to an elaboration of the theory of set complexes, Forte’s grand design of nine years earlier. We shall now discuss this theory with reference to both the book and the 1964 article. The changes that Forte made to it did not affect its essential features, which we will be discussing here. A set complex consists of a number of families of subsets and supersets. Consider two PC sets that complement the twelve-tone chromatic. We shall label them X and X '. In example 6.14, their elements are unspecified. Black noteheads indicate PCs in X; white note-heads indicate PCs in X '. Basically, to the set complex about X and X ' belongs any PC set A that satisfies one of four statements: 1.1 A ⊇ X 1.2 A ⊆ X 2.1 A ⊇ X ' 2.2 A ⊆ X '

In any of these cases, A is in a set-complex relation to X and X '. From this it follows that a set complex is the union of four families: the family of subsets and the family of supersets of a given set; and the family of subsets and the family of supersets of the given set’s complement. It should be noted that the above four statements reflect Forte’s 1964 presentation of the PC set complex. In 1973, he assumed subsets and supersets to be “proper” (see chapter 2, p. 45), excluding an A that is equal to X or X '. He even stipulated the exclusion of all A of the cardinal numbers of X and X '.27 Example 6.14 is in agreement with both definitions of the set complex. (Hence the substitution of “⊂” and “⊃” for “⊆” and “⊇.”) In set-complex theory, labels like X, X ', and A not only denote single PC sets, but also all transpositions of these sets, and all transpositions of their inversions as well. In other words, the set-complex relation is a relation between classes of PC 27. See Forte 1973: 93–95. This does not follow inevitably from the four relations that define the membership of a PC set complex. When A is a proper subset or a proper superset of X, it may have the same cardinal number as X '. (It may even be equal to X '.) This is logically impossible only when the cardinal number of X is 6.

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inclusion Set X'

Set X

1.1

A⊃X

1.2

A⊂X

2.1

A ⊃ X'

2.2

A ⊂ X'

(1.2 + 2.2) T/ Tn I

Example 6.14. Set-complex relations. In all cases A is member of the set complex about X and X '. Below: a subcomplex relation.

sets. As a set-class, A may satisfy two of our four statements. For example, one PC set of class A may be included in a PC set of class X, while another includes, or is included in, a PC set of class X '. A then belongs to a more exclusive complex, the so-called subcomplex about X and X '. Every member of this subcomplex satisfies a statement of number 1 (1.1 or 1.2) and a statement of number 2 (2.1 or 2.2). The set-complex and subcomplex relations are symmetrical, that is, they extend from A to X/X ' and from X to A/A'. This can be shown by replacing each of the above four statements by an equivalent statement: 1.1 X ⊆ A 1.2 X ⊇ A 2.1 X ⊇ A ' 2.2 X ⊆ A '

The statements of number 1 have simply been reversed. Those of number 2, however, have traveled a longer way, under the license of what Forte called the “natural association of inclusion and complementation” (Forte 1973, 94). This

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is what he meant: the complement of any subset of any set A is a superset of the complement of A. Thus, from the reversal of the original statement 2.1 (“X ' ⊆ A”) we can deduce that X ⊇ A'. Similarly, from the reversal of the original statement 2.2 (“X ' ⊇ A”) it follows that X ⊆ A'. It was this property of the inclusion relation that led Forte to the development of the PC set complex: By property 3 it is possible to construct about each of the distinct sets [that is, A, A', X, or X'] and its complement a symmetrical array of sets to which they are in the inclusion relation. Such an array will be called a set-complex, and the complementary pair about which the complex is arranged will be called the reference-pair. (Forte 1964, 162)

Example 6.15 shows the (“total”) set complex about the reference-pair consisting of “5-33 and “7-33” (above and middle).28 It brings out very clearly the symmetry between the family of supersets of “5-33” and the family of subsets of “7-33” by virtue of the assignment of the same ordinal numbers to complementary set-classes. The smaller families of subsets of “5-3” and supersets of “7-33” are shown to be equally symmetrical; Forte’s “property 3” ensures that for each subset of “5-33” (e.g., “4-24”) there is a superset of “7-33” (in this case, “8-24”) Furthermore, all set-classes that bear an inclusion relation to both members of the reference-pair—and that thus appear in two of the four families instead of in one—are entitled to the membership of the (“most significant”) subcomplex about “5-33/7-33” (ex. 6.15, bottom). Example 6.15 dates from 1964. What would have happened to it if Forte had included it in The Structure of Atonal Music nine years later? He would have left out all set-classes of cardinal numbers 2 and 10, which by then were no longer represented on his list of prime forms. Furthermore, all set-classes of cardinal numbers 5 and 7, including the duplicates of “5-33” and “7-33,” would have disappeared from the chart. However, the PC set complex did not only develop with respect to the number of restrictions on its membership. A striking feature of its 1973 presentation was the emphasis on analytical procedure. This emphasis was key to resolving an issue of vital importance: in what practical way could the PC set complex serve as the “framework for the description, interpretation, and explanation of any atonal work,” as Forte hoped? Once an analysis had yielded a number of important set-classes, the suggested procedure could not be more straightforward. The first step was to check each pair of these set-classes for a set-complex relation (labeled “K”) or subcomplex relation (labeled “Kh”). Each member of such a pair should then belong to the

28. The set-class with Forte name “5-33” is that of the whole-tone pentachord. The prime forms of both referential set-classes are contained in examples 4.2 and 4.8. Note that Forte uses “K” and “Ks” as labels for “set complex” and “subcomplex.” (He would replace “Ks” by “Kh” later.) “N” and “M” are variables, like our A and X.

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Example 6.15. The set complex (above) and subcomplex (below) about “5-33/7-33.” An illustration from Allen Forte’s article “A Theory of Set-Complexes for Music,”Journal of Music Theory 8/2 (1964): 169. Used by permission of Duke University Press and Yale University.

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211

(a) Mm:

PC sets:

1:

C

Set-classes (Forte):

C♯

2–4:

D

4–5:

D♯

E

E♭

E

C

4–3 F F

G♭

B♭

B

5–4

B

5–7

2–5:

8–5

6–7:

F#

8:

E♭

9–10:

C

C♯

E

D

G

A♭

A

5–1

F

3–1 G♭

B

11:

G♯

12:

B♭

G

5–5

A

A♭

11–12:

3–1

13:

C♯

D

E♭

E

B

5–2

(b) 3–1 4–3

K

8–5

Kh

4–3

8–5

5–1

Kh

Kh

K

5–2

Kh

K

K

5–4

Kh

K

Kh

5–5

Kh

K

Kh

5–7

Kh

Kh

Example 6.16. (a) PC sets of the main segments of Anton Webern’s Bagatelle for String Quartet, Op. 9, no. 5. (b) A set-complex analysis of the Bagatelle

set complex or subcomplex about the other. The second step was to group the results in a matrix. A very small example is shown here as example 6.16b. It depicts the relations between the set-classes represented by the principal segments of Webern’s Bagatelle, Op. 9, no. 5 for String Quartet. (Table 6.16a. shows the PC sets of these segments.) Such a matrix was a basis for assertions about different types and degrees of coherence. One would note the relative density of set-complex relations in Webern’s Bagatelle, and especially the involvement

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of “3-1” in subcomplex relations with all but one of the other set-classes. This involvement reflects the tendency toward a clustering of minor seconds in this piece.29 In another work, the set-classes might all be members of the set complex about one of them, but not have many mutual relations of this kind. Or a work might not appear to be particularly coherent in this sense at all. Let us dwell for a moment on this procedure. When carrying it out, what is our purpose? Suppose we have selected a number of tone combinations in a work for further investigation, on the basis of the recurrence of their set-classes, or on account of their importance in other respects. So far, the only thing we have is a set of guided observations, like “A and B have the same PC set,” or “A and B have the same similarity index as C and D.” There is a fair chance that these will not connect to reveal a larger pattern of relationships. Yet it is the possibility of such a pattern that motivates our procedure. We want to ascertain whether larger structural entities can be built from smaller ones, so as to widen our analytical scope. The interaction of these larger bodies of musical material may help us explain whole works or (most often) sections thereof.30 One aspect of set-complex theory is especially intriguing: a small entity can represent a larger one. More specifically, a PC set can stand for a set complex or subcomplex, as the “head of a family” (to quote Gottfried Weber). This means that it is a potential source of coherence over a larger span of music. For a PC set to realize this potential, it should be in a set-complex relation with all (or most) other set-classes represented in that span. It is then called a nexus set. On its first appearance, in The Structure of Atonal Music, this term seems to serve as a mere substitute for the older “reference-pair,” but later on Forte used it to refer to a PC set that exerts a dominance over concurrent sets on account of the number of its set-complex relations (Forte 1973, 101, cf. 113–14). The above will sound familiar to the reader who is well versed in the history of music theory pedagogy and in traditional methods of music analysis. PC set (complex) theory aims to proceed smoothly from elements to complex forms. 29. Set-class “3-1” is the class of chromatic trichords. Its prime form is (0,1,2). The set-complex analysis does not record the important tendency towards equal distribution of PCs in Webern’s Bagatelle. This is what the arrangement of PC sets in example 6.16a reveals. The same example also shows the instantiation of {E♭/D♯,E} as a subset of four PC sets. 30. Examples of the application of set-complex theory to larger works include Forte’s own analysis of Stravinsky’s The Rite of Spring (Forte 1978) and Janet Schmalfeldt’s study of Alban Berg’s Wozzeck (Schmalfeldt 1983). Forte traces the harmonic materials in The Rite of Spring to the set complexes about three “prominent” set-classes of cardinal number 7 and four set-classes of cardinal number 8. Schmalfeldt, in a more hermeneutic approach, assigns the “significant” components of the harmonic language of Berg’s opera to three set complexes, which in her view support the main psychological threads in the drama. The quotation marks in the foregoing sentences are not intended to cast doubt on the judgment of these authors; they should remind the reader of the critical role of the decisions preceding the set-complex analysis.

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This smoothness is important in theories of harmony and musical form, which provide ways to assure a clear connection between the small detail and the larger whole of a composition. Harmonic functions can be attributed to chords, sequences of chords, phrases, and sections. Similarly, certain formal patterns underlie themes as well as movements of compositions. Can the nexus set be compared with the tonic triad, as another instance of a concept that provides a link between different structural levels? And, consequently, does the PC set complex owe something to the concept of a key? In the previous section, we referred to Gottfried Weber’s definition of a key as a “family of chords.” It would probably not be right to regard this expression as an anticipation of twentieth-century musical thought. More likely, PC-set (complex) theory has expanded on elements of earlier music-theoretical vocabularies. Therefore, it is less instructive to look for vestigial remnants of the tonal system in the PC set complex than to study the tonal system—the system of keys and key relationships—from the perspective of set-complex theory. However contrived such an exercise may seem, it enables the concepts of key and complex to shed light on each other without falsely suggesting an organic development between the two. Let us take another look, then, at Weber’s chart of key relationships (ex. 6.13). Each key is represented by a letter name. We can also take this letter as referring to that key’s tonic chord. Thus, we read the chart as a topographical survey of all major and minor chords, each of which serves as the tonic of a key, and is surrounded—or at least one-half surrounded—by the consonant triads belonging to that key. This reading matches with the eighteenth-century intuition that keys are closely related when the tonic chord of one exists in the other.31 If we were to define an “in-key relation” between two consonant triads—requiring one triad to be a diatonic member of the key of the other—this relation would, of course, be symmetrical, like the set-complex relation. A complication, however, is the presence of the major dominant triad in a minor key family, which is more natural (or frequent) than the presence of the minor key’s tonic in the family of its major dominant. To maintain the symmetry, we would have to include the minor subdominant as a member of a major key family (ex. 6.17).32 Another complication is the almost complete intersection of the families of a major key and its relative minor. This may lead to a wrong assignment of tonic function (that is, if we equate “tonic” and “nexus set”). For example, if an analysis of Bach’s chorale setting in example 6.12 would follow the same lines as that 31. Earlier on, I quoted Reicha on this subject. For more, see Lester 1992, 214–17. 32. A rule of correspondence between these two harmonies is provided by harmonic dualism. According to Oettingen (1913) one changed the mode of the Unterdominante (i.e., subdominant) in major and its mirror image, the Oberregnante (i.e., dominant) in minor so as to reinforce the relation with their tonic and phonic Klänge (“klangs”). These transformations added a leading tone for the fundamental of the major triad (C) and its structural equivalent, the “phonic overtone” of the minor triad (G), respectively. See further below.

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(a) f

F

d

D

b

B

g♯

b♭

B♭

g

G

e

E

c♯

e♭

E♭

c

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a

A

f♯

g♯

A♭

f

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d

D

b

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D♭

b♭

B♭

g

G

e

f

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d

D

b

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g♯

b♭

B♭

g

G

e

E

c♯

e♭

E♭

c

C

a

A

f♯

g♯

A♭

f

F

d

D

b

c♯

D♭

b♭

B♭

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G

e

(b)

Example 6.17. (a) The family of chords in C major, projected on Weber’s chart of key relationships (b) The family of chords in C minor

of Webern’s Bagatelle Op. 9, no. 5 in example 6.16b—using an “in-key relation” instead of a “set-complex” relation—the F♯-minor chord would turn out to be the “nexus set” instead of the tonic A major. The only chord in this excerpt that does not belong to the A-major family (C♯ major) is still a member of the F♯-minor family. Therefore, F♯ minor has the most in-key relations among the chords in example 6.12. To prepare the next step in this comparative study, we should note once more that relations between keys exist in various degrees. Some keys are closely related, others more remotely so. Furthermore, in the context of a musical composition, one key may be secondary to another. For this to be seen in the light of set-complex theory, it seems proper to use the notion of transitivity. Consider the family of subsets of X. As will be recalled, if such a family includes A, it also includes the family of subsets of A (see ex. 6.8a). In turn, each subset of A has a similar family, which is included by the families of subsets of A and X, respectively. Now, in a sense, we “move away” from X as we traverse subsets of successively lower cardinal numbers, and as the number of families that exercise a claim on these subsets increases. This may give us a sense of “distance.” It takes some mental effort to speak about key relations in terms of families of subsets and supersets—let alone in terms of subsets or supersets that involve their own families—but Arnold Schoenberg has given us more than a hint of how to achieve this. Schoenberg’s theory of monotonality presents us with a

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great expansion of Weber’s “family of chords.” Each member of this family now supports its own family, under the central governance of the tonic. For example, F major is a member of the family of chords of C major; and B♭ major is a member of the family of chords of F major. Seen thus, B♭ major is also a member of the C major family, albeit a more remote one than F major. However, distance is not the only measure of relationship within a family. The inclusion relation is transitive, but a family relation—like a set-complex relation, or our in-key relation—is not. Given that A ⊆ X, there are members of the family of subsets of X that do not belong to the family of subsets of A. This means that a family of subsets (or supersets, for that matter) has subdivisions, each of which intersects with other families. Apart from a sense of distance, then, family relations provide a sense of direction, depending on the clarity of these subdivisions. Although Schoenberg’s theory employs geographical imagery (“regions”), its tendency toward a unification of tonal space may somewhat undermine our orientation. Obviously, most chords from the region of the “Mediant Major” (in C major: the family of chords of E major) do not belong to the region of the Submediant Minor (in C major: the family of chords of A minor); but since these regions are both integral to the territory of one tonic, one may seriously ask whether there still exists a clear line of demarcation between these regions. In Schoenberg’s tonal space an in-key relation may yet be transitive. Finally, the involvement of complement-related families in one set complex seems to have no parallel in tonal theory. Yet the picture changes when we express the relation between such families in a more general way—as two non-equal families with progenitors X and X ', which are arranged symmetrically about an axis (see ex. 6.15). For each member A of the family of X there is a unique member A' of the family of X ', such that the relation from A to X extends in reverse from A' to X '. The members of both families are thus related by inversion (as defined in chapter 3). This holds true, not only for the subsets of X and the supersets of X ' in a set complex, but also for the members of the families of major and minor keys in the context of nineteenth-century harmonic dualism. In that context, the minor triad was the mirror image of the major triad—a chord composed of the same intervals measured down from the fifth. Proceeding from this inverse relation, theorists like Arthur Joachim von Oettingen (Harmoniesystem in dualer Entwicklung, 1866) and Hugo Riemann (e.g., Skizze einer neuen Methode der Harmonielehre, 1880) constructed the minor tonality as a system of harmonic progressions in contrary motion to those of the major tonality. This placed the subdominant of the minor family in exact opposition to the dominant of the major family, and likewise the subdominant of major in opposition to the dominant of minor. In example 6.18, the relations between the major and minor families are depicted so as to convey the parallels with the set complex in example 6.15. By showing two pairs of families, this example indicates that the choice of the “progenitors”—i.e., the actual tonics—was arbitrary. Whereas Oettingen contrasted the

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(a) Oettingen: E♭

A♭

D♭

c

f

b♭

F

C

G

d

a

e

G

C

F

e

a

d

F

C

G

d

a

e

(b) Riemann:

Examples 6.18. Harmonic dualism: the symmetrical alignment of major and minor keys (cf. ex. 6.15).

major and minor tonalities of the same Hauptton (C, as the fundamental of C major—or “c+”—and the “phonic overtone” of F minor—or “co”), Riemann chose to stay within the collection of natural tones, setting off C major (“c+”) against A minor (“oe”). The structure of the set complex is certainly reminiscent of harmonic dualism; however, the meaning of the resemblance is unclear. Although harmonic dualism allowed modal mixture to occur (see note 32), it did not, of course, envisage a merger of the major and minor tonalities into one family—like the merger of four families of subsets and supersets into one set complex. In fact, all parallels observed between set-complex theory and the key system—as interpreted by theorists of different generations—exist exclusively on a conceptual plane. They tell us little about the musical nature of the set-complex itself. In one essential respect, the set complex is even at odds with notions of tonality: its constituents are set-classes, not pitch-class sets; set-complex theory deals with nomadic subsets and supersets, not settled ones. One may conceive of the set-complex as a family of pitch materials; but these pitch materials owe their structural identities to relations other than their inclusion in that family. However, there are two reasons to take note of the abstract similarities between set-complex theory and the tonal system. One is the later development of PC set genera as tightly knit families with more distinct harmonic profiles than the set complex. This bespeaks the ambition to imbue PC set theory with a sense of tonality-like hierarchy. Forte’s PC-set genus of 1988 consists only of a family of supersets. And from this family he omitted some “black sheep”—most significantly the members whose complements do not include the three-tone progenitor (Forte 1988a, 192; see also note 13). In the following year, Richard

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Parks presented a genus that combines two families: the subsets and supersets of what he called a “cynosural set.” This role was assigned to the diatonic (Forte name “7-35”), whole-tone (“6-35”), and octatonic (“8-28”) scales among others. Parks selected these scales not only for their historical significance, or for the role they play in the music of Debussy (the object of his study), but also for their symmetrical properties. Because of these, they accommodate all subsets under transposition alone, making it possible, as Parks writes, “to associate the sound of a given set to its genus” (Parks 1989, 58). The other reason to pinpoint elements of tonal thought in set-complex theory is that these may help achieve continuity with the music-theoretical past— not some peripheral movement, but the grand tradition. Rather than relegating such elements to the category of the “incidental,” or of the “outer form,” we can talk about them in terms of their effect on the interpretation of music. We thus regard them as integral to the theory from a performative perspective. The following chapter will develop such a perspective.

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Chapter Seven

“Blurring the Boundaries” Analysis, Performance, and History Music theory, it was argued in chapter 1, can be a discipline at the service of music historians, providing them with the conceptual apparatus to analyze and assess music from the past. But music theory is also a part of music history, presenting itself for analysis and assessment. This double identity can be viewed as a special case of a problem fundamental to historiography: the problem of a historian’s own historicity, which renders an objective truth outside history inherently impossible. Historians trained in modern hermeneutics know about this. They know that their reading of a source is at best a mediation, a give-and-take between two historical viewpoints, and that as such it is temporally conditioned.1 This explains why history is, on a fundamental level, at odds with science. For the result of a scientific experiment to be declared valid, it should not be temporally conditioned, but, on the contrary, be obtainable under all circumstances. It should make no difference at which time, in which location, and by whom the experiment is carried out. This requirement of repeatability goes against the very nature of historical research.

What Music Analysts Do What music analysis has in common with the historian’s account of past events is its historicity. The analysis results from a mediation between an observer and an object, and between the worlds they represent. It tells us something about the music and its historical background, while at the same time reflecting the craft, the experience, the interests, and the expectations of the analyst. Even a hundred music students supplying the harmonies of a four-voice chorale setting by

1. This renders synoptically what Hans-Georg Gadamer, in Wahrheit und Methode (1960), called a “fusion of horizons” (Horizont-verschmelzung). A “horizon” delineates the world in which a person operates, and the means by which he understands that world. In Gadamer’s view, to understand the other does not mean to empathize with him fully, but rather to integrate one’s horizon with his. Thus, in historical writing, one integrates one’s horizon with the historical horizon.

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Johann Sebastian Bach with exactly the same function labels do not alter the fact that such an analysis represents a historically determined state of affairs: harmonic functions having become a special focus of music theory; the professional education of musicians having been assigned to institutions and coordinated in training programs; composition, performance, and analysis having drifted apart as musical disciplines; and works from different periods and of different styles having been integrated into a “classical” canon. In any case, this state of affairs was largely unknown to Bach. This example shows how complex analyzing music can be from a historiographical point of view. It may not only involve the historicity of an observer and an object, but also the historicity of the methods used. People who care about the scholarly status of music theory and analysis, who are sensitive to the criticism that these disciplines lack historical perspective and uphold the “fiction” of self-contained musical structures, may argue that this “fiction” results from conditions also delimiting the writing of history, or indeed any process of signification, like the reading of a poem, the performance of a symphony, or the staging of a play. In other words, they may not be willing to suppress their interest in an aspect of music only because it was presumably unimportant to the composer and his contemporaries. Thomas Christensen (1993b), for example, wanted to safeguard the music analyst from a permanent obligation to represent musical works by means of contemporary modes of understanding (without giving up the obligation to be aware of these modes of understanding).2 He argued against Joseph Kerman, Gary Tomlinson, and Leo Treitler, who had been critical of music analysis in its role of an academic specialization. But the scholar whose position especially bothered him was Richard Taruskin. Like other theorists, Christensen could not reconcile Taruskin’s notorious campaign against the orthodox historicism of “authentic” performance practice with his repudiation of analytical tools not contemporary with the object analyzed. Taruskin had played the card of historical correctness with characteristic eloquence in his debates with music theorists such as Allen Forte (Taruskin 1979, 1986; see further below), Matthew Brown, and Douglas Dempster (Taruskin 1989, 1995): What one hopes to achieve by means of analysis is not merely a taxonomy of musical configurations, but insight into praxis: methods, routines, devices of composition. Since one cannot claim to understand a praxis unless one can state the theoretical basis on which it rests, one is going to be interested in historical viewpoints as well as any more efficient ones we may now propose. (Taruskin 1989, 157) As for music analysis, if it is not to be merely another reinforcement of transcendentalist myths—if, that is, it is going to tell us something we don’t already “know”—it should provide us with information (yes, historical information) 2.

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Christensen referred to Gadamer’s model of interpretation (see note 1).

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about composing techniques . . . If we are not interested in learning that (and many have by now so declared themselves) then analysis has nothing to tell us but comforting bedtime stories, for which just as many retain a seemingly insatiable appetite. (Taruskin 1995, 24)

Taruskin thus denied the music analyst the freedom of interpretation he granted to the performer. The latter’s primary task was to please and move “an audience in the here and now” (1995, 23). This view of the performer’s mission had already triggered a polemic from the champions of “authenticity.” Performers who prided themselves on their historical research were given to understand that: Research alone has never given, and is never likely to give . . ., enough information to achieve that wholeness of conception and that sureness of style—in a word, that fearlessness—any authentic, which is to say authoritative, performance must embody. (Taruskin 1982, 343)

Christensen pointed out the difference in treatment of performers and analysts in Taruskin’s writings. And he tried to draw Taruskin’s analyst into the firing line of his own criticism: Why, . . . one wonders, is Taruskin not led to conclude that historical contextuality in music analysis is but another form of “authenticity”? The search for historically indigenous modes of analytical interpretation, after all, is based upon an “authentistic” aesthetic just as historical performance practice is. (Christensen 1993b, 23)

The answer to Christensen’s question is quite simple: in Taruskin’s view, scholars, more specifically analysts, had no choice. They did the right thing, following the “authentic” course. It was their duty to search for “historically indigenous modes of analytical interpretation.” They were not in the position to communicate an artistic experience. Taruskin thus drew a line between musical practice and the scholarly study of music, which he considered subject to different rules of conduct vis-à-vis the musical work. Music analysis was consigned to the side of scholarly study; its goal was to yield knowledge. However, one cannot play down the role of signification in the analytical literature. In the first decades of the twentieth century, the analyses of Ernst Kurth, Heinrich Schenker, and Arnold Schoenberg were intended to stress certain values that music—the best music, in the opinion of these authors—embodied. Kurth regarded music as a direct expression of the inner forces of human life. Schenker devised a concept of organic unity which only the works of a group of great—and, coincidentally, for the most part German—masters could illustrate. And for Schoenberg, aesthetic value came from an immutable substance pervading an ever-changing musical surface. These men did not analyze with a view to stating knowable facts about a musical composition, but to highlight it as

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a work of art, to celebrate its wholeness and inner logic; in a way, they acted as performers. Analysis can be seen as a mode of representation alongside concert performance. Like the latter, it reinforces the image of a self-contained musical whole—of Opusmusik, as Hans-Heinrich Eggebrecht (1975) called it—in an environment in which it stands out in bold relief.3 Can this view also be applied to current analytical practices? Both Schenker and Schoenberg have left their marks, while Kurth, on the other hand, has not had a lasting influence. Schenker’s method—gradually stripping down a musical text to a basic progression, the Ursatz—is still being used today. As noted in chapter 1, his name has become attached to a dominant analytical tradition in the United States, which has spread to Great Britain, Australia, and Canada. Schenkerian analysis has changed in the academic environment that adopted it. American music theorists have downplayed Schenker’s unedifying polemic, and his exclusive commitment to an “aesthetically superior” repertoire. They have decoupled his method from his “unfortunate excursions into the realm of the political, social, and mystical” (Babbitt 1952, 264). And they have taken the Ursatz-concept—with its Romantic connotations, and the uniformity it imposes on the deep levels of musical structures—for granted, stressing the value of the analytical process at the expense of its final outcome.4 William Rothstein (1990) has documented the method’s assimilation to the norms of scholarly discourse maintained at American universities. And Robert Snarrenberg (1994) has reported the transition from a “natural” to a “scientific” imagery reflecting this assimilation in the literature, a transition that has caused people to associate Schenkerian analysis with academic dryness and formalism. This association may be justified in a number of cases, but it does not apply to everyone, and not to Schenkerian analysis in itself. Notwithstanding its transformation, it is still essentially a practice meant to elevate musical works of art. A Schenkerian analysis is the written, graphed-out, or spoken counterpart of the concert performance, from which one should not expect historical information, but an artistic interpretation. Schoenberg’s influence on twentieth-century analytical discourse has been no less strong, if a little more diffuse than Schenker’s. It has been felt in the field of classical form—due to his posthumously published Fundamentals of Musical Composition (1967)—in analyses focusing on large-scale motivic coherence—due 3. One may think of other environments that do not allow musical works to be perceived as tightly organized statements. And, of course, there is music to which the concept of a “work of art” is not relevant at all. See Cook 1989a (10–70) and Stockfelt 1993 for some reflections on the interdependence of musical structure, space, and mode of listening. 4. This line of defense was taken by Pieter van den Toorn: “[W]hat counts are the musical issues that are brought to the fore, not the product itself, its ‘solution’ of a musical puzzle, or the fact that alternative readings may suggest themselves as equally valid. The latter are, indeed, encouraged as an essential part of the process.” (Van den Toorn 1995, 98)

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to the motivic analyses of his own music, and of earlier music—and, more generally, in the very close alliance between theory and contemporary composition in the years after the Second World War, an alliance to which PC set theory ultimately owes its existence. Schoenberg had fewer sides to screen from the public gaze and from scholarly attention than Schenker. His uneasy modernism—which seems to have grown organically from a deeply felt commitment to the musical tradition—was more agreeable than Schenker’s forthright conservatism. Schoenberg’s impact as a composer testified to the relevance of his analytical observations, although his analyses may actually have been meant as justifications of his own music.5 In any case, the message communicated by these analyses was that a full appreciation of the music required total immersion in the score. By pointing out the intricacy of a movement by Brahms or Beethoven, by shedding light on the recurring intervallic configurations unifying it on a deep structural level, Schoenberg argued the necessity of a textual interpretation.6 Taking the score as representing the work, and searching it for unifying elements, has been common practice in academia for years. PC set theory, for example, has extended Schoenberg’s viewpoint that the outer appearance of things may conceal their sameness, and that it is their sameness that matters, even when it is not obvious at a first hearing. PC set theory considers a PC set as being the essence of a chord, a melody, or a motif, or any other musical entity involving traditional pitches, so that when two such entities have their PC sets in common—or when their PC sets are of the same equivalence class—they are equated, by supplying them with identical labels. When the PC sets are not equivalent, one may measure the degree by which their relationship is close or distant (the common ground among the various similarity measures). A difference between PC sets, however, is valued as such only when it comes under the heading of PC set complementation (a category still involving intervallic similarity). PC set theory can thus be seen as dealing with a concept of unity based on recurrence and variation. Like Schenker’s theory of tonal unity (a concept based on prolongation), it has transformed an aesthetic creed into a set of analytical tools. Not surprisingly, then, it has met with the same kind of criticism that was meted out to the Schenkerians. It was considered to be insensitive to musical 5. See also chapter 5, p. 137. I am referring to the analytical sections in Schoenberg’s papers “Brahms the Progressive” (1933; Schoenberg 1984, 98–441), and “Composing with Twelve Tones” (1941; Schoenberg 1984, 213–45), and to the analysis of his own Four Orchestral Songs, Op. 22 (1932; Schoenberg 1965). 6. Schoenberg may have gone much further than that. The American composer Dika Newlin remembered this characteristic utterance of her teacher, Arnold Schoenberg: “Music need not be performed any more than books need to be read aloud, for its logic is perfectly represented on the printed page; and the performer [i.e., the player], for all his intolerable arrogance, is totally unnecessary except as his interpretations make the music understandable to an audience unfortunate enough not to be able to read it in print.” (Newlin 1980, 164)

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experience, and ignorant of what may render a work unique. Roger Scruton’s judgment was particularly severe: The dry pseudo-science of the language draws our attention only to what is most lifeless in the music, and seems to forbid us to hear its expressive power. (Scruton 1997, 415–16)

The professionalization of music analysis, through the systematic application of Schenkerian theory and PC set theory, was deplored as degenerate by those who saw a natural affiliation of analysis with criticism, like Scruton, Kerman (1980, 1985), or Treitler (1980). Yet, this professionalization has never been—and will never be—able to change the fact that music analysis is essentially a matter of signification, that it gives meaning to an object, even if analytical techniques are governed by fixed protocols enforcing conformity without regard to the individuality of that object. Even the strictest codification of analytical practice cannot diminish its expressive purport. Of course, this does not preclude complete failure. Analysts are fallible to the same extent as performers are. As Carolyn Abbate wrote, “[verbally] couched interpretations of music are performances and can only be more or less convincing” (Abbate 1991, xi). The question is: how does an analysis convince us as a performance, quite apart from the empirical or historical evidence that it may provide? For one thing, it should demonstrate knowledge and skill, the latter comprising both the power of observation and the ability to arrange the various observations into a structured statement. For another, it should convey an experience, that is, the impact the musical work has made on the analyst. The balance is rather delicate; the problems some people have with Schenkerian theory and PC set theory can be attributed to the upsetting of this balance by an extreme systematization of analytical techniques. But a highly sophisticated analytical apparatus need not be an impediment to analytical understanding (no more, in any case, than a highly developed technical proficiency impedes a performance). If it is handled with all due awareness of the analyst’s aesthetic creed, and with all due concern for the compositional routines and the conventions of hearing that the musical work follows or breaks, it may be the key to profound insights. However, an analysis should also be convincing as an act. That is, one should be made to believe that the musical work reveals itself through the analysis. There is a growing literature on this aspect of the discipline, an objective that is as vital as it is difficult to achieve. With reference to writings of David Lewin, Fred Maus and Jonathan Dunsby, among others, Nicholas Cook (1999, 252) speaks of an emerging “performative epistemology” of music theory, according to which analyses produce, rather than trace, musical meaning. This has led to several rereadings of the analytical literature. As an example, Cook discusses the performative content of Schenker’s analysis of Beethoven’s Ninth Symphony:

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In its wealth of striking literary images, in its fervent advocacy of the “absolute music” tradition, and in its attempt to erase Wagner from that tradition, as well as in its documentation for all time of a now defunct style of performance, Schenker’s book constitutes an irreversible event in the reception history of the Ninth Symphony. (Cook 1999, 254)7

The word “event” takes us back to the beginning of this chapter. Cook does not present Schenker’s analysis as a commentary from the sidelines of music history, but as a moment of music history. Music analysis is treated by him as an activity and as such as an essential part of musical life itself. Consequently, it is not only judged by its accuracy, but also by its capacity to engage people and to interact with other ranges of activity (like composition, performance, and history). Analysis may not only reveal how music was composed, and how it should be played or heard; it is also a process that can be enjoyed in its own right. By the same token, a musical work is as much a possible object of an analysis as an incentive to analyze. Any representation of this work is induced by reciprocal interaction with a particular craft. The musical work enables people to express themselves and to display their talents. People do not just mediate, unselfishly passing on musical works to their contemporaries. They also appropriate, using them to serve personal, artistic, scholarly, pedagogical, or political goals. When we analyze works from the past, we make them contemporaneous with our own present. A history of music analysis, then, is concerned with degrees of mediation and appropriation in the products of that discipline. The writing of such a history is not just for practitioners. Nor is it meta-history, if such a thing exists at all. The history of music analysis is a branch of music history—and, ultimately, of history— focusing on a particular aspect of musical life: a practice that may either be closely allied with composition and performance, or pursued independently. Music analysis history can be put on a par with music performance history, as it relates a musical practice to its cultural environment. It allows us to see the larger complex of which this practice is part; to see how musical meaning arises from this complex, and is acted out—”performed”—by the analyst. As such, it is of interest to non-specialists if we for the moment disregard the expertise its subject requires. Before we start to study aspects of PC set theory’s interaction with its environment, it is appropriate to emphasize the egalitarian spirit of the present discussion. It may be disturbing to some readers that analysis is treated on a level with performance, and that analysts and performers are supposed to construct, rather than just to channel, musical meaning. The work of historians, as stated, is likewise constructed and creative. The problem is a confusion concerning disciplines that we have learned to distinguish so carefully. Scholars are not automatically artists; yet, earlier on, we have declared them capable of taking artistic 7. See Cook 1995 for a more extensive reading, along the same lines, of Schenker’s analysis.

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action (even worse, by exploiting the work of others). Performance is not the same as composition. Yet, what we have taken as a model for music analysis is not performance as the reproduction of a text—in this case, a score—but performance as an event, in which the performer takes charge of the text.8 This might be an invitation to think of a score as an “ideal” performance (like the score of Cassandra’s Dream Song by Brian Ferneyhough), an arrangement as an analysis (like Webern’s orchestration of the Ricercar from The Musical Offering), a recording as a score (like Miles Davis’s 1964 version of My Funny Valentine), or a composition as a historical essay (like Berio’s Sinfonia). Stretching the point, we might say that there is only one range of musical activity, involving various disciplines; a continuum of actions, reactions and reflections, deploying sounds, tones, words, and/or gestures. What is the reason for suggesting this? Why should we blur the traditional boundaries between these disciplines? In the previous chapters, we have traced the evolution of a conceptual framework pertaining to a vigorous analytical practice. Up to this point, we have seen that PC set theory was driven by a major development in the field of twentiethcentury composition, that is, twelve-tone serialism. However, it is unlikely that serialism was the only driving force for PC set theory. First, PC set theory has never found a firm footing in continental Europe, even though serialism was for a time just as pervasive there. Second, although its foundations were laid by a composer and theorist of serial music—Milton Babbitt—PC set theory has never shown an exclusive commitment to such music. On the other hand, twelve-tone serialism has certainly served as the principal paradigm of PC set theory, sometimes with far-reaching consequences for the music to which it was applied. For example, we have seen how serial concepts were redefined through PC set theory so as to become applicable to non-serial music. In view of the origin of these concepts this music was deemed “atonal” or “non-tonal,” even if there was evidence to the contrary, for example, in Stravinsky’s The Rite of Spring. Allen Forte’s analysis of this piece (Forte 1978) might have constituted an “irreversible event” in its reception history, to paraphrase Nicholas Cook, if Richard Taruskin (1979) had not seriously questioned Forte’s stance of “phenomenological virginity” (Taruskin’s words). Forte took up the gauntlet a few years later (1985). The ensuing exchange between the two authors, in the British journal Music Analysis, about a short passage from the Ritual of the Rival Tribes, certainly constitutes an irreversible event in the history of music analysis. It made the papers, even if only because of its unusual fierceness.9 The discussion

8. For a critique of the still-persistent idea of performance as reproduction, and the radical negation of this idea in the work of music philosophers like Stan Godlovitch, Robert Martin and Christopher Small—which involves the subordination of musical works to performances—see Cook 2001. 9. It was reported about in the New York Times (“Should We Care Who Wrote It?” by Donal Henahan; May 17, 1987, 102). For another commentary, see Hinton 1988.

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8va

3

W.W. Tpt.

4-18:[2,3,6,9]

4-7:[11,0,3,4]

4-17:[11,2,3,6]

4-7

4-18:[9,10,1,4] 4-18:[10,11,2,5] 4-8:[6,7,11,0]

(4-18)

Example 7.1a. A passage from Stravinsky’s Rite of Spring (Rehearsal No. 60). An analysis by Allen Forte, The Harmonic Organization of The Rite of Spring. Copyright © 1978 by Yale University. 8va

p.t.

p.t.

p.t.

3

ant.

B: I

V*

I

V (VII)

I

**

V

G: I

VII ***

* (or "modally", VII) ** Stravinsky spells this note B *** "anomalous"

Example 7.1b. The same passage. An analysis by Richard Taruskin (1979).

Example 7.1c. Taruskin’s model: Esprit Philippe Chédeville, Musette, La Sincère. Richard Taruskin, “Letter to the Editor,” Music Analysis 5/2–3 (1986): 314. Used by permission of Blackwell Publishing.

centered on the question of what were the most appropriate grouping criteria for the tones sounding at Rehearsal No. 60. Forte characteristically took his cue from recurring set-classes (such as his “4-18”), whereas Taruskin, in an equally characteristic manner, provided an example from common (tonal) practice, which could have served as a model for the passage (ex. 7.1). This “event” shows that to talk of music theory as depending on actual music needs a critical attitude, and is indeed very problematic with respect to PC set theory. This theory has not only developed through a dialogue with a musical repertoire, with respect to which it has tried to attain an optimal level of accuracy, but also through solving problems inherent to its own methodology. In other words, it has also been self-driven, as for example the interest in segmentation as a topic of theoretical speculation illustrates. Segmentation—the parsing of a composition into significant musical units—is an important analytical

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issue, especially when the music does not seem to suggest obvious criteria for the process on a first hearing or reading. (It was actually the heart of the matter in the debate between Forte and Taruskin.) It is one thing to find relations between PC sets; quite another to claim that these relations assure coherence. This required a supplementary theory of segmentation, including typologies of possible segments and procedural guidelines (Forte 1973, 83–92, Hasty 1981, Schaffer 1991). PC set similarity, as we have seen in chapter 5, is another topic that emerged from the self-imposed limitations of the theory; it compensated for the rigid relational concepts PC set theory had adopted from twelve-tone theory (like transposition and inversion). However, the decision to represent musical structures by PC sets—and PC sets, in turn, by frequencies of interval classes (the interval vector)—has deeply informed the notions of similarity that were applied to these structures. If a theory of music is not entirely music-driven (which is not as odd as it sounds), if it exploits musical works as much as it explains them, then what is it that makes this theory and its concomitant analytical practice flourish? Why, and to whom, is it gratifying? These are important questions, for, whatever its motivation, an analytical practice causes music to resonate with certain meanings and to persist through time. It can even provide a “safe haven” for written music that would otherwise have little chance to survive. Thus, music theory and music analysis, while using the riches of great music, reward music with reflective thought enhancing its depth.10 To return to the earlier question: we do not blur the boundaries traditionally dividing musical and scholarly activity, composition and performance, etc. These boundaries are already blurred, allowing various practices to interpenetrate.

Historical Models and Topical Meanings There are various notational devices to help us clarify music, like chord symbols, harmonic reductions, voice-leading graphs, and form charts using letters. Furthermore, we can draw on an analytical vocabulary offering many angles from which to view musical compositions. However, these devices and vocabulary are also attributes of a skilled activity seeking to maintain and improve itself by means of music. We can describe this activity with regard to its effects, to the way in which it reacts with the music, thus stressing its performative dimension. Subsequently, we can put this activity into historical perspective, with regard to the goals it serves and the competence it requires. 10. Joseph Kerman (1983) has called attention to the role of musical discourse in the complex mechanism by which canons of respected musical works arise. For a more recent study of this topic, considering craft, repertory, criticism, and ideology as principles of canon formation, see William Weber (1999).

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Let us consider the notational devices and the vocabulary of PC set theory. What effect do they have on music? Is there a property or quality they accentuate? PC set theory can be seen as basically providing an inventory of relations between PC sets. However, this is not what defines it as a theory. PC set relations are in fact properties of the algebraic structure that has been chosen as a model for pitch structure. PC set theory is a theory only in so far as it claims that some of these relations can be, and have been, exploited in musical compositions.11 What it does is to group notes together and to encode the resulting combinations in order to substantiate this claim. Michael Friedmann, whether he was conscious of it or not, has described this kind of analysis as the listener’s active interference with the musical structure: To perceive . . . a musical structure, one must first segment it from its surroundings in the musical continuity, so that it becomes a unit with sufficient internal integrity to be independently scrutinized, and then identify characteristics that can later be used to liken or contrast the unit to others. (Friedmann 1990, xxii, my italics)

This is worth a moment of contemplation. What are the grouping criteria? They are very hard to pin down. Joseph Straus probably gave an accurate characterization of the general practice when he advised his students “to poke around in the piece, proposing and testing hypotheses as you go” (Straus 2000, 51). And what does the encoding involve? As we have seen, the analyst notates the PCs represented in a musical segment, and assigns the segment to a class. Most often this is a Tn/TnI-class, the segment’s membership of which is certified by the “prime form” (Forte) of its PC set or by its interval vector. The predominance of this class type in the notation and vocabulary of PC set theory is no doubt a token of its common utility. On the other hand, using transposition and inversion to identify classes of PC sets means the generalization and perpetuation of the timebound, since the way in which these musical operations are defined—that is, without regard to order and contour—reflects a specific phase in the history of composition (see chapter 3). Paradoxically, this is the phase of twelve-tone serialism. The paradox is that the “unorderedness” of PC sets can only be observed in a context in which order is central. 11. The question of whether composers like Arnold Schoenberg or Igor Stravinsky actually composed with PC sets—decades before their theoretical formulation—has been subsumed under the “intentional fallacy” by Forte in his exchange with Taruskin: “I submit that we can never know with any certainty what ‘the composer thought he was about’ and that to attempt to do so to validate an analysis is an empty pursuit” (Forte 1986, 335). However, if PC set theory does not address the question of how, and by whom, PC set relations were used, it cannot not claim anything except that the music of Schoenberg is composed of pitches under twelve-tone equal temperament (a statement entailing all the relations between PC sets). Ethan Haimo (1996) criticized Forte for invoking the intentional fallacy, and provided analyses of Schoenberg’s music taking into account Schoenberg’s compositional thought, as evidenced by his manuscripts and writings.

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PC set theory fundamentally conceives of music as composed of segments that are different in some ways, but connected by recurring intervallic patterns or qualities. In other words, it claims that music achieves coherence by a continuous process of variation. This process is sometimes concealed by the articulation of the musical surface. The phrases and rests, the changes of register and instrumentation—in short, the salient features of a musical composition—may not always help to identify the “relevant” segmentation. Or one may find several competing segmentations, each of which involves different PC sets. Such ambiguities are not valued negatively, though. On the contrary, the prevailing view is that conflicting segmentations produce rich musical structures requiring a deep understanding, as the following statements illustrate: Schoenberg often conceals the main structural components of his music behind primary configurations. (Forte 1973, 92) It seems most reasonable to begin by identifying the primary segments, that is, those segments clearly articulated as distinct musical units. But it would be dangerous [sic] to stop here—to do so is to assume that meaningful relationships can exist only at the surface level. We must look for other, less obvious, relationships. (Beach 1979, 15) The presence of ambiguity in segmentation . . . allows for many interrelated lines of development to take place and this makes it possible for the music to achieve great structural richness and depth. (Hasty 1981, 60) Some of these groupings may span across rests or phrasing boundaries, and that is okay. A rich interaction between phrase structure and set-class structure is a familiar feature of post-tonal music. (Straus 2000, 51)

Not every conflict is to be valued as aesthetically enriching, though. In A Guide to Musical Analysis (1987), Nicholas Cook demonstrated an analytical application of PC set theory as expounded by Forte (1973), using Arnold Schoenberg’s Little Piano Piece, Op. 19, no. 6. Cook—a scholar from the English tradition; not a “pure” set-theorist, but somebody deeply interested in music analysis and its pedagogy—had set himself no easy task. On the one hand, he wanted to explain why the set-theoretical approach was so appealing to some. On the other, he had to interest the uninitiated by rewarding their immediate perceptions about the piece. Now, as we can infer from the above-quoted statements, analysts felt attracted to PC set theory because it transcended those immediate perceptions. Therefore, Cook’s analysis is a rather unusual one, dealing only with the PC sets of discrete musical units.12

12.

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Cf. the analysis of Opus 19, no. 6 made by Forte (1973, 97).

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Cook’s segmentation is reproduced here as example 7.2a. Cook claimed to have based it on “surface features like texture, rhythm and dynamics” (Cook 1987a, 124). At first sight, this seems certainly acceptable. The first three segments can be seen as having distinct formal functions: segment “A” presents an idea, segment “B” repeats and elaborates this idea, and segment “C” carries it further. The segments “D,” “E,” and “F” build a less cogent succession, but the framing silences and contrasting textures give support to Cook’s segmentation. (Moreover, segment “F” seems to serve as the recapitulation in a ternary form, as it brings back the opening chords.) The PC sets of these six segments have interesting relations. For example, “E” includes “D” and the complement of “F.” “D” is also included in the complement of “A” (under inversion and transposition), and the complement of “F” is also included in “B” (under transposition). Eventually, Cook shows that all the PC sets belong to one set-complex about “E.” One of Cook’s more interesting observations is that “C” stands apart from the other segments, although it belongs to the same set-complex. It is not related to the two surrounding segments (“B” and “D”) in the same way that these segments are related to one another. In other words, it interrupts a chain of logical connections (Cook 1987a, 130). This is a statement about the musical content of Opus 19, no. 6, and that statement is supported by Cook’s neatly drawn Figure 62 (reproduced here as ex. 7.2b). What Cook wanted his readers to see is that PC set theory is capable of producing such statements. But what is their value? On closer investigation, “C” is not a complete segment based on the surface features used by Cook. The two silent beats in measure 4, followed by the third appearance of the chord {A4,F♯5,B5} (a chord marking the beginning of the segments “A,” “B,” and “F”), suggest that “C” should actually start three beats earlier, that is, on the last beat of measure 4. Mark Delaere (1993, 82) has interpreted the entire first half of Opus 19, no. 6 as a classical sentence, with the enlarged “C”-segment functioning as the continuation phrase. The accelerating activity in this segment validates Delaere’s comparison, but this activity is obscured by Cook’s segmentation. Is it revealing to speak of a conflict between “set-class structure” and “phrase structure” inherent in this work, a conflict by which it achieves “structural richness and depth”? Cook’s analysis does not pose such a distinction. That is why, in this case, the conflict is not enriching but just problematic. To speak of “C” as an interruption is inconsistent with its function, which is that of a culmination. Admittedly, Cook’s interpretation is not unattractive. The idea of breaking up an otherwise logical sequence of events is conceptually simple and artistically potent. It is therefore worth being considered as an explanatory model for a piece like Opus 19, no. 6, which is enigmatic, leaving much scope for interpretation. What is more, “interruption” is also an important concept in Schenkerian theory, as Cook does not fail to point out. It is the cadence on the dominant at which the fundamental line of a composition comes to a “premature” halt (see example 1.5). This historical background adds some weight to Cook’s interpretation. However,

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Example 7.2a Nicholas Cook’s segmentation of Schoenberg’s Little Piano Piece, Op. 19, no. 6. © 1913, 1940 by Universal Edition A.G., Wien/UE 5069.

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Example 7.2b. PC set relations between the segments identified by Cook. A Guide to Music Analysis, by Nicholas Cook (1994). By permission of Oxford University Press.

Cook’s interpretation is actually dictated by his starting point. As already stated, he sought to demonstrate the analytical theory of Allen Forte. According to this theory, segment “C” cannot begin in measure 4, because this would raise the cardinality of its PC set to 10, a number that Forte had decided not to take into account (as Cook points out on page 125; see also chapter 4, p. 107). Consequently, segment “C” is brief, and its pitch content suggests a discontinuity. This just shows that an analytical theory imposes certain a priori conditions on the analysis of a work; the music is made to respond to the analyst’s perspective. In any case, now that we have discussed what music analysts do, we may ask what they want to achieve. Cook’s purpose was primarily educational: to teach an analytical skill, not as an “esoteric specialism,” but as a part of the “takenfor-granted professional equipment of the historical musicologist and the ethnomusicologist” (Cook 1987a, 3). He wanted to show his readers how one goes about analyzing with PC sets, using the surface of Schoenberg’s piano piece as a sample “playing field.” As we have seen, the more ardent set-theorists have pursued their aims further. They speak of discovering relations beneath the musical surface, relations that might even contradict the articulation of that surface. They analyze musical pieces as complex and multivalent networks. Such an activity is certainly not without precedents or historical models. The crystallization and elaboration of classical musical forms from 1750 onwards, the subtle interaction between the tonal and thematic strands in, for example, sonata forms, tonal harmony’s capability of generating various structural levels (by “prolonging” tones and harmonies, or by moving into subordinate key areas), and the intricate motivic relationships in the music of Beethoven, Brahms, and Schoenberg, have all been factors in the rise of music analysis as a discipline in its own right. As such, they also account for the biases of that discipline. One cultivates those fields that have yielded well. Analytical methods

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“blurring the boundaries” a



233

b

Example 7.3. Rudolph Réti’s reading of the opening measures of Mozart’s Symphony No. 40, K. 550: the first three notes of the bass (a) form “a literal transposition of the symphony’s main motif” (b). Rudolph Reti, The Thematic Process in Music. By permission of Faber and Faber Ltd.

are based on historical canons—sets of works that respond to those methods— which their practitioners often seek to enlarge. Felix Salzer (Structural Hearing, 1952), for example, has tried to expand the horizon of Schenkerian theory by adding the music of, among others, Monteverdi, Ravel, Hindemith, Bartók, and Stravinsky to what was originally one of the most exclusive analytical canons.13 Rudolph Réti (The Thematic Process in Music, 1951) and David Epstein (Beyond Orpheus: Studies in Musical Structure, 1979) followed in Schoenberg’s footsteps, analyzing Classical and Romantic compositions in terms of the recurrence and development of small intervallic cells. Thus, Réti could represent Mozart’s Symphony No. 40 as a musical statement almost as motivically tight as Schoenberg’s Four Orchestral Songs, Op. 22 (ex. 7.3; Réti 1961, 116). And Erwin Ratz (Einführung in die musikalische Formenlehre, 1951) reformulated the theory of musical form, with its inherent orientation towards the Viennese classics (Beethoven in particular), in such a way that it also encompassed the music of Johann Sebastian Bach. Seen thus, it is not strange that the canon of PC set theory has been expanding beyond the serial and proto-serial repertoires of the second Viennese school, as noted in chapter 2. Its productiveness in one field entitled it to move on to other fields, just like Schenkerian theory and the theory of musical form. Forte’s article “Pitch-Class Set Analysis Today” from 1985, which reports the use of PC set theory in a wide variety of musical repertoires, reminds the reader of a showcase of trophies. One such “trophy” is this quotation from an article on Debussy’s prelude Brouillards, by Richard Parks. It argues that PC sets can serve as a resource for the analysis of the music of this composer: To a rather large extent, Debussy’s music (of which Brouillards may be considered typical) reveals, in its pitch resources, combinations which exhibit characteristics lying beyond traditional notions of harmony, voice leading, and a referential tone and sonority (tonic). (Parks 1981, 134, quoted in Forte 1985, 35)

13. Schenker’s own canon consisted of the works of the great German and Austrian masters from Bach to Brahms, supplemented with those of Domenico Scarlatti and Frédéric Chopin. In recent years, Schenkerian theory has been applied to Japanese koto music (Loeb 1976), jazz (e.g., Larson 1998), and American popular song (e.g., Forte 1995).

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When music is analyzed, it is often implicitly measured against a historical model: a work—or a set of works—that have been shown to respond to the method applied. Whether the choice of model is appropriate is sometimes a moot point. The works of Debussy, like those of Schoenberg’s middle period, have sometimes been analyzed with tools specifically designed for the tonal music of the common-practice period.14 In other words, they have been represented as the end of nineteenth-century practice rather than the start of twentieth-century practice. Forte and others have criticized such analyses for neglecting what is novel—and, in their view, essential—in these works. (Hence Forte’s aversion to Taruskin’s analysis of that famous passage from The Rite of Spring, which he expressed in the same article; see ex. 7.1b–c.) Parks’s words on Debussy must have been very welcome indeed. It may seem that music analysis is involved in a large, ongoing historical project: grouping works by identifying threads running through the history of music composition. However, in view of the fact that music analysis is interwoven with its own cultural environment, we may ask why analysts engage with particular historical models—for example, the sonatas and symphonies of Beethoven, or Webern’s Movement for String Quartet, Op. 5, no. 4 (the textbook example of “free atonal” music). What has attracted them to these works? The previous discussion perhaps suggests a cynical answer to this question: these works underpin the relevance of their enterprise. This answer is not only cynical but also incomplete. There is no doubt reciprocity between the analytical armamentarium and particular musical repertoires. However, this reciprocity need not inhibit individual creative thought. Music analysts have broken fresh ground of their own accord, and they have received impulses from their cultural environment. An analysis may reflect current orientations in composition and performance, or the culture and concerns of an institution; it may borrow insights from other research disciplines; or it may be affected by developments in the fields of technology, teaching, and education policy. In short, the analyst’s perspective is subject to change, and so is what appeals to analysts in music. This means that what they value in a historical model is of topical interest. The reception history of the music of Debussy and Schoenberg, to which I just referred, is an example of how music can take on different meanings in the light of different historical models (in those cases, tonal versus non-tonal). In the 1980s and 1990s, approaches emphasizing the self-containment of the musical work of art, which had dominated the music-analytical scene in the 1960s

14. Hugo Leichtentritt’s attempts to analyze Schoenberg’s “atonal” music as an extension of nineteenth-century tonal practice (Leichtentritt 1927, 436–57, 1928) are documents of historical interest. Many have followed in his footsteps, writing analyses dealing with harmonic constraints in works of Schoenberg. For a historical overview, and some additional analytical observations, see Schuijer 1996. For a persuasive tonal analysis of Debussy’s music, see Pomeroy 2003.

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and 1970s—and which had been targeted by Joseph Kerman—had to compete against a growing interest in music as a social and cultural phenomenon, an interest seen in a preference for scholarly criticism15 above scientific inquiry. This shift of focus has also resulted in new, or renewed, historical models. For example, gender studies have brought to the attention of music analysts repertoires that were either neglected (music by women), or considered of limited analytical interest (popular music). They have also helped vocal genres—like opera, oratorio, and song—rise through the ranks of a discipline traditionally concerned with issues of instrumental music. Studies of “intertextuality” in music were inspired by Harold Bloom’s theory of literary influence, which sees the history of Western poetry as driven by a struggle of younger poets against their predecessors, and the individual poem as an attempt to subdue an older poem. The music of Brahms, Schoenberg, and Stravinsky—already firmly established in the canons of music analysis—epitomized the anxiety about a rich, but at the same time endangering, past, and figured prominently in this new branch of study (Straus 1990b, Korsyn 1991). Each of these examples reflects a state of affairs comprising a lot more than a score, a set of analytical techniques, and a perceptive mind—and this is true for PC set analysis as well. We need to determine what has led to an analytical practice outside of all the cues it has taken from actual music, or from the history of music analysis. The following chapter will focus on two conditions that have encouraged the use of PC set theory in the United States. The first is the availability, from the 1960s onwards, of the computer as a research tool in the humanities. The computer was welcomed not only because of its practicality, but also because of its apparent capability to transcend the human mind. The second condition is the American university itself, which has not only offered training to and employment for music theorists, but has also been a forum where they could meet and exchange ideas and views in an intellectual atmosphere.

15. The term “criticism” was used by scholars like Leo Treitler and Joseph Kerman to denote a way of writing about music that is historically and technically informed and conveys an artistic experience. According to a definition from Kerman himself, dating from 1965, “criticism” is “the way of looking at art that tries to take into account the meaning it conveys, the pleasure it initiates, and the value it assumes, for us today.” (Quoted in Kerman 1985, 123)

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Chapter Eight

Mise-en-Scène From a narrativistic point of view, history is a staging of the past.1 This book has placed PC set theory, an evolving theoretical discourse on music, in the foreground, without adding much scenery. It has only made casual reference to its social, cultural, and technological setting. Now that we are about to bring two crucial elements of that setting onstage—the computer and the university—we should realize that their mere presence does not provide an explanation for their influence on music theory. They have exerted that influence because music theorists reacted to them, probably for a complex of reasons. This chapter seeks to investigate these reasons, with particular emphasis on the roles of Allen Forte and Milton Babbitt.

A: Formalization and Forgetfulness The Computer as a Role Model Analyzing with PC sets can be laborious. The process of coding tone combinations and establishing their relations in terms of their intervallic structures, the operations that connect them, or the complexes to which they belong, is timeconsuming. Moreover, it is extremely prone to inaccuracy. The slightest error can result in the wrong classification of a PC set, and this, in turn, can have distorting consequences for an analysis. PC set relations do not necessarily follow syntactical rules, so that, in the absence of detailed information about the composition process, we have little that can help us verify them. Accuracy is the only foundation on which the analyst can build. PC set theory has done a lot to regulate the extraction of information from scores and to ensure its correct classification. It has provided very precise coding protocols, and it has mapped out a considerable number of musical relations in the abstract. But what is most important, its notation also allows the electronic storage and manipulation of this information. This has not been a response to the

1. I owe this expression to a number of writers. For example, Susan Sontag (1980, 116) used it to describe the workings of memory. Here, it is intended to state that what is told about the past cannot be separated from how it is told. The impact of this idea on historiography has been described as a “narrativistic turn” (cf. Ankersmit 1990, 23–33).

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cognitive demands PC set theory made upon analysts; composers and music analysts did not start to use computers because the number of PC sets was so vast, because their interval vectors were so hard to memorize, or because their relations were so intricate. On the contrary, it was by means of computer technology that this information had become available in a comprehensive form in the first place.2 Few people realize that PC set theory was actually devised for computer-aided analysis—today, this fact is ignored in most analytical textbooks. John Rahn’s Basic Atonal Theory, for example, contains a few chapters exploring a single piece by way of instructions and assignments. In spite of the integer notation and the mathematical vocabulary, these instructions and assignments require only a score, a keyboard, and audio equipment. With regard to set type identification, Rahn comforts his readers: With a little practice, using the shortcuts, you will soon find yourself doing this quickly in your head. For example, learn to identify {7,8,1} as “up 1, up 5” in normal form ({7,+18,+51}) which, starting on zero, produces {0,+11,+56}. The inversion will be “down 1, down 5,” and the reverse of this upwards will be “up 5, up 1” for normal form, which, starting on zero, gives {0,+55,+16}. Obviously [0,1,6] is more normal than [0,5,6] so the Tn/TnI-type is [0,1,6]. (Rahn 1980, 82)

The following quotation has been taken from the 2000 edition of Joseph Straus’s Introduction to Post-Tonal Theory. The student is taught here to relate the numerical notation of a PC set found in the eleventh song from Schoenberg’s Book of the Hanging Gardens, Op. 15 to a tonal musical shape: The four-note gesture is a member of set class 4-17 (0347). [Straus uses both Forte name and prime form to identify PC sets.] It is easy to visualize this melody as a triad with both major and minor thirds, although, as we will see, it occurs later in the song in a variety of guises. (Straus 2000, 61)

Michael Friedmann, in his Ear Training for Twentieth-Century Music, proceeds from formal definitions (“DEFINITION 5.1: Z-related set classes”; Friedmann 1990, 74) to ear-training and sight-singing exercises (“EXERCISE 5.2: Distinguishing Z-related set classes”; ibid., 75). The following exercise should be used to train the skills of segmentation and set-class identification: 2. In the academic year 1963–64 Hubert Howe had worked in Princeton on a project “involving the application of digital computers to music theory and musicology.” One of the results of this project was “a list of pitch structures of sizes 2 through 10 together with their complements, interval-content, and another list which shows the number of transpositions with n common tones, for each possible n with each size PS.” To achieve this result, Howe had used a program written in FORTRAN for an IBM 7094 computer. Allen Forte published his first list of PC sets in the same year, but he only mentioned the program by which he had mapped out the set-complex relations that were the topic of his research. This program was written for a similar computer installed at Yale (Howe 1965, 45, 56; Forte 1964, 182, note 21).

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As with trichords, the reading off tetrachord types should be practiced with any one-voiced example, first by reading off adjacent tetrachords (the first four pitch classes of a melody, then the next four, and so on), then by overlapping tetrachord types (notes 1,2,3,4, then 2,3,4,5, then 3,4,5,6, and so on). (ibid., 100)

Even when, in these textbooks, PC set theory is presented as a set of practical tools rather than as a compositional theory, it seems to be dealing only with the question of how pieces of music have been put together, how they “work,” not with the questions that spurred its development in the early 1960s: how to transfer conventionally notated music to the computer, and how to retrieve new and interesting information about that music from the computer memory. Now that PC set theory is found in pedagogical programs, now that music students learn to analyze with PC sets, to sing them, and to identify them by ear, we are hardly ever reminded of those seminal questions. In the mid-1960s the computer was a hot issue in the humanities. This was the time that scholarly communities became aware of its potential as a research tool. In the United States, the years between 1964 and 1967 saw a burst of conferences on the subject, where computer applications in a number of disciplines, including archeology, history, and literature, were demonstrated. The momentum was felt in music research as well: computers were used in the spheres of electronic composition, style analysis, structural analysis, music education, bibliography, and thematic indexing. Computer centers at universities functioned as meeting places where unlikely alliances were forged. “One of the virtues of a university computer center,” the composer James Randall wrote, “is that it seems at once to create, because of its numerous functions, an interdependence among people of widely divergent positions, interests and skills.” (Randall 1965, 85). At Princeton University, Randall himself, Godfrey Winham, Hubert Howe, Tobias Robison, and Eric Regener—all of whom were interested in the use of the computer as a tool for sound generation and electronic composition— helped the historical musicologists Lewis Lockwood and Arthur Mendel, who were carrying out a computer-aided investigation of stylistic properties of the masses of Josquin Desprez. Lockwood (1970) described how he and Mendel had the 18 masses key-punched manually from the Smijers edition in a computer language called IML (“Intermediate Music Language”). This arduous task was carried out by paid students.3 Lockwood and Mendel were also provided with programs for proof-reading and playing the music thus stored. The higher aim, however, was to obtain answers to questions about musical 3. At the time, the punched card devised by Herman Hollerith (1860–1929) was still a common input medium. It was prepared by typing characters on a keypunch, which appeared as holes on the card. When scanned by a card-reading device, the patterns formed by these holes generated a series of electronic impulses, which could be processed by a computer.

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structure and style. Such questions were translated into a computer language for music information retrieval (MIR) that had been developed by Michael Kassler and Tobias Robison. According to Robison, this language enabled an investigation “fundamentally concerned with the occurrence of discrete surface phenomena in the music.” He explained that: [these] events, which might be certain chords, patterns of notes, rhythms, pitches and so on, all share the characteristic that they can easily be sought out and found by short sequences of MIR instructions. It is thus a simple matter to write a MIR program which will look . . . for these phenomena. The results of such searches can then be used in several ways. (Robison 1967, 127)4

For example, a typical question that could be handled by MIR was “to search a piece of music for all examples of a rising second followed by a rising third” (P. H. Smith 1967, 20). This question illustrates how the computer could help people establish contacts and discover similarities between their scholarly orientations. It was the kind of question that theorists of a more serial bent raised in connection with the music of Schoenberg, Berg, and Webern. However, it was equally relevant to an exploration of Josquin’s musical style, which should match Knud Jeppesen’s “handmade” study of Palestrina’s—as Lockwood said, “in a tiny fraction of the time and in a more flexible, more exhaustive, and more reliable manner.5 So, apart from speed and accuracy, the computer provided a sense of community among a differentiated group of music scholars. It also helped musicology to develop self-confidence. Musicology was still a very young academic discipline in the United States,6 but with respect to the use of the computer it was as old as other branches of the humanities. Music was an area of research considered 4. For another description of MIR, and its relation to other early input languages for music, see Kassler 1966. Kassler, a student of Babbitt, later developed a computer program carrying out certain tasks involved in a Schenkerian analysis. 5. Lockwood 1970, 24. Lockwood’s statement reflects the situation in 1965. He and Mendel were dealing with many other questions as well, including questions of musica ficta, modality, and text setting. Judging from later reports (Mendel 1969 and 1976), much of the initial optimism about the rewards of using computers had ebbed away by around 1970. 6. The years between 1930 and 1960 saw a rapid increase of activity in American musicology, but it took some time before it was recognized as an independent academic discipline. The first American chair in musicology was established in 1930 at Cornell University. Four years later the American Musicological Society was founded. In 1951, the AMS became a constituent member of the American Council of Learned Societies. According to Charles Seeger, “departments of music in colleges and universities and conservatories of music knew little or nothing of European musicology and were not friendly to attempts to introduce it [in the 1930s].” (Seeger 1969, ix). In his book The Place of Musicology (posthumously published in 1957), Manfred Bukofzer wrote that that “musicology . . . entered the American university by the back door, by way of established nonmusic departments,” and he characterized its recognition as an academic discipline as a “slow and somewhat roundabout process” (Bukofzer 1957, 3).

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both challenging and promising in this connection: it was considered challenging because it demanded a specific language to represent music in all its complexity; and promising, because, from the structuralist point of view that was en vogue at the time, it seemed to lend itself to formalization without losing information. Moreover, it was expected to yield numerous data that would otherwise be hard to retrieve in the same quantities. Music would thus allow computer technology to show off its great power. The involvement of data processing techniques contributed to the academic status of both musicology and music analysis; the ability to employ the computer counted as a measure of a discipline’s maturity. When summing up the advantages of the new technology, hardly anyone failed to mention the mental exercise it demanded. It was obvious that the computer could not just solve every problem right away. In order to be “understood” by it, a problem had to be broken down into a number of smaller questions. Each question had to be converted into an algorithm, a sequence of instructions. Of course, the range of these instructions was dependent upon the categories in which prior information—for example, a musical score, a literary text, the census data of a city, or a list of artifacts in an archeological site—was stored. This meant that data, if not already formulated for the computer, underwent a considerable transformation before an answer could come into view. Some denounced the strong influence that the computer—a mere tool, after all—exerted on the scholar’s perspective, narrowing it down to the measurable. Many others, however, welcomed the opportunity it offered to test their research questions and methods, and to meet its demands of exactitude and consistency, thus raising the standard of accuracy of scholarly research. Those who incorporated the new technology into their work stressed the methodological virtues of working with computers, even if they were aware of the dangers of scientistic reductionism. Lewis Lockwood was one of them: [T]he logical organization of computer investigation not only invites us but compels us to frame our small-scale questions in precisely defined terms, and thus to formulate our large-scale problems in ways more exact than seems likely to be the case if we were merely making tentative steps towards answers that in large measure appeared to be hopelessly beyond reach. (Lockwood 1970, 23)

The musicologist Jan LaRue, whose specialization was the study of musical styles, saw a new age dawning: May I recommend the computer to you as an instrument without human prejudices. It has its own prejudices, numerical and procedural. But these often act as stimulances and correctives, as healthy balances and supplements to human attitudes. With this new aid, the coming generation of musicologists should develop a style analysis that is comprehensive rather than selective, broad rather than personal, and rich in musical insight. (LaRue 1970, 197)

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According to these authors, the computer held a mirror up to the scholars, making them aware of their limitations. However, it also let them overcome these limitations. The computer served them as a role model. By letting it perform certain complex tasks, one hoped to gain a better understanding of these tasks, and, at the same time, of the processes in one’s own mind. For example, analysis programs would probe the consistency of common analytical methods. Likewise, composition programs, such as those developed by Lejaren A. Hiller and Leonard M. Isaacson, shed light on fundamental processes in existing music. Thus, it was thought, the computer would not only assist, but also educate its users. When we answer the question of what stimulated scholars to apply the computer around 1965, we should not overlook the fact that the computer industry had taken an interest in the evolving “market” of the humanities. It provided funds and equipment for scholars who worked with computers. The International Business Machines Corporation (IBM) was frequently mentioned on sponsor lists of conferences and research projects, together with universities, and institutions like the Rockefeller Foundation and the National Science Foundation. Apart from supporting current research projects, it tried to increase the awareness of data processing techniques at universities. IBM employed the musicologist Edmund A. Bowles as a manager of professional services in its Department of University Relations. In this capacity, Bowles—himself a specialist in medieval music—developed contacts with scholars in the United States (whose equal he was) and acquainted them with the computer as a research tool. In 1964 and 1965, for example, Bowles organized a series of six regional conferences in the United States. The universities that hosted and sponsored these conferences received financial support from IBM, as Bowles wrote in his preface to a published selection of conference papers.7 He reported that some 1200 people had visited at least one of the conferences. This number, and the rapid succession of the conferences, testify to the intensity of IBM’s marketing: The Use of Computers in Humanistic Research (Rutgers, the State University of New Jersey; December 4, 1964) Computers for the Humanities? (Yale University; January 22–23, 1965) The Digital Computer as a Servant of Research in the Humanities and Arts (University of California, Los Angeles; April 15–16, 1965) Humanistic Studies and the Computer (Georgetown University, Washington DC; October 21–22, 1965) Computers and Research in the Humanities (Purdue University, October 29–30, 1965) 7. This collection was published by Prentice Hall in 1967, under the title Computers in Humanistic Research: Readings and Perspectives.

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The Impact of the Computer on the Humanities and the Arts (Boston University, November 4–5,1965) The company also supported conferences and symposia devoted to single topics. For example, it hosted a Literary Data Processing Conference, which was held on September 9–11, 1964, and a symposium on input languages for representing music, on May 10, 1965. And it did even more. It granted financial support to the journal Computers and the Humanities, founded in 1966,8 it enabled the American Council of Learned Societies (ACLS) and New York University to establish a National Center for Bibliographic Data Processing in the Humanities,9 and it sponsored fellowships that the ACLS provided for computer-oriented research in the humanities.10 Obviously, research projects involving computers were beginning to generate funds at that time, and many opportunities were offered to share the results with others. Apart from the Josquin project in Princeton, several other computer-aided ventures in the field of music attracted attention: stylistic investigations of the 16th-century Italian frottola repertoire (Lincoln 1967), the 16th-century chanson repertoire (Bernstein and Olive 1969), and Joseph Haydn’s symphonies (LaRue 1967); the bibliographic RILM project conducted by Barry Brook (Répertoire International de Littérature Musicale; Brook 1967), the generation of a complete list of all-interval twelve-tone series (Eimert 1964, Bauer-Mengelberg and Ferentz 1965), and last, but not least, a project concerned with the structure of non-serial atonal music. The initiator of that project was Allen Forte.

Computer Language and the Structure of Atonal Music Forte’s project comprised three levels of activity: encoding (representing a musical score in machine-readable language), parsing (dividing the encoded music into smaller components), and interpreting (determining relations between those components). For encoding, the first level of activity, Forte used a language that had been developed by the multi-talented Stefan Bauer-Mengelberg for the purpose of automatic music printing. Bauer-Mengelberg, a mathematician, lawyer, and conductor, worked as a programmer for IBM, and was Leonard

8. In the first issue’s editorial (“Prospect”), the contribution of “the major computer manufacturer” is gratefully acknowledged (p. 1). 9. “News and Notes,” in Computers and the Humanities 1 (1966–67), 11. 10. For a statement on the ACLS Program for Computer Studies in the Humanities, see Lieb 1966. Allen Forte held a fellowship in the ACLS when working on a score-reading program during the academic year 1965–66 (see further below). His report includes a footnote to the effect that this fellowship was sponsored by IBM. (Forte 1966, 363)

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Sehr langsam (

= ca 50)

espress. 3

mit Dämpfer

Geige

Klavier

5

col legno weich gezogen

sempre

äußerst zart 3

3

rit. pizz.

3

3

Example 8.1a. Anton Webern, Piece for Violin and Piano, Op. 7, no. 1. © 1922 by Universal Edition A.G., Wien/UE 6642.

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Example 8.1b. Forte’s representation of Webern’s score in Ford-Columbia (DARMS). Each graphic symbol in the score is represented by one or more items from the character set of an IBM Type 29 keypunch or typewriter. Numerical values indicate positions on the staff. (Odd numbers indicate lines, even numbers spaces; the numbers “21” and “29” refer to the first and fifth lines, respectively.) The music is arranged per staff from top to bottom of a page in the score. Allen Forte, “A Program for the Analytic Reading of Scores,” Journal of Music Theory 10/2 (1966): 334. Used by permission of Duke University Press and Yale University.

Bernstein’s assistant at the New York Philharmonic before he became president of Mannes College of Music in New York in 1966. His “Ford-Columbia” input language—also known as DARMS (“Digital Alternative Representation of Music Scores”)—was very successful. It was straightforward with respect to standard music notation, using an ordinary typewriter character set; furthermore, it could cope with complex music, and was adaptable to various purposes.11 Example 8.1a shows Webern’s first Piece for Violin and Piano, Op. 7. FordColumbia made such a piece of music appear as a single character string (ex. 8.1b; Forte 1966, 334). In this form it was ready for a computer program written in SNOBOL (an “acronym” for “String-Oriented Symbolic Language”). This language executed operations on free character strings. For example, it could divide a string, or paste strings together; it could look for and take out substrings, transpose them, substitute them, erase them, etc. Allen Forte was an expert user of this language. He published a manual for SNOBOL3 in 1967 (Forte 1967b). In his own project, it served as the basis for a score-reading program, the purpose of which was to extract segments from music encoded by keypunch in the Ford-Columbia language. This was the second level of activity in this project. 11.

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See Bauer-Mengelberg 1970 for a brief introduction.

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(a)

‘Primary segments’ :

PSGT 1

[]

(Cello)

PSGT 2

[]

(piano RH)

PSGT 3

[]

(piano LH)

(b)

‘secondary segments’ 1-5:

PSGT 1 SSGT 3 SSGT 2

SSGT 1

PSGT 2

PSGT 1 SSGT 5 SSGT 4 PSGT 2

(c)

‘secondary segments’ 5-8:

PSGT 1 SSGT 8 SSGT 7

SSGT 6

PSGT 3

PSGT 1 SSGT 5 PSGT 3

(d)

‘secondary segments’ 9-12:

PSGT 2 SSGT 10 PSGT 3

SSGT 11 SSGT 9

SSGT 12 PSGT 2 PSGT 3

Example 8.2. Anton Webern, Three Short Pieces for Cello and Piano, Op. 11, no. 1, mm. 2–3. A segmentation following the rules of Forte’s score-reading program. The example shows secondary segments (SSGT) resulting from the interaction between three different pairs of primary segments (PSGT). © 1924, 1952 by Universal Edition A.G., Wien/UE 7577

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The way in which the score-reading program operated—Forte has described this in much detail (Forte 1966 and 1967a)—was constrained by the properties of SNOBOL. First, the instrumental or vocal parts had to be reassembled from the Ford-Columbia character string, which obscured them (as ex. 8.1b shows); new strings had to be created from substrings of the original character string. Second, to each code that implied a temporal position—i.e., a note or a rest—a position value had to be assigned, so as to enable the location of musical segments. So-called primary segments were extracted from character strings that represented single parts. They were defined as one-dimensional substrings delimited by rest codes. In other words, any succession of notes in one part beginning after a rest and continuing until the next rest counted as a primary segment. Example 8.2a shows the three primary segments extracted from measures 2–3 of the first of Anton Webern’s Three Short Pieces for Cello and Piano, Op. 11. Next, the program took all pairs of character strings, each member of which represented a single part, and scanned them for secondary segments, which resulted from the interaction between two contiguous primary segments. Each of the coinciding substrings of these primary segments, the union of these substrings, and the remaining substrings of each segment counted as a secondary segment (ex. 8.2b through 8.2d). Further programmed operations on the strings of primary and secondary segments resulted in configurations involving more than two parts. In the segmentation process, the pitch codes of the input language—a space code, or the combination of a space and an accidental code—were replaced by PCs, while other characters were deleted. In an output string, a segment like SSGT 9 in example 8.2d would take the following form: (P72*1*4*5*8*9 P80*7),

where “P72” and “P80” are position values with which the subsequent PCs are associated. The segmentation shown in example 8.2 differs greatly from the segmentation of the same excerpt in Forte 1970, which actually dates from 1965 (example 8.3; this example also includes the first measure of the piece). For my segmentation I have followed the protocols of the SNOBOL-based score-reading program, which was not available in 1965. Forte’s segmentation is probably hand-made and, as a consequence, involves some intuitive judgments. This may explain why it is the better one. For example, it makes a lot of sense to distinguish the cello’s B–B♭ motif (11,10), the melody in the right hand of the piano part (3,0,1,7), and the chord {4,5,8,9} as primary segments. What, by contrast, causes the segmentation of example 8.2 to be so awkward? The blame has to be placed on the Ford-Columbia input language. Flexible though it may have been in its application, it did not really lend itself to analytical purposes, especially not when the music had a piano part. Example 8.1b shows that Ford-Columbia “read” a score

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Example 8.3. Mm. 1–3 of Webern’s Opus 11, no. 1 as segmented by Forte in 1965. Barry S. Brook, ed. Musicology and the Computer. New York: City University of New York Press, 1970. Used by permission.

staff by staff. This meant that the left and right hands of a piano part were fed into the computer as separate entities. Example 8.2 is in strict compliance with this procedure. As a result, the G♯3 (8) of PSGT 2 (right hand) is unfortunately separated from the tones of PSGT 3 (left hand) with which it forms a chord. Of course, one can always rearrange a score before encoding, moving notes from one staff to another. As we shall see, however, Forte had reasons to refrain from such interventions. Having identified a large number of segments, Forte could proceed to the third level of activity, and trace relations between these segments. To this end, he had written a sequence of programs in another language, called MAD (for “Michigan Algorithm Decoder”). After determining the PC set of each segment, these programs would (in Forte’s own words): (1) determine the class to which each set belongs; (2) list and count all occurrences of each set-class represented; (3) compute, for each pair of set-class representatives, an index of order similarity; (4) determine the transpositioninversion relation for each pair of set-class representatives; (5) list, for each set-class represented, those classes which are in one of three defined similarity relations to it [R0, R1 and R2; see chapter 5] and which occur in the work being examined; (6) summarize in matrix format the set-complex structure of classes represented in the work; (7) accumulate and retrieve historical and other informal comments in natural language. (Forte 1966, 357)

This enumerative description maps out a large part of the terrain of PC set theory; several lines of development have departed from here. Some of these lines

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were short-lived, while others have continued, owing to the work of Forte himself and many of his students. “PC set,” “normal” or “prime form,” and “interval vector” have become household concepts for many music theorists, but in the mid-1960s they met a very specific need: the need for definitions of musical relations that a computer program could recognize. A comparison of two musical segments, such as those obtained by Forte’s score-reading program, now had to involve an algorithm, a series of reductive operations applied to both with a view to reveal possible correspondences between them.12 Forte seems to have worked on the three levels of his project in reverse order. In 1964 he presented his work on PC set relations, enabled by MAD (Forte 1964, 1970). The score-reading program was developed in 1965 and 1966 (see footnote 10). No doubt, the implementation of this program was the most problematic part of the project. First, it was apparently written with a very specific repertoire in mind: ultra-brief atonal compositions, a music of small gestures and evocative silences.13 It is difficult to imagine what the program would have done with long chains of chords and melody. Second, the score reading program applied parsing rules that were context-independent and utterly rigid. Obviously, the formulation of such rules was constrained by the properties of the computer language, but Forte regarded these constraints as salutary rather than restrictive. As we have seen, he was not the only humanities scholar who greeted data processing techniques in the 1960s, but he was one of those showing the fewest reservations. The cause to which he had devoted himself—the recognition of the structural integrity of the non-serial atonal repertoire—demanded a fresh outlook, an attitude unimpaired by traditional notions and personal biases. Forte believed that the computer could enforce such an attitude, and thus contribute to a proper understanding of the “revolutionary” music of Schoenberg, Berg, Webern, and others: [T]he choice of atonal music as the object of research . . . brings with it the need for an extensive revision of outlook. For example, conventional descriptive language (“melody,” “harmony,” “counterpoint”) is not very useful, and may even be a significant hindrance to new formulations, since it is oriented towards older music. When, in addition, problems of structural analysis are stated in terms accessible to computer programming, many conventions are set aside and many familiar concepts are rejected after serious scrutiny. (Forte 1966, 331)

12. For algorithms yielding normal and prime forms of PC sets, see chapter 4, pp. 102–6. For the algorithm on which the interval vector is based, see chapter 5, p. 132. 13. Forte’s published demonstrations involve music of Anton Webern: Piece for Violin and Piano, Op. 7, no. 1 (Forte 1966; see ex. 8.1), Short Piece for Cello and Piano, Op. 11, no. 1 (Forte 1970; an excerpt of this piece is shown in ex. 7.2), and one of the Bagatelles for String Quartet, Op. 9, no. 5 (Forte 1966).

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249

Added to this came a more general epistemological viewpoint. More than once, Forte has expressed the opinion that a musical analysis should have a clearly defined object, providing all the evidence that could possibly support the analysis. A valid statement about a musical work—a statement beyond a conjecture or an opinion—should restrict itself to what verifiably constitutes that work: the score. When Forte spoke about the “completeness,” “consistency,” and “testability” of an analysis, as he did as late as 1985, the score was still his sole reference point (Forte 1985, 42). This view of the object of a musical analysis—which was informed by positivist epistemology14—fit with his efforts to computerize the analytical process: The musical score . . . constitutes a complete system of graphic signs and properly presented for computer input, may be analyzed by a program as a logical image of the unfolding musical events that make up the composition. (Forte 1966, 332)

Can we disentangle PC set theory from the history of computer applications in the humanities? Can we use it without reference to that history? Of course, PC set theory has evolved. It has left the rigidities of its initial “technological” phase behind. In The Structure of Atonal Music, for example, a primary segment is defined loosely as “a configuration that is isolated as a unit by conventional means.” Forte here refers not only to units demarcated by rest codes—as he did in 1966—but also to “rhythmically distinct melodic figures” and “beamed groups.” Secondary segments are now called “composite segments,” and the way in which they are formed is left more open as well (Forte 1973, 83). However, Forte still suggests a systematic and easily programmable procedure to obtain less obvious, but possibly significant segments from a score: sequential imbrication, or overlapping. (This is the procedure introduced as an aural exercise by Michael Friedmann in 1990: “notes 1,2,3,4, then 2,3,4,5, then 3,4,5,6, and so on.”) Joseph Straus’s advice “to poke around in a piece” to obtain meaningful segments is a long way from such attempts to formalize the analytical process. Notwithstanding its greater flexibility now, PC set theory bears the marks of its origin as a computer project. To begin with, the similarity and set-complex relations (5 and 6 in Forte’s enumeration; see above, p. 247) had never been defined before the start of this project. And relations that had already made their appearance in the analytical literature—such as the (unordered) transposition-inversion relation (4 in Forte’s enumeration)—became more important, since the computer could detect them so well. Furthermore, PC set theory presupposes a practice of close reading of the score. Although this practice already existed before computer programs were applied to music analysis—we have seen how much this meant to 14. For the relationship between atonal music theory and positivistic philosophy, see Davis 1995. More on this in the next section of this chapter.

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Schoenberg (see chapter 7, p. 222)—it was sanctioned, and reinforced, by the special conditions that the computer imposed on the study of music. All in all, PC set theory seems to have received considerable impetus from the computer. The reason why this is not mentioned in textbooks anymore is the role of music analysis as it has been sketched in chapter 6, more specifically its kinship to performance. A successful performance makes us forget about its own contingent nature. An analysis can also successfully represent a musical work of art, but by transcending a mere exposition of musical techniques. What is more, it has to transcend its own history. Therefore, it is understandable why the canonical textbooks of PC set theory and PC set analysis are silent on the naive reception of computer technology in the 1960s. A discussion of this would run counter to the purpose of these textbooks: to present an authoritative body of music theory.

B: Contemporary Music, Science, and Education Babbitt’s Positivism The existence of a “distinctly American version of music theory,” presenting itself as a “legitimate academic discipline rather than as a service discipline for conservatories and university music schools” (McCreless 1996, paragraph 5), is generally acknowledged. And there is no doubt that PC set theory is a product of music theory’s adaptation to an academic environment under the sway of positivistic philosophy, as James Davis (1995) has shown. Positivism dominated the American universities of the Ivy League in the 1950s and 1960s. Davis sees the following tendencies as characteristic of this philosophical orientation: [A] rigid empiricism which leads to an emulation of science in both methodology and terminology; a rejection or avoidance of metaphysics; the use of linguistic and logical analysis; and, partly as result of these characteristics, the avoidance of subjective interpretation in the process of analysis. (Davis 1995, 508)

Milton Babbitt—who, while he was working on twelve-tone theory, laid much of the groundwork for PC set theory—was affiliated with Princeton University, where one of the European proponents of logical positivism, Carl Hempel, taught philosophy of science. Inspired by Hempel, Rudolf Carnap,15 and others, 15. Rudolf Carnap was one of the founding members of the famous Viennese Circle in the 1920s, but Babbitt liked to see him as a member of a “Viennese triangle,” together with Schoenberg and Schenker (Babbitt 1987, 17 and 1999). Carnap fled the Nazi terror and settled in the United States (like Hempel, who came from Berlin), where he was appointed at the University of Chicago. Later, he spent two years in Princeton, where Babbitt had the opportunity to meet him (Babbitt 1999, 47).

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Babbitt wrote a few methodological papers (“Past and Present Concepts of the Nature and Limits of Music,” 1961;16 “The Structure and Function of Musical Theory,” 1965; “Contemporary Music Composition and Music Theory as Contemporary Intellectual History,” 1972), in which he argued that any music theory, while bearing on sense-experience, should comply with the requirements of a formal theory. He defined such a theory as “a deductively interrelated system of laws, statable as a connected set of axioms, definitions, and theorems, the proofs of which are derived by an appropriate logic” (Babbitt 1961b: 399). Babbitt was an adherent of the doctrine of the unity of science, a doctrine to the effect that all sciences are representatives of a single universal science. Although this “universal science” was first and foremost physics, Babbitt wanted to subject statements about music to its regime as well: [T]here is but one kind of language, one kind of method for the verbal formulation of “concepts” and the verbal analysis of such formulations: “scientific language” and “scientific” method. Without even engaging oneself in disposing of that easily disposable, if persistent, dichotomy of “arts” and “sciences” (or, relatedly, “humanities” and “sciences”)—that historical remnant of colloquial distinction—it only need be insisted here that our concern is not whether music has been, is, can be, will be, or should be a “science,” whatever that may be assumed to mean, but simply that statements about music must conform to those verbal and methodological requirements which attend the possibility of meaningful discourse in any domain.17 (Babbitt 1961b, 398)

According to Babbitt, theories of music had failed to meet these requirements in the past. More specifically, they had failed to define properly their empirical domains (the musical repertoires to which they applied), and to choose properly their primitives (Babbitt 1961b, 399). For example, harmonic theory had never succeeded in making a reasonable case for the overtone series as the ultimate cause of the “triadic-tonal system.” It is obvious that the first five overtones form a structure corresponding to that of a major triad—a potential tonic—but how could one infer from this that chords tend to appear in particular successions, or that minor triads can assume the role of tonic? Babbitt’s criticism did not target the assumption of a “natural” cause for music, but rather the inability to base a theory without gaps and internal contradictions on this axiom (Babbitt 1965a, 57–58). 16. This paper was delivered at the Eighth Congress of the International Musicological Society in New York. The title was not Babbitt’s own; it had been given to the session in which he spoke. Another paper was delivered by the German musicologist Heinrich Hüschen. 17. This quotation shows Carnap’s influence on Babbitt’s thinking. It was a life-long enterprise of Carnap to construct a meta-language of science by way of an analysis of the concepts, laws, and theories of the special sciences. (Der logische Aufbau der Welt, 1928; Testability and Meaning, 1936; The Methodological Character of Theoretical Concepts, 1956.)

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Babbitt made an exception for Schenkerian theory, which he helped to gain academic credibility at a time when it was not yet commonly known among music scholars in the United States. In one of his few published statements about Schenkerian theory—a review of Felix Salzer’s Structural Hearing—he argued that the Urlinie (actually the Ursatz) was an integral part of the empirical domain of this theory, while at the same time “it was completely acceptable as an axiomatic statement (not necessarily the axiomatic statement) of the dynamic nature of structural tonality” (Babbitt 1952, 260). And he referred to his own twelvetone theory as a music theory evolving logically from two very simple perceptual assumptions: “the capacity to perceive pitch-class identity and non-identity, and interval-class identity and non-identity” (Babbitt 1961b, 402). These assumptions enabled the reduction of the chromatic universe (PITCH) to a set of twelve PCs (PITCHCLASS). Furthermore, they enabled the representation of musical transformations by operations on this finite set of numbers. The consequences of these operations, in terms of the variance and invariance of PCs and PICs, could be derived from finite group theory. For example, one could deduce from a PC set how many of its elements remained invariant under a transposition operation. The number of PCs held in common by a PC set A and its transposition Tn(A) is equal to the frequency of the PIC of transposition (n) in that set.18 Babbitt (1960) extended this rule to a PC set A contained in a twelve-tone series S, showing that for A to recur in the Tn-form of this series, S should also include B = T12−n(A). He cited the series of Schoenberg’s Third String Quartet, Op. 30, in which each of the six disjunct dyads maps onto another, or onto itself, under T6 (ex. 8.4a and 8.4b). While enhancing the scientific image of music theory, this also proved rewarding material for textbooks. In Serial Composition and Atonality, George Perle identified two more operations causing the same dyads to interchange (T3I and T9I; Perle 1962, 129). Joel Lester used the Schoenberg example in an introductory text for undergraduate students on analytical approaches to twentieth-century music—a text drawing on several analyses by Babbitt. With reference to the beginning of the second movement of Schoenberg’s quartet (ex. 8.4c), Lester wrote the following assignment: [The] intervals between first and second violin are pairs of pitch-classes from P0 . . . Write out series forms P0, P6 [T6], I3 [T3I], and I9 [T9I]. What happens to the intervals in P0 in these series-forms? What is there in the construction of the series that allows these relationships to arise? (Lester 1989, 204)

This provided students with a basic awareness of the relation between the structure of a series and the effects of particular operations performed on that series—an awareness, in other words, of a system expressing itself in a different 18.

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See chapter 3, note 21, and chapter 6, note 25.

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9

PIC:

6

5

5

3



253

6

Example 8.4a. The series of Schoenberg’s Third String Quartet, Op. 30, divided in disjunct dyads.

9

PIC:

6

5

5

3

6

Example 8.4b. The T6-form of the same series. “[T]he interval succession determined by disjunct dyads is 9,6,5,5,3,6 [in PICs]. The interval between the identical 5’s is 6, between the complementary intervals 9 and 3 is also 6, and interval 6 is its own complement. So, under the application of t = 6, the pitch content of disjunct dyads is preserved and ex. 2 [here (b)] can thus be regarded as a permutation of the dyads of ex. 1 [here (a)].” (Milton Babbitt’s commentary; Babbitt 1960, 251) Adagio (

Vln. I

= 60) 2

3

4

Vln. II

Va.

Example 8.4c. The beginning of the second movement of Schoenberg’s quartet. Used by permission of Belmont Music Publishers, Pacific Palisades, CA 90272.

way in each new twelve-tone composition. The excerpts from Schoenberg’s Opus 30 thus came to serve a traditional pedagogical purpose. Originally, however, this material was supposed to provide ammunition for the professionalization, in an academic environment, of music theory. It was a means for composers and music theorists to be at the forefront of scientific research.

Anxieties about Musical Discourse Babbitt’s deep involvement in the positivistic philosophy of science can be viewed from a historical or a contemporary angle. One may point to the long history of the relationship between music theory and science, or refer to ideas and experiences that gave Babbitt the incentive to extend this philosophy to music theory.

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What do we learn from the first approach? As one of the seven “Liberal Arts,” music was an established part of the late-Medieval university curriculum. It was concerned with numbers and their ratios representing harmonic intervals. These were believed to contain the key to the order of the universe. Therefore, music was a truly scientific discipline (in the Aristotelian sense), transcending what was only practically useful, and providing a model for understanding the physical world at large. As Penelope Gouk (1999, 2002) has shown, music continued to play this role far into the seventeenth century—for example for Robert Fludd, Johannes Kepler, and even, to a lesser extent, for Isaac Newton. Babbitt, one might say, has renewed music theory’s old claim to the status of a science. He has attempted to reposition music theory in the vanguard of intellectual life. This is certainly a viable perspective from which to view Babbitt’s mission. Needless to say, Babbitt was a composer and a teacher of composition, not a twentiethcentury theoros in splendid isolation from musical practice. And he was scornful of those who represented the relationship between music and science in the past, like Marin Mersenne or Jean-Philippe Rameau.19 However, his own invocation of a source of authority beyond the canon of Western art music—a source of authority that not only constrains and validates musical thought, but also brings it into alignment with other fields of human endeavor—can be seen as an echo of this relationship. It reflects the concern with ultimate foundational principles that continued to exist in music theory when, from the eighteenth century onwards, it became more of an aesthetic endeavor than a scientific one.20 Babbitt spoke out so strongly against Mersenne and Rameau—his seniors by 328 and 233 years respectively—because, in his experience, the authority still accorded their theoretical works frustrated the advancement of contemporary music, and impeded the recognition of music theory as a contemporary intellectual discipline. It annoyed him beyond measure that “music in the form in which it would appear pertinently in the documentation of the intellectual activity of our time, as discourse on music by competent professionals, has not been admitted to membership in that activity” (Babbitt 1972, 157).

19. Babbitt had no time for Mersenne and Rameau. He called Rameau’s Génération harmonique “a collection of usually unsatisfactorily formulated definitions, unconnected protocol sentences containing these defined terms, and superfluous insular constructs.” And he blamed Mersenne for justifying the rule of the number six (the correspondence between the interval content of the major triad and the first six divisions of the vibrating string) by the number of planets that were known at the time: “Beyond the intimations of the cosmic scope and affinities of music, there is the implication that certain classes of objects hierarchically ‘justify’ others, and the pressure of the still persistent numerological fallacy (the assumption that two different exemplifications of the same number class therefore possess other properties in common).” (Babbitt 1961b, 399, 400) 20. For example, acoustics provided backing for the harmonic theories of Riemann (1880) and Hindemith (1937), and psychology for Kurth’s dissection of Romantic harmony (1923). Krenek (1937) found support for a fundamental revision of music theory in Hilbertian axiomatics.

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The role of the “competent professional” is crucial, and a recurring topic in Babbitt’s writings. In his view, musical discourse should be limited to those with musical competence. Showing a lack of such competence made one’s opinion irrelevant. For this excessively protective attitude, Babbitt took another cue from the sciences: the ongoing process of specialization. In the paper that earned him the widely held reputation of an “alienated modernist”21—“Who Cares if You Listen?,” first published in High Fidelity in 195822—he wrote: The time has passed when the normally well-educated man without special preparation could understand the most advanced work in, for example, mathematics, philosophy, and physics. Advanced music, to the extent that it reflects the knowledge and originality of the informed composer, scarcely can be expected to appear more intelligible than these arts and sciences to the person whose musical education has been even less extensive than his background in other fields. (Babbitt 1958, 39)

Specialized knowledge was a measure of the dignity of music. We can see how this fit with Babbitt’s concerns as a composer, a point to be discussed below. But what kind of knowledge would regain the musical specialist the communal respect he once had earned on account of the “universality” of his insights? How could music theory be recognized as a modern intellectual discipline if its practitioners were not answerable to a community larger than their own? On the one hand, it was necessary for Babbitt to disentangle himself from the foundationalism inherent in Western music theory, and clear the way for musical experimentation: [T]he informed musician . . . has been obliged to recognize the possibility, and actuality, of alternatives to what were once regarded as musical absolutes. He lives no longer in a unitary musical universe of “common practice,” but in a variety of universes of diverse practice. (Babbitt 1958, 38) 21. A characterization attributed to Joseph Kerman by Martin Brody (1993, 180). Kerman had written about the “sense of modernist alienation” that Babbitt’s utterances, like those of Schoenberg and Adorno, betrayed (Kerman 1985, 101). 22. Apart from its being first published in a periodical for music listeners, the article was reprinted in The American Composer Speaks (1966), edited by Gilbert Chase, and in Contemporary Composers on Contemporary Music (1967), edited by Elliott Schwarz and Barney Childs. Babbitt and his advocates later stressed that the provocative title was not his own, but was imposed by the editor. However, Babbitt did not change it when the article re-appeared in the above-mentioned anthologies. It was only in a much later reprint, in Richard Kostelanetz’s Esthetics Contemporary (1978), that the original title was used (“The Composer as Specialist”). Whatever its origin, the catchphrase “Who Cares if You Listen?” has exerted too great an influence over the reception of Babbitt’s text to remove it from history. And, in all fairness, the gist of Babbitt’s argument was hardly less provocative. Babbitt urged the necessity of “a total, resolute, and voluntary withdrawal from this public world to one of private performance and electronic media, with its very real possibility of complete elimination of the public and social aspects of musical composition.” (Babbitt 1958, 126)

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On the other hand, he had to create a new strand of foundationalism to warrant a meaningful discourse on music. Martin Brody, in his article “‘Music for the Masses’: Milton Babbitt’s Cold War Music Theory” (published in the Musical Quarterly in 1993), has traced Babbitt’s insistence on the use of “scientific language” back to this dilemma: Babbitt did not want to invoke absolute criteria for musical practices or compositional techniques as such. He regarded them as constrained, not by any doctrine on the nature of music, but by rules of discourse. Therefore, Brody speaks of a “metatheory,” a theory not of music itself, but of its conceptualization. No a priori limits were set to the range of musical invention, but statements that characterized, explained, justified, or criticized music were to meet absolute criteria of scientific rigor. Thus, in the words of Brody, Babbitt managed “to refashion a conservative orthodoxy out of a radical, if anxious, acknowledgement of cultural relativism” (Brody 1993, 167). How far this acknowledgement really went, and whether it could hold out in spite of the “conservative orthodoxy,” is a moot point. How flexible was Babbitt’s metatheory? Could it embrace “a variety of musical universes,” or was it mostly pitched toward his own brand of music? We might ask with Brody why Babbitt imposed such stringent rules on what should be “an open conversation among the participants of a musical culture.” Additionally, we might ask why the forbidding complexity of Babbitt’s own writing should be the corollary of the application of these rules—a question that Brody does not raise in his article. These questions shift the focus away from Babbitt’s place in intellectual history—from his defining the terms for a new rapprochement between music and science—toward his cultural context. This, then, is the second approach to an interpretation of Babbitt’s scientism, an approach that may help to iron out some of the remaining incongruities. For one thing, it is quite amazing that language has been so central to the thought of this composer. Its importance as the last basis of communal discourse in a fragmented musical world does not explain the fierceness with which Babbitt has called for its purification, a fierceness that must have made it difficult for many readers to take his side. Joseph Kerman (1985, 101) perceived “an undertone of distress, even rage” in Babbitt’s writings, “erupting into repeated assaults and innuendos directed against various predictable targets.” According to Brody, these emotions reflected Babbitt’s deep-seated aversion to nationalism, proletarianism, and mass culture as exemplified by the cultural politics of the Soviet-Union—a cultural politics that had outspoken proponents in the United States in Babbitt’s formative years, the 1930s.23 However, we must also take into

23. See Brody 1993. The 1930s were a period in the cultural history of the United States in which notions like “high art” or “autonomous art,” etc., came under great pressure. First, there was much dissatisfaction with America’s cultural dependence on foreign nations. Composers like Aaron Copland argued a turn to the country’s own musical

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account Babbitt’s experiences in the public domain as a composer of demanding music with no roots in common practice.24 For such a composer, language was much more than just a medium of conversation. It was something that could make him or break him: Perhaps there have been eras in the musical past when discourse about music was not a primary factor in determining what was performed, published, therefore disseminated, and—therefore—composed, . . . when—indeed—the compositional situation was such as not to require that knowing composers make fundamental choices and decisions that require eventual verbal formulation, clarification, and—to an important extent—resolution. But the problems of our time certainly cannot be expressed in or discussed in what has passed generally for the language of musical discourse, that language in which the incorrigible personal statement is granted the grammatical form of an attributive proposition, and in which negation—therefore—does not produce a contradiction; that wonderful language which permits anything to be said, and nothing to be communicated. (Babbitt 1965a, 50)

Babbitt felt his own music, the music of his students, and of the composers whom he admired—like Schoenberg, Webern, Stravinsky, Varèse, Sessions, or Dallapiccola—to be extremely vulnerable to misconceptions. And the effect of such misconceptions could be disastrous in a cultural society in which contemporary art music was weakly represented. Since this music obviously did not “speak for itself,” language could either be a threat to its survival or rescue it from extinction. This may have motivated the density and armored rhetoric of Babbitt’s written language—a language that does not comfort, but overpowers; that does not provide access to music, but rather surrounds it with a barricade of abstractions.

vernaculars, which involved a rejection of the imported European “grand tradition” of art music. Second, the economic depression following 1929 was a serious blow to the institutions that had been fostering Western art music, such as orchestras and opera companies. Artists became reinvolved with social issues (Alexander 1980, 152ff.). Third, mention should be made of the Popular Front, the broad coalition of working-class and middleclass parties against Fascism that was founded in 1935 under the auspices of the Moscow Comintern. For some time, the American branch of this movement successfully pursued a cultural politics that emphasized the national history and traditions of America. (Alexander 1980, 192ff, and Guilbault 1983, 17ff.) 24. These experiences are recalled in papers (Babbitt 1987, 1988, 1989) and interviews (especially Duckworth 1999). Sources of frustration include the economics of performance, the uncompromising rehearsal schedules of orchestras, the level of journalism, the lack of publishing opportunities, many fruitless attempts to apply for a Guggenheim Fellowship (eventually awarded to Babbitt in 1960), and the lack of interest on the part of the general public.

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The University as a Refuge? Whereas Babbitt’s methodology served to delimit what counted as knowledge about music, his language may have restricted the number of those involved in the production and the assessment of such knowledge—thus creating a caste of specialists. However, it could not do this without the backing of an institution to which the provision of knowledge was already assigned—a much-respected and essentially meritocratic institution with a highly developed “training and certification machinery” (Jencks and Riesman 1968, 45). Babbitt’s view of the university as a “home for the ‘complex,’ ‘difficult,’ and ‘problematical’ in music” (Babbitt 1958, 126), an environment at some distance from normal society, in which market and consumer pressures were not all-pervading, indicates that his theoretical writings—with their overt appeal to a learned audience beyond the discipline of music—were a call for protection. In any case, he has never made a secret of his affection for this institution, which he regarded as the modern source of artistic patronage: Perhaps, then, I should begin by bringing you the latest tidings from our sector: serious musical composition is still alive and well, and living in our universities, including my own. Or, in the interest of historical subjectivity, I could with equivalent candor report that serious musical composition is fighting for its life, is well on its way to extinction, and is still breathing only in a number of well-heeled nursing homes for the musically self-indulgent, of which my university is fortunately one. (Babbitt 1972, 152) [The] survival [of serious music] seems unlikely when the conditions necessary for that survival are so seriously threatened. These conditions are the corporal survival of the composer in his role as a composer, then the survival of his creations in some kind of communicable, permanent, and readable form, and finally, perhaps above all, the survival of the university in a role which universities seem less and less able or willing to assume: that of the mightiest of fortresses against the overwhelming, outnumbering forces . . . of anti-intellectualism, cultural populism, and passing fashion. (Babbitt 1987, 163)

This raises the question of how society has taken care of composers—or not. There have been many complaints about the unwillingness of concert agencies, orchestras, publishing houses, record companies, and broadcasting corporations in the United States to bring the work of living American composers of “serious” music into the public realm. Babbitt was not the only one to seek refuge in the university. In 1965, a society was formed to serve the interests of composers whose public musical activities—such as performance, publication, and professional discussion—were mostly confined to the university: the American Society of University Composers. Its founding members were unanimous in their rejection of the working conditions of composers in the public domain:

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[An] environment where music is regarded as entertainment, where professional standards are set by non-professionals, and where writing about music is dominated by a belief in amateurism, is inadequate to our professional requirements.25

Babbitt was among these founding members, as were several of his closest allies, such as Benjamin Boretz and Donald Martino. But the list also included the names of other composers, for example Norman Dello Joio, Otto Luening, and Gunther Schuller; concern for composition as an art was shared by many people. Even a music historian like Nicholas Tawa—a known critic of art for art’s sake, and an avowed adversary of Babbitt’s “self-perpetuating circle of serialists”—observed that “the rejection of contemporary American composers and their works reached epidemic proportions in the three decades after World War II,” and that the composer was not the only culprit of this development: [T]he United States, in social, economical, political, and cultural ways, has managed to defeat its native art composers by imposing conditions on them that are impossible to meet. Among these conditions are the equating of artistic value with a huge audience, large monetary profits, and only amusement rather than also the enlightenment of listeners. (Tawa 1995, 179, ix, x)

Between 1959 and 1970, the programs of American symphony orchestras were surveyed annually by Broadcast Music, Inc. (BMI) and the American Symphony Orchestra League. Oliver Daniel, who was responsible for the publication of these surveys, aired a scathing judgment on the symphonic repertoire in the United States: No other country so ignores the creative efforts of its own citizens. And nowhere is the generation gap more noticeable. Because so little attention is given to the work of our young composers, they are turning toward smaller instrumental combinations and electronic instruments.26

Still, we must keep a sense of perspective about this. It would be going too far to say that American contemporary art music has virtually been banned from American cultural life—a point so persistently hammered home by Babbitt. It has received support through intertwined channels of public and private funding, and it has benefited from the work of devoted organizations such as the Composers’ Forum (founded 1935), the American Music Center (founded 1939), and the American Composers Orchestra (founded 1977). One problem, however, is that the efforts to provide footholds for American composers

25. 26.

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Quoted from an advertisement in Perspectives of New Music 4/1 (fall–winter, 1965). Quoted in Hart 1973, 417.

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of contemporary art music have had no self-sustaining effect. In other words, for these composers to reach an audience in the public arena, the combination of special funds and “missionary” activity has remained a sine qua non. Another problem is that this combination usually does not bring advantage to everyone. Funding, commissioning, and honoring can only be selective, leaving some composers “in the cold.” Either you should contribute to the goal of the funding agency, or fall within its scope. Substantial contributions to the dissemination of new music have been made by the Rockefeller and Ford Foundations (established in 1913 and 1936, respectively). These philanthropic agencies have helped to launch large projects, such as the “Composers in Public Schools” Program (Ford Foundation, 1959–69) and the “Orchestra Residencies” Program (Rockefeller Foundation, 1982–92). They have also channeled funds to performance and composer service organizations, so as to broaden the scope for contemporary art music (Wiprud 2000, 8). In other words, they have invested in the building of a community receptive to musical innovation—an approach in keeping with the humanitarian ideals upon which they were founded. Individual fellowships for scholars and artists (including composers) have been provided by the John Simon Guggenheim Memorial Foundation (established in 1925), but these fellowships have tended to fall into the hands of the already successful.27 Federal support comes from the National Endowment for the Arts, an organ of the National Foundation on the Arts and the Humanities, which was established by the United States Congress in 1965. Its task was to spread funds over the full range of native activity in the arts. Regarding music, it has focused on organizations and institutions, rather than individuals, especially in the sphere of performance.28 The flow of funds to new music has sometimes followed a tortuous course. For example, Babbitt’s Philomel, for soprano and tape (1964), resulted from a commission of the American Music Center, which in turn was supported by the Ford Foundation; and his Concerto for Piano and Orchestra (No. 1, 1985) was written for the American Composers Orchestra, which had received a grant from Francis Goelet—a New York-based real estate developer, and one of America’s private patrons of new music (Wiprud 2000, 3, 13).

27. They were intended for “scholars and artists of proven abilities” from the beginning. Senator Simon Guggenheim’s second Letter of Gift (June 7, 1929) stated so explicitly. 28. Such initiatives are listed in The National Endowment for the Arts 1965–2000: A Brief Chronology of Federal Support for the Arts (2000). Examples include the Audience Development Project of 1967 (“to fund presenters of local concert series for young or unknown artists,” 14–15), the Opera-Musical Theater Program launched in 1979 (“to help broaden the concept of music theater and to make this art available to a larger audience,” 29), and the Meet the Composer Program of the same year (“to help young artists work with composers in residence,” 30).

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In view of the supply of funds from government agencies and non-profit organizations—which are eligible for tax-exempt status in the United States29—we might ask to what extent Babbitt’s doom-laden perspective on the condition of serious contemporary music outside the university was justified. Babbitt spoke warmly of an organization that was established in 1952 by the German-born wine importer Paul Fromm: the Fromm Music Foundation. He considered this a rare example of a foundation “devoted to the contemporary, mainly American, composer,” and gratefully endorsed its founder’s belief that music “was not a ‘performing art’ but a creative art.”30 This distinction is revealing. Apart from the implication that performing arts are “non-creative,” and therefore probably subservient to other arts, it shows that Babbitt regarded composition as the most vital aspect of a musical culture, the source of all other musical activity. That is why, in his view, the composer’s craft warranted special protection. Babbitt did not so much deny that American society provided opportunities for contemporary composers, as regret that these opportunities were not tailored to their own needs. Indeed, one can say that many funding programs have required an effort on the part of composers to find their own audience, or to establish their own contacts with performers, rather than letting them write music and taking care of everything else. This is not surprising; funders, understandably, want to ensure that commissioned works meet with a positive response. However, Babbitt expected a greater commitment from society to support the composer. This expectation was most probably strengthened by the successes of his European contemporaries—Pierre Boulez, Herbert Eimert, Karlheinz Stockhausen, and others with whom he shared an interest in serial control, mathematics, and science. These composers had contracts with major publishers, most notably Schott in Mainz and Universal Edition in Vienna.31 Some of them occupied key positions in musical life. For example, Eimert was on the staff of the West German State Radio (WDR, for Westdeutscher Rundfunk) in Cologne, which he made into an important platform for new music.32 The 29. See Section 501(c)(3) of the US Internal Revenue Code of 1954. 30. Babbitt 1988, 3, 4. The Fromm Foundation ran a commissioning program and sponsored the first, and also other, performances of the works that had resulted from this program. It brought composers and performers together in the Tanglewood Festival of Contemporary Music (from 1964), and it sponsored the Seminars in Advanced Musical Studies at Princeton University (1959, 1960). These seminars were intended as the American counterpart of the famous Darmstadt Summer Courses for New Music (see further below), and gave birth to the journal Perspectives of New Music in 1962. 31. Universal Edition had also issued the journal Die Reihe, which served as the principal forum for European serialism between 1955 and 1962. An American edition of this journal was published between 1958 and 1968. The lacuna it left in relation to American developments was filled by Perspectives of New Music (see note 30). For the competitive relationship between both journals, see Kerman 1985, 102, and Grant 2001, 2–3. 32. This forum was the famous “Musical Night Program,” which Eimert had launched in 1948. At the time, the radio station’s name was Nordwestdeutscher Rundfunk.

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WDR acted as a patron of Stockhausen, who worked at the electronic studio that Eimert and Werner Meyer-Eppler had founded there in 1951. Boulez’s contributions to the debate on cultural politics in France, together with his brilliant international career as a conductor, caused the French government to appoint him director of the musical department of the Beaubourg Center, a museum of modern art and a center of creativity bequeathed to the French people by President Georges Pompidou.33 This was in 1974. Under the leadership of Boulez, the musical department became known, independently, as the Institut de Recherche et de Coordination Acoustique/Musique (IRCAM), where musical research and experimentation took place on a permanent basis. Babbitt could only dream of being so prominently involved in a rich nation’s struggle for cultural prestige. As an American composer, he considered himself to be at a disadvantage, and at times he could barely contain his bitterness about this: The Continental composer may be obliged to court program directors of radio stations, but he also can expect to encounter even conductors—for example— who not only share his nationality and cultural background but have the authority and professional ambition to perform works other than the riff-raffish displays of evanescently flashy timbral patinas which make life easy for the American conductor, his performers, his audience, and his employers, and yet are to count as a generous gesture to contemporary music. The European composer can even expect to find a devoted publisher to relieve him of many of all those onerous and demeaning tasks and obligations which attend the preparation of a composition for performance and publication. Any American composer who anticipates the time when, if not just for eminence at least for advanced age, he may receive such treatment, is very likely to be disappointed. . . The Continental composer may guide his professional behavior with a view to enjoying the material rewards of genuine celebrity; the American composer of highly “cultivated” music, with no illusions as to what are the cultural heroes in a people’s cultural democracy, may attain bush-league celebrity, with many of the disadvantages of materially genuine celebrity and but few of the advantages. (Babbitt 1989, 110)

Babbitt was right insofar as the composers in Europe (whom he was hardly willing to consider his equals) seemed to have received unconditional support for their work. In the words of Paul Griffiths, “their sole responsibility was to 33. Boulez’s outspoken articles and interviews, and his successful organizational initiatives—such as the Domaine Musical series of concerts in Paris (1954–73)—had increased his celebrity. After the French government had refused to act upon his proposals for a reform of musical life, in 1966, Boulez turned his back on officialdom in his home country. The subsequent breach ended with his appointment at the Beaubourg Center in 1974.

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create” (Griffiths 1995, 36). However, Babbitt may have been unaware of the sometimes very special conditions that warranted this “unconditional” support. The history of the Darmstadt Summer Courses for New Music is most illustrative in this respect. This annual contemporary music festival, which featured concerts, lectures, and workshops, provided another major platform for the European avant-garde, but it had come about, in 1946, under auspices of the American occupying military government in Germany. The initiative of a local administrator, it had been adopted as part of a program to rebuild the German cultural infrastructure, and at the same time infuse it with American cultural and political values. American music played a much more prominent role in the Summer Courses than was suggested by their reputation as a vehicle for the group around Stockhausen and Boulez34—the “Darmstadt group” as it is called in Donald Grout’s widely read A History of Western Music (1980, 740). This throws an interesting sidelight both on the complaints about the marginal status this music held in the United States itself, and on the denigrating remarks about American composers with which Boulez shocked readers of the New York Times in 1969: Americans are jealous—I’m not sure if that’s the right word—thinking the Europeans are taking attention away from them. The Americans do operate under a severe handicap, of course; they have no strong personality in the field. If they were strong enough to establish their personality on the world, they would see that no national favoritism exists.

Boulez directed the bulk of his criticism at the university affiliations of American composers: European music is not connected with the university. There is no ivory castle for us . . . The university situation is incestuous. It is one big marriage in which the progeny deteriorates, like the progeny of old and noble families. The university musician is in a self-made ghetto, and what is worse, he likes it there.35 34. Amy Beal (2000) has shown that the director of the Summer Courses, the former music critic Wolfgang Steinecke, received financial and material support on request from the Music and Theater Branch of the American military government. And American officials, like the composer Everett Helm, had a hand in the organization of the Summer Courses: “In addition to supplying money . . ., Helm helped Steinecke contact American composers and obtained scores of American music from the United States. During the fifties he remained actively involved with the [Summer Courses] by lecturing frequently on American music.” (Beal 2000, 114) Milton Babbitt taught in Darmstadt no earlier than 1964. 35. Boulez interviewed by Joan Peyser for the New York Times (“A Fighter From Way Back”; March 9, 1969, 19). Boulez followed in the footsteps of Stravinsky, who had warned young composers, “Americans especially,” against university teaching in his conversations with Robert Craft (Stravinsky and Craft 1959, 132). Boulez later became a vigorous champion of the music of Elliott Carter.

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Were American composers a more appropriate target of the “ivory tower” accusation than their European counterparts? One cannot reproach somebody for seizing the best opportunities that are available in his profession. Boulez et al. no doubt have achieved a stronger presence on the international music scene— they have kept in touch with the public world to which Babbitt had bidden farewell in “Who Cares if You Listen”—but America’s cultural politics in Germany in the aftermath of the Second World War had been a factor in their rise to fame. We know that Babbitt “liked it there.” He felt at home at the university. He had positioned himself as someone who spoke the language of contemporary science. As such, he claimed to have a more thorough understanding of Schoenberg’s twelve-tone music than Schoenberg himself. Babbitt thought that Schoenberg “like many other great innovators, was not . . . entirely aware of the implications of his own discovery.”36 And he scorned the European serialists of his own generation particularly for what he saw as their methodological shortcomings, while leaving aesthetic issues unaddressed: Mathematics—or, more correctly, arithmetic—is used, not as a means of characterizing or discovering general systematic, pre-compositional relationships, but as compositional device, resulting in the most literal sort of “programme music” . . . The alleged “total organization” is achieved by applying dissimilar, essentially unrelated criteria of organization to each of the components, criteria often derived from outside the system, so that—for example—the rhythm is independent of and thus separable from the pitch structure. (Babbitt 1955, 55)

Babbitt and Boulez both saw the university as separate from the larger world. But they fundamentally disagreed in their assessments of this relation. Whereas Boulez offhandedly depicted the university as insulated—an “ivory castle,” a “ghetto”—Babbitt rather saw it as a “fortress.” For him, the university was not a greenhouse of impractical knowledge but an outpost of a developing society. Paradoxically, the idea that the university offered a relatively safe haven for innovative composers arose from the interaction of American universities with society at large. These universities have a long history of societal involvement.37 They are regarded as directly contributing to the progress and prosperity of the nation, which naturally involves a responsiveness to problems and questions 36. Babbitt 1950, 266. Babbitt expressed this view again many years later, in the lectures he delivered at the University of Wisconsin, Madison, in 1986: “Schoenberg never understood the generality of the principles involved.” (Babbitt 1987, 14) 37. Clark Kerr, president of the University of California from 1958 till 1967, and an acknowledged chronicler of American higher education, has described this history (Kerr 1995). It began with the Morrill Act of 1862, which authorized grants of public land for the establishment of universities contributing to the industrial and agricultural development of the United States. These “land grant” universities (e.g., Rutgers, Cornell University,

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raised from outside the groves of academe. From early on, they have established or accommodated professional schools, so as to ensure the transmission of new knowledge to the applied professions. Of course, this utilitarian image of the university has not gone unchallenged. There has also been the image of an institution devoted to the disinterested pursuit of truth. For example, Allan Bloom thought of the university as a vital counterbalance to the regime of public opinion in a modern democracy.38 But the disappointment of this classic man of letters with the way in which the university had developed in the United States— especially with its growing neglect of the canon of Western literature and intellectual thought—is a natural result of its relative openness to influences from outside the campus. This openness has long distinguished American universities from European ones.39 In the words of Francisco Ramirez, a specialist in comparative higher education, “American universities are more likely to historically develop as universities embedded in civil society while European ones are more likely to be buffered from society and linked to the state” (Ramirez 2002, 257). The employment of composers by universities is a testimony to this “embeddedness” rather than to the recognition of music as a scientific discipline in its own right. American composers have been affiliated with universities since long before Milton Babbitt gave us the archetype of the “university composer.” The first Chairs of Music were occupied by John Knowles Paine (Harvard, 1875– 1906), Horatio Parker (Yale, 1894–1919), and Edward MacDowell (Columbia, 1896–1904). The American university, due to its spectacular growth in the twentieth century (both in the size of enrollments and in the range of programs offered), has become a home not only for the sciences but also for the creative arts. It is “the place where post-secondary education ought to occur, regardless of its activity or goal.” (Risenhoover and Blackburn 1976, 7).

the University of California at Berkeley, and the University of Wisconsin), offered a mixture of academic and vocational instruction, and much of their research was intended to meet practical needs. Other instances of societal involvement include the participation of universities in programs of military research during and after the Second World War, and the G.I. Bill (1944) enabling war veterans to enroll at colleges and universities (which necessitated a diversification of programs). The final stage of the history described by Kerr is that of the “multiversity,” a large but disunited knowledge factory accommodating various communities and serving a disparate clientele. 38. “It is to prevent or cure this peculiar democratic blindness that the university may be said to exist in a democracy, not for the sake of establishing an aristocracy, but for the sake of democracy and for the sake of preserving the freedom of the mind.” (Bloom 1987, 252) 39. Dutch scientists and scholars who traveled in the United States between 1900 and 1940, such as the historian Johan Huizinga, the pharmacologist W. Storm van Leeuwen, and the chemist Hugo Rudolph Kruyt, were very surprised to see the role of vocational instruction at American universities, the pragmatic orientation of scientific research, and the tendencies towards the popularization of science (Van Berkel 1990).

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Composers were not recruited as researchers, but as educators who could transfer hands-on experience to students of music and musicology. The musicologist Edward Lowinsky stressed the value of their presence on the faculty: composers, he wrote in 1963, “impart to musical training an acuteness, a liveliness, and standards of musical quality that spring from their constant translation of musical craft into musical art in their own work” (Lowinsky 1963, 47). In the same year, Russell F. W. Smith, then associate dean of general education at New York University, expressed a similar view with reference to the recruitment of artists in general: [W]e need scholars in a university because students are given their best chance if they learn philosophy from philosophers, sociology from sociologists, and biology from biologists, not from historians and appreciators of philosophy, sociology, and biology, so they have their most real introduction to the arts from artists, not from historians or appreciators of the arts. (Russell F. W. Smith 1963, 69)

A high value was placed on the contribution of artistic practices to academic programs. It was considered important to nourish these practices and to keep them within reach. This was the main reason to offer doctoral degrees—such as the DMA (for “Doctor of Musical Arts”) or PhD (for Philosophiae Doctor)—in composition from the 1950s onwards. These degrees were not so much intended to reward “scientific accomplishments” as to create job opportunities for composers. As certificates of eligibility, they were especially required by less distinguished colleges, which could not afford to hire composers who had already proven themselves by commercial success (Beglarian 1971, 465). The first PhDs in composition were awarded to students of Babbitt at Princeton University, after the degree program had been established there in 1961.40 Whether this brought academic recognition to Babbitt’s program is a matter of doubt. The musicologist Arthur Mendel, who chaired Princeton’s music department from 1952 till 1967, gave remarkably shallow reasons for the existence of a PhD in composition: (1) it was more difficult for composers than for musicologists to achieve publication, so they actually had more need for the upper-grade license; (2) there was already a Master’s degree for composers, and it would be illogical not to offer a Doctor’s degree as well; (3) no other degree than that of PhD was offered in Princeton in any other field, and it was not desirable to create an entirely new one for composers (Mendel 1963). This suggests that the degree was actually a sort of compromise resulting from the multiple functions and goals of the American university.

40. These students were required to hand in a composition and an analytical or theoretical essay, but over the years the emphasis shifted toward the essay. Examples of such essays include: Godfrey Winham, “Composition with Arrays” (1964), Michael Kassler, “A Trinity of Essays” (1967), and Benjamin Boretz, “Meta-Variations: Studies in the Foundation of Musical Thought” (1970).

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Another indication that composers did not owe their university positions to their image as “scientific researchers” is the relatively small number of serialist composers working at universities. After the Second World War serialism became associated with scientism. However, the commonly held view that serial music thus naturally secured a home within the American university—at the expense of other musical orientations—has come under scrutiny. An article by Joseph Straus in the Musical Quarterly (1999) provides statistical information that suggests otherwise. Straus examined composers who taught in universities from 1950 through 1969. He discovered who earned recognition by performances, recordings, and publications, and who received important grants or prizes in this period. From these data he inferred that the American contemporary music scene was not dominated by composers of serial music. Even at the four alleged hotbeds of musical modernism (Harvard, Yale, Columbia, and Princeton), these composers represented a small, if high-profile, minority. Certain aspects of Straus’s research invite criticism. For example, it is a bit odd to draw one’s information about the stylistic orientations among university composers in the period 1950–69 from the 1970–71 edition of the Directory of Music Faculties in Colleges and Universities.41 Furthermore, even if one wants to denounce the “myth of serial tyranny,” its existence cannot be brushed aside as resulting from gossip, a lack of knowledge on the part of journalists and historians, or the “provocative rhetoric” of a few serial composers (Straus 1999, 333–34). Musical discourse does not cover music like a layer of dust. It takes part in the chain of actions and reactions constituting the history of music—which is exactly the reason Milton Babbitt was so anxious about it.42 Notwithstanding these objections, Straus deserves credit for pouring the cold water of statistics over the heated claims of favoritism and straitjacketing. One of the things he has convincingly shown is that the American academy at large has had no special interest in furthering serial composition per se. It has been indifferent to the emulation of a scientific attitude by composers of serial orientation. It is unlikely that it has ever required composers to serve the progress of music under the same methodological regime that prescribes how scientists should serve the growth of knowledge. If there has been a trend toward equating composition with scientific research, this concerned the academic freedom— specifically, the freedom from extensive supervision—allowed to composers and researchers alike. The composer’s task has been to maintain and teach an artistic practice, as part of the university’s involvement in cultural life. In this regard, 41. See the caption of Figure 2 in Straus’s article. Straus refers to this figure as follows: “By 1970, at the end of the period under study and after twenty years of presumed serial dominance, one might expect to find at least a majority of these . . . composers to be working in the serial idiom. The facts however are somewhat different.” (Straus 1999, 307; my italics) 42. For a substantial criticism of Strauss’s article, see Shreffler 2000.

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the American university is better depicted as a complement of society rather than as a refuge from it. It is not surprising that Boulez, coming from Europe, failed to take the history and societal context of the American university into account when criticizing its role as employer of composers (and performers). What is surprising is that Babbitt, as an advocate and beneficiary of this relationship, seems to have misunderstood its role as well.43 His scientific rigor was largely self-imposed, and his view of the university as a “fortress” ignored this organization’s permeability to the surrounding society as much as Boulez’s image of an “ivory castle.” This raises a number of questions, the very last this study will deal with. Babbitt is credited by many with pioneering the American theoretical discourse on music, and with bringing academic respectability to it. How could he have been so successful if, as the foregoing discussion suggests, this was largely his own preoccupation, stemming from his concerns as a composer, and not so much that of the academic environment in which he was working? Furthermore, what is the extent of his influence? Did this theoretical discourse develop in alignment with his own writings, or has it been shaped by other factors? As for PC set theory, we have identified one such factor earlier on: the rise of computer technology. But we shall consider one more: the practice of teaching music theory.

A Democracy of Learning No doubt, Babbitt’s work has been instrumental in the birth of what is called “contemporary” music theory by some (like McCreless 1996, 1997), and “professional” music theory by others (like Kraehenbuehl 1960, and Babbitt himself; 1987, 121)—a discipline that, judging from these characterizations, was intended to obliterate old-fashioned and dilettantish music theory. However, Babbitt’s view of music theory as an empirical-theoretical system should not be seen as the Open Sesame! to the research departments of American universities.44 Nor did his view become a paradigm for all research endeavors in the field. Babbitt’s scientistic conception of the discipline certainly elicited responses. It was developed further, or modified, in methodological papers by younger colleagues like Benjamin Boretz (1969/1973), John Rahn (1979a, 1979b), and

43. The gap between Babbitt’s ideal university and the realities of academic life appears from his remark that “the university’s deadly deficiencies . . . are the extent to which it reflects the ‘real world’ outside the university.” (Babbitt 1970, 65) 44. Accepted wisdom has it that the establishment of the doctoral program at Yale meant American music theory’s breakthrough as an academic discipline. However, it was not a breakthrough on all fronts. As McCreless (1997, 38) has noted, the discipline has gained entrance more easily to university-based music schools and liberal arts colleges than to research universities.

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Matthew Brown and Douglas Dempster (1989). And there were theories that obviously, if perhaps tacitly, took it as their starting point. Fred Lerdahl’s and Ray Jackendoff’s application of generative linguistics to the structure of classical Western tonal music is a case in point. Starting from the notion of “grouping”—that is, from an experienced listener’s capacity to organize sound signals into larger entities (motifs, phrases, themes, etc.)—they developed a theory of musical understanding (A Generative Theory of Tonal Music, 1983). This theory consisted of logically connected rules of musical grammar, rules according to which Lerdahl’s and Jackendoff’s “expert listener” would decide how to understand a musical composition. Whatever its shortcomings, it was a serious attempt to combine a formal organization with a focus on the empirical world.45 But not all the work that was published in the wake of music theory’s advent as a research discipline was influenced by Babbitt’s attitude, not even when that work seemed to show signs of obedience to the rules of scientific explanation. It certainly did not hold true for PC set theory, which could be formalized, but was hard to operationalize. In other words, it was hard to provide empirical indicators for the importance of PC sets, which the theory sought to underpin. This problem has been raised in previous chapters. In chapter 3, it was shown that the assumed independence of PC sets from musical articulation seriously hampers their verification. Chapter 4 dealt with the concept of PC set equivalence, which served to unite what could be heard as different, and thus to represent a musical composition as coherent. But a concept like that does not help to explain why we hear a musical composition that way, at least not according to the rules of scientific explanation invoked by Babbitt. This would entail the condition that PC sets correspond to perceptual units—that is, to groups of notes that are heard as belonging together—whereas they often serve as a kind of heuristic tool to find relations that the ear might have missed. “Coherence” is not a perceptual category if the elements by which a piece of music coheres are not directly perceived—still less if their relations are as abstract as the PC set similarity relations discussed in chapter 5. In short, PC set theory is not a theory of music cognition, like Lerdahl and Jackendoff’s “generative grammar” of tonal music. As a consequence, it should not be judged from this viewpoint. (This answers Fred Lerdahl’s criticism of the theory, discussed in chapter 1.) Nor did PC set theory provide what Richard Taruskin demanded: insight into past compositional practices (once again, if we take scientific criteria into

45. It would be going too far to see Lerdahl’s and Jackendoff’s work as a response to Babbitt’s methodological papers. Their theory was intended as a musical counterpart of Chomskian linguistics, and was spurred by Leonard Bernstein’s enthusiasm for a similar approach to music. I refer the reader to the 1973 Charles Eliot Norton Lectures of Bernstein at Harvard University, which were published in 1976 under the title The Unanswered Question.

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consideration). What could positively indicate the role of the PC set as a device of composition if we limit our inquiry to the “music itself” (which is usual, if not a matter of principle, in music analysis)? One might think of the frequent occurrence of a particular set-type in a work, or of harmonic progressions that involve particular PC set relations (like the inversion relation in ex. 3.13). But these criteria are utterly ambiguous. How frequent is “frequent”? And what are the conditions in which these set-types occur? The PC set {1,4,5,7} was shown to appear a number of times in the Gavotte from Schoenberg’s Suite, Op. 25 (see ex. 2.9); but then it consists of the first four elements of a recurring twelve-tone series: a fairly trivial case of a PC set leaving a strong mark on a piece of music. If PC set theory claims that PC set relations form the common theoretical basis of a body of twentieth-century art music larger than serial twelve-tone music, it should enable us to validate that claim. This requires the consultation of sources outside the domain of the theory, which is restricted to scores. Seen thus, PC set theory is not an explanatory theory, but the codification of an analytical practice. As such, it bears no substantial relation to modern science, even though it appeals to a scientific sense of objectivity and consistency. It is true that it incorporates what one might call a hypothesis about pitch organization (that atonal pieces are structured around groups of notes that can be related to each other), but this “hypothesis” is not supposed to be tested—at least not by analysts. On the contrary, the hypothesis actually serves to guide the analysis. As Straus writes in one of the first chapters of his Introduction to Post-Tonal Theory: “When we listen to or analyze music, we search for coherence. In a great deal of post-tonal music, that coherence is assured through the use of pitch-class sets.” (Straus 1990a, 26) Such a statement hardly invites an empirical test. But the perhaps obvious verdict of circularity should not be made without taking into account the pedagogical orientation inherent in PC set theory. It allows one to analyze musical works which are otherwise closed off. It has generalized and extended structural concepts like “motif,” “chord,” and “series”—with their legacy—so as to render them applicable to a large repertoire of music from the twentieth century and beyond. Music analysis at large belongs more naturally to the world of education than to the world of science; music analysis in the United States is no different, in spite of the great attraction science has exerted upon American music theorists. It is the principal aim of an analytical theory that it be taught, that the insights it offers trickle down to the classrooms where new generations of musicians, scholars, and teachers are bred—much as Babbitt’s highly involved study of twelve-tone invariants in Schoenberg’s Third String Quartet reached a student audience through the manuals of Perle and Lester. Book-length theories of musical structure—like the study of Lerdahl and Jackendoff mentioned earlier, or like William Caplin’s Classical Form (1997)—often suggest a didactic purpose. They are set up progressively—starting with analyses of small fragments and then proceeding to larger and more complex structures; they have abundant musical examples; they show a deep concern with the

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consistency and clarity of the applied terminology; and the pace of the argument is relaxed, encouraging the interested novice to persevere. If such a publication is still too much a theoretical treatise, it may be followed by another one better suited to the classroom. Heinrich Schenker’s theory has been presented to a student readership in books like Introduction to Schenkerian Analysis (Allen Forte and Steven Gilbert, 1982) and Analysis of Tonal Music (Allen Cadwallader and David Gagné, 1998). And the books of John Rahn (Basic Atonal Theory) and Joseph Straus (Introduction to Post-Tonal Theory) provided easier access to PC set theory than Forte’s own The Structure of Atonal Music.46 The professionalization of music theory has been described as a process of emancipation. According to Patrick McCreless, it was in the late 1950s and early 1960s that American music theory could “stake a claim for admission to the modern academy,” after it had thrown off “the shackles of its old pedagogical self.” McCreless speaks about the birth of the music theorist “out of the music theory teacher” (McCreless 1996). In reality, however, the discipline has never truly disentangled itself from the practice of teaching. PC set theory, for example, does not embody music theory’s emancipation from a teaching tradition; rather, its existence is a measure of the importance, endurance, and flexibility of this tradition. The new formulations that PC set theory offered—whether they resulted from the application of computer technology to music analysis or from the adoption of scientific language—found their way into pedagogical contexts, where they were used without specific reference to the conditions that had engendered them. They were absorbed, and transformed, by a practice with a much longer history. This does not mean that PC set theory’s formulations failed to affect the teaching practice in the United States; they affected it greatly. They helped revitalize music theory instruction—a need that was very urgent in 1959, when David Kraehenbuehl—a former student of Hindemith at Yale University, and the first editor of Journal of Music Theory—wrote: Theory has fallen into a state of almost universal academic disgrace. In many schools, it is the program in which those unfortunates who have failed in every practical musical activity are placed; in others, the hours devoted to it, obviously a waste, are quite reasonably pared down; in still others, it is maintained as a kind of traditional house-keeping service which no one has had the courage to improve or eliminate. (Kraehenbuehl 1959, 31)

46. This does not mean that Rahn and Straus merely reproduced the contents of The Structure of Atonal Music. Rahn extended Forte’s theory (operations on pitch sets, multiplication), but also deleted a considerable portion of it (set-complex theory, the intended crown on Forte’s achievement). Moreover, his nomenclature differed from Forte’s. Straus included the most successful elements of Forte’s theory in a general survey of analytical concepts and techniques applied to post-tonal music.

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And it was still urgent in 1965, to judge from Howard Boatwright. In the view of this composer, most of the theory texts in use at teaching institutions had “neither kept up with the present nor incorporated much of our improved understanding of the past.” He called on those involved in the teaching of music theory in colleges and elsewhere to “eliminate anachronistic practices . . ., to make [theory] serve more meaningfully in developing general understanding of our musical heritage from the distant and recent past, and to connect it more realistically with the events of the present” (Boatwright 1965, 48). In this period, the topic of music theory instruction was discussed in a number of forums. Periodicals such as Journal of Music Theory and College Music Symposium devoted space to these discussions.47 A thread running through them was music theory’s failure in dealing with music of the present. Music education at large was felt to be serving the preservation of past heritages rather than their evolution. This was the main reason why, in 1963, the Music Educators National Conference launched the so-called “Contemporary Music Project for Creativity in Music Education.” Enabled by a grant of $ 1,380,000 from the Ford Foundation, it investigated music programs in public schools and colleges, it organized workshops and seminars for music educators, and it conducted experiments with new approaches to music teaching with a view to “modernize and broaden the quality and scope of music education on all levels.”48 Apart from a greater focus on contemporary composition in music education—for example via the “Composers in Public Schools Program” mentioned earlier—the Contemporary Music Project aimed at an integration of the various fields of knowledge and skill with which students of music had to be acquainted. Thus, a general training in the practice of composition for all student musicians was proposed. Aural training programs would draw on a greater variety of musical materials and benefit from advances in the field of learning psychology. Courses in music history would support the study and performance of music, while music analysis, apart from being taught systematically in separate classes, would inform music instruction in any class at any level. The Contemporary Music Project lasted until 1969, but continued to inspire curricular experiments afterwards (Wason 2002, 71). It went a long a way toward the kind of reform that David Kraehenbuehl had proposed in 1959. Kraehenbuehl saw music theory as an area of creative thought, interacting with the music that was currently composed, played, and heard, as well as with other domains of knowledge. “Such a

47. “On the Nature and Value of Theoretical Training: A Forum,” Journal of Music Theory 3/1 (1959); “The Professional Music Theorist: A Forum,” Journal of Music Theory 4/1; “Symposium: The Crisis in Music Theory Teaching,” College Music Symposium 5 (Fall 1965). 48. Quoted from the introduction to Comprehensive Musicianship: The Foundation for College Education in Music: A Report of the Seminar sponsored by the Contemporary Music Project at Northwestern University, April 1965. Washington: Contemporary Music Project/National Music Educators Conference, 1965, 3.

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dynamic theory would always be welcome in the training of any practicing musician” (Kraehenbuehl 1959, 32). The issue of Journal of Music Theory in which these words appeared also contained Allen Forte’s introduction to Heinrich Schenker and his theory, which was cited in chapter 1 (“Schenker’s Conception of Musical Structure”). Although Schenker’s work was already known through the teaching and writings of Adele Katz, Felix Salzer, Roger Sessions, and Milton Babbitt, it is this article that marks the beginning of its march through the American academy.49 Forte was 32 years old, and about to be appointed at Yale University. His article was actually a programmatic statement, as the reader could infer from the subtitle (“A Review and an Appraisal with Reference to Current Problems in Music Theory”). As such, it is highly relevant to the present discussion. In Forte’s view, one of the main attractions of Schenkerian theory was “the discovery of a fundamental principle, which . . . [opened] the way for the disclosure of further new relationships, new meanings.” (Forte 1959, 3). This fundamental principle was not the Ursatz—which Babbitt had given axiomatic status seven years earlier—but the concept of structural levels. Apart from being “far more inclusive” than the Ursatz (9), this concept lent itself to pedagogical applications. It involved a repeatable analytical procedure: the successive reductions leading from the surface to the fundamental structure. Unlike Babbitt in 1952, Forte stressed the fact that Schenker was a music teacher, and that “almost all his writings are intended to instruct—in the most practical sense of that term.” He brought the educational significance of Schenkerian theory to the notice of his readers, claiming that it led to “an understanding of the total [musical] work in all its complexity” (5). And he particularly addressed its relevance to the concerns of practicing musicians. It would bring music theory teaching into a natural relationship to performance, a relationship of which Schenker had said: If I had my way, every instrumentalist would have theory as his major study. It is not enough for him to play mechanically, as though he has Czerny exercises before him. They say they have practiced (especially the ladies). But what is the use of that? What Geist makes their practicing vital? In painting and poetry, Czerny exercises do not exist. (Schenker quoted by Forte 1959, 4–5)

Forte also touched on the relevance of Schenkerian theory for musicologists, with reference to Schenker’s autograph studies and his editorship of the sonatas of Carl Philipp Emanuel Bach and Beethoven. For Forte, Schenkerian analysis meant the reconciliation of diverse kinds of musical activity; it meant comprehensive musicianship. Since this was one of the main objectives of the later Contemporary Music Project, it is worth exploring a bit further the relation between 49. This is acknowledged by several writers, such as Kerman (1985), Rothstein (1990), and Snarrenberg (1994).

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this project and the rise of Schenkerian theory in the 1960s, to the point where the issue involves the relation between music theory as teaching subject and music theory as an academic discipline. Forte’s article on Schenker prefigured the Contemporary Music Project in two ways. It not only stated that the highest degree of aesthetic response was achieved by a combination of practical skill and theoretical insight—following the example of Schenker’s teaching—but it also presented Schenkerian theory as based on a general model for understanding all music. Thus, Forte tried to establish a line of continuity between the music of the past and that of the present, another objective of the Contemporary Music Project. Forte was well aware that the avowed conservative and doomsayer Heinrich Schenker would not have danced to this tune. So, he picked his words carefully: It seems contradictory that Schenker’s work should contribute significantly to the solution of certain problems in advanced contemporary music. Yet, his general concept of structure, apart from his specific formulations of triadic tonal events, lends itself to modern thought regarding music. (Forte 1959, 26)

This general concept of structure was the “totally organized work” (26), in which every detail contributed to the experience of the whole. Although at first sight this concept does not seem to be in conflict with Schenker’s own ideas, Forte’s rendering of an analysis by Schenker—his analysis of Robert Schumann’s song “Aus meinen Tränen sprießen” (ex. 1.5)—has struck some readers as very different in tone and style from its model. William Rothstein calls Schenker’s voice “that of a prophet, pronouncing sacred mysteries from on high,” whereas he sees Forte as “the cool taxonomist, concerned above all with rationalism and clarity” (Rothstein 1990, 198–99). Robert Snarrenberg (1994, 51) speaks of a “scientistic transformation of Schenker,” which turns works of genius into passive subjects of dissection. While all of this is to some extent true, the most radical departure from Schenker consisted in Forte’s tendency to generalize. He suggested that Schenkerian theory could be subsumed under a more inclusive concept of musical structure, a concept independent of tonality, triadic or otherwise.50 It is when he speaks of “essential relationships” (Forte 1959, 26), or “similarities at other than

50. The more inclusive, the more insightful: this seems to have been the motto guiding the attempts to conceive of musical structure in the most general terms. On the one hand, this generalization enabled comparative studies across boundaries of style and genre; on the other, it took the focus away from the composer and historical context (see also chapter 7, note 11). In these two respects, Forte’s agenda can be characterized as a genuinely structuralistic one.

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the surface level” (27) in “advanced contemporary,” “non-triadic,” or “problematic modern music” that PC set theory comes into view.51 Forte’s “general concept of structure” also appeared on the agenda of the Contemporary Music Project—perhaps by his own agency. The documentation of the project includes the proceedings of a Seminar on Comprehensive Musicianship, which was held at Northwestern University in Evanston, Illinois, from April 22 through 25, 1965. The purpose of this seminar was to lay down the aims of the project with regard to college education in music. Forte was one of the participants. At the time, he was already deeply involved in the computeraided investigation of the atonal music of Schoenberg, Berg, and Webern. Less than two weeks earlier, on April 10, he had presented the first results of this investigation at a meeting at the Rockefeller Center in New York (cf. Forte 1970). Now he was invited to deliver a position paper on “The Role of Music Theory in the Development of Musical Understanding.” Furthermore, he was a member of a group of participants assigned to make recommendations with respect to training programs in music analysis and aural skills.52 Most of these recommendations dealt with the necessity of expanding the musician’s frame of reference (“Materials may be drawn from any era or culture”), achieving an integration of musical skills (“the repertoire used in aural training should be actively related to the repertoire heard in performance”), and of developing effective means of communication (“the musician should . . . have the widest possible acquaintance with the various relevant formulations of systematic music theory”). For the present discussion, however, the following brief passage is important: A basic prerequisite for the analytical training process is a generalized framework for the study of musical materials, which might be in the form of a systematic description of musical structure cutting across stylistic barriers and approaching music from a less narrow point of view than the now generally current.53

51. In his article, Forte used all these labels to designate a repertoire of twentiethcentury music that he regarded as one of the five great challenges for music theorists of his day. His brief sample analysis of Debussy’s Prelude La Cathédrale Engloutie (1910)—a piece he apparently considered as belonging to that repertoire—still made use of “Schenker-derived” techniques. Besides the analysis of this and other “problematic modern music,” the challenges he mentioned were: “constructing a theory of rhythm for tonal music,” “determining the sources and development of triadic tonality,” “gaining information about compositional technique,” and “improving theory instruction.” 52. Other participants in this group included the composers Leslie Bassett (University of Michigan, Ann Arbor) and Robert Cogan (New England Conservatory), and the music educator Charles Leonhard (University of Illinois, Urbana). 53. All quotes have been taken from the proceedings of the seminar (Comprehensive Musicianship, 15–17; see note 48).

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These words seem too abstract to have ever elicited action; however, when they were put to paper, action was already under way. The passage can be read as an introduction to the projects Forte had already embarked on. It stressed the need of a generalized analytical vocabulary as a remedy to the narrow-mindedness of teaching programs offering an inventory of past practice. We know that Forte put little trust in the knowledge and skills handed down by tradition. In his position paper, he had said that music theory owed regained respectability to computer technology, the use of which had helped widen the scope of the discipline. Now, his aim was to bridge the gap between advanced research in music theory and music theory as taught in the classroom: The implications and responsibilities of advanced research with respect to education often are totally disregarded by persons deeply involved in such work, just as the implications of these recent developments appear to have been overlooked in the teaching of music theory at the classroom level. (Forte 1965b, 39) The study of skills and techniques . . . must be informed by the highest level of scholarship, for the task of leading to an understanding of complicated art music requires a knowledge of the role of systematic generalization and a comprehension of the significant characteristics of musical abstractions and of symbolic processes in general. (Forte 1965b, 40)

Forte thus aligned his program for the development of music theory at the college level—including Schenkerian theory and PC set theory—with a broad education policy aiming at maximum participation in what was called “the information revolution.” It is interesting to compare his words with those of Ole Sand, who headed the Center for the Study of Instruction of the National Education Association, and was invited to speak about general trends in curriculum and instruction: Probably the most immediate factor forcing change upon education is the explosion of knowledge—the “information revolution.” Furthermore, because scientific and scholarly work is now quite extensive and many are engaged in it, the rate of revision is swift. Teaching the disciplines in this situation clearly requires teaching something more permanent and pervasive than a catalog of factual knowledge, although some facts are essential, and it is clear that there is a continuing need for drill and repetition for learning basic information. Educators are not only concerned with the amount of knowledge students possess but also with students’ lack of understanding about what they presumably know. Since about 1955, a vivid awareness of this latter problem has led some scholars and researchers to explore ways of selecting, organizing, and teaching available information to make it intelligible and more usable. In general, the recent studies shift the balance in learning from inventory to transaction. The structure of a discipline, its methods of inquiry, and the styles of

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thinking of its scholars and specialists offer important keys to this educational task. (Sand 1965, 73)

A “generalized framework for the study of musical materials,” as recommended by the Seminar on Comprehensive Musicianship, can be seen as a response to the changing conditions of learning pointed out by Sand. Apart from the cultural and artistic values it may have involved (Forte’s “totally organized work”), apart from its role in retaining a sense of continuity between the past and the present, such a framework was expected to assure the distribution of new knowledge through a system of mass education. Thus, it would serve the “democracy of learning” so firmly anchored in the American self-image. For American society mass education is more than the product of social forces and demographic change. Mass education entails the idea that everybody should have access to knowledge, wield decision-making power, and be able to change their status. Mass eduation is “the United States’ assurance that democracy may thrive,” to borrow a word from Samuel B. Gould, once a university chancellor and propagator of educational television (Gould 1971, 355–56). This idea points to the rise of PC set theory in the United States. PC set theory is not only the product of the scientification of music theory, but also of a commitment to education. Whoever rejects its formalism and reductionalism, and finds its language inappropriate, should bear in mind that these aspects serve the purpose of equal opportunity. They enable anybody who is interested, and perseveres, to develop an expertise in some of the art music of the post-1900 era and to hand it down to others. It may be hard to believe that the mathematical borrowings and the more arcane relational concepts are interesting to those outside a highbrow elite; but whoever thinks they are, therefore, incompatible with an idealistic notion of mass education, should be reminded that this notion does not exclude the emergence of such elites. Rather, it allows various levels of prior ability and acknowledges various levels of achievement. This is what European composers (like Arnold Schoenberg, Ernst Toch, and Paul Hindemith), who had settled in the United States during the Nazi period and assumed teaching positions at colleges and universities, found so difficult to cope with. As Alan P. Lessem has shown, they objected to the numbers of students they were required to teach, and to the quantification of their achievements: [The American democratic ideal] was . . . irreconcilable with the belief, so much a part of the meaning of ‘culture’ in its nineteenth-century middle-class European milieu, that significant artistic achievement was given to only a chosen few, from whose hands the multitude gratefully received secondary benefits. And it was a belief to which many of the émigrés still clung. (Lessem 1988, 11)

PC set theory has never become—and may never become—a part of the standard equipment of the “musically literate”; but it is itself equipped to meet the

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demands of mass education in music, in the sense that it provides a conceptual framework the intrinsic logic of which could be taught to many. An answer to the question of why PC set theory has become so vital to the image of American music theory should not ignore this aspect; it may even have been the key to the theory’s success. On the other hand, it may have precluded its triumphal progress in Europe, where the aristocratic notion of culture that Lessem described is more deeply embedded than in the United States. Finally, then, we can answer the question left over from the previous section: how did Milton Babbitt bring music theory so successfully into the American academy? Babbitt has not been successful in every respect. Music theory has not, on the whole, developed into something scientifically viable—that is, from his original empirical-theoretical viewpoint. In other words, it has not quite secured its place among the academic disciplines on his conditions. While being informed by an academic discourse, it has kept in touch with its “old pedagogical self”: a discipline at the service of music education. PC set theory is one manifestation of this double mindset. It testifies to Babbitt’s influence, but at the same time it reflects the pedagogical tradition from which he had tried to cut music theory loose; it may well be that Babbitt’s own intellectual legacy owes much to the vitality of this tradition.

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———. 1982. “‘To Worship that Celestial Sound’: Motives for Analysis.” Journal of Musicology 1/2:153–70. Van den Toorn, Pieter C. 1983. The Music of Igor Stravinsky. New Haven, CT: Yale University Press. ———. 1995. Music, Politics, and the Academy. Berkeley: University of California Press. Varèse, Edgar. 1967. “The Liberation of Sound.” In Contemporary Composers on Contemporary Music, ed. Elliott Schwarz and Barney Childs, 196–208. New York: Holt, Rinehart, and Winston. Vicentino, Nicola. 1555. L’Antica Musica ridotta alla moderna prattica (facsimile of the edition of 1555), ed. E. Lowinsky. Kassel: Bärenreiter, 1959. Vogler, Georg Joseph. 1776. Tonwissenschaft und Tonsetzkunst. Mannheim: Kuhrfürstliche Hofdruckerei. Wason, Robert W. 1985. Viennese Harmonic Theory from Albrechtsberger to Schenker and Schoenberg. Ann Arbor: UMI Research Press. ———. 1988. “Progressive Harmonic Theory in the Mid-Nineteenth Century.” Journal of Musicological Research 8:55–90. ———. 2002. “Musica Practica: Music Theory as Pedagogy.” In The Cambridge History of Western Music Theory, ed. Thomas Christensen, 46–77. Cambridge: Cambridge University Press. Weber, Gottfried. 1817. Versuch einer geordneten Theorie der Tonsetzkunst zum Selbstunterricht mit Anmerkungen für Gelehrten, vol. 1. Mainz: Schott. Weber, William. 1999. “The History of Musical Canon.” In Rethinking Music, ed. Nicholas Cook and Marc Everist, 336–55. Oxford: Oxford University Press. Webern, Anton. 1960. Wege zur neuen Musik, ed. Willi Reich. Vienna: Universal Edition. Weigl, Bruno. 1925. Harmonielehre. Mainz: Schott. Wilcox, Howard J. 1983. “Group Tables and the Generalized Hexachord Theorem.” Perspectives of New Music 21/2:535–39. Winham, Godfrey. 1964. “Henry Weinberg, Three Songs (1959).” Perspectives of New Music 2/2:106–11. ———. 1970. “Composition with Arrays.” Perspectives of New Music 9/1:43–67. Wiprud, Theodore. 2000. “On the Money: New Music Funding in the United States.” NewMusicBox 2/4 (2000), http://www.newmusicbox.org/page.nmbx?id=804. Wittlich, Gary. 1974. “Interval Set Structure in Schoenberg’s Opus 11, No. 1.” Perspectives of New Music 13/1:41–55. Zarlino, Gioseffo. 1558. Le Istitutioni harmoniche (facsimile of the edition of 1558). New York: Broude Brothers, 1965.

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Index An italicized page number indicates a figure or table.

Abbate, Carolyn, 23n18, 223 absolute pitch-interval class (APIC), 48, 82–83, 99, 151, 175, 191n13; definition of, 40. See also interval class, Forte’s definition of absolute value, 40 Adorno, Theodor W., 255n21 aggregate, twelve-tone, 92 Albert, A. A., 86 algebra, 84, 86 algorithm, 117–18, 102, 105, 119, 120, 132, 240, 248. See also APIC vector; normal form; normal order; prime form all-interval series, 116, 242 American Composers Orchestra, 259 American Council of Learned Societies (ACLS), 242 American Music Center, 259 American Musicological Society (AMS), 239n6 American Society of University Composers, 258–59 American Symphony Orchestra League, 259 Analytic Approaches to Twentieth-Century Music (Lester 1989), 20, 22n17, 252, 270 Ankersmit, Frank, 236n1 APIC. See Absolute pitch-interval class APIC vector, 48, 52, 62, 88, 98n18, 99, 108, 125–26; algorithm, 132; as a basis of equivalence, 92–93, 103; definition of, 48; similarity measures based on the, 132 144–46, 148–49, 151–155, 160n21, 163, 165, 168–69, 173–175. See also Z-relation

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Armstrong, David M., 190 Ashby, Arved, 116n28 “Aus meinen Tränen sprießen” (Schumann, Dichterliebe, Op. 48, no. 2), 14, 16, 274 Babbitt, Milton, 9–10, 18, 26, 31, 36, 43, 44n17, 47, 58, 92, 94–97, 98, 105, 116, 145n12, 206, 225, 236, 239n4, 278; Carnap’s influence on, 251n17; on contemporary music culture in the USA, 257–59, 261; group multiplication table, 51; and the language of science, 35, 251–58, 264, 268–69, 278; on Mersenne, 254; metatheory of, 256; on Rameau, 254; relation to his European contemporaries, 261–64; on Schenkerian theory, 221, 252, 273. See also combinatoriality; communality and contextuality; equivalence; integers; inversion; normal form; partial ordering; positivism; Princeton University; retrograde; retrogradeinversion; transposition (T); twelvetone serialism; university Babbitt, Milton, works by: The Function of Set Structure in the Twelve-Tone System (published 1992), 33, 55, 67, 87, 96n14; “Set Structure as Compositional Determinant” (1961), 88, 102; “Twelve-Tone Invariants as Compositional Determinants” (1960), 55, 69, 88, 252–53, 270; “Who Cares if You Listen?”(1958), 255n22, 264. Bach, Carl Philipp Emanuel, 273

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index

Bach, Johann Sebastian, 66, 202–3, 205, 213, 219, 233; Art of Fugue, The, 65, 66; Nun lob, mein Seel, den Herren, BWV 17/7, 202, 203, 205, 213; Well-Tempered Clavier, The, 65, 66 Bacon, Ernst, 33, 43, 117, 122 Baker, James M., 2n2, 45 Bartók, Bela, 78, 233; Music for Strings, Percussion, and Celesta, 78, 79; String Quartet, No. 3, 78. See also interval expansion; multiplication (M). Baryphonus, Henricus, 30 “basic set,” 44n16, 68, 94, 96n14, 105 “basic shape,” 10, 12 bass position, 41, 100, 121. See also harmonic inversion basse fondamentale (Rameau). See fundamental bass Bauer-Mengelberg, Stefan, 116n29, 242– 43, 244n11. See also all-interval series; Ford-Columbia (input language) Beach, David W., 2n2, 32, 229 Beal, Amy C., 263 Beethoven, Ludwig van, 222, 223–24, 233, 234, 273 Benjamin, William E., 3, 24. See also segmentation Berg, Alban, 78–79, 91, 116, 126–28, 152, 212n30, 239; Five Orchestral Songs after Picture Postcard Texts by Peter Altenberg, Op. 4, 78 (no. 1), 126, 127, 128 (no. 2); Lulu, 79, 80 Berger, Arthur, 200–2 Berio, Luciano, 75–76, 225 Bernard, Jonathan W., 2n2, 33, 43, 116n27, 123 Bernhard, Christoph, 53 Bernstein, David W., 67n16 Bernstein, Lawrence, 242 Bernstein, Leonard, 244, 269n45 Birkhoff, George D., 86 Bloom, Allan, 265 Bloom, Harold, 235 BMI. See Broadcast Music, Inc. Boatwright, Howard, 272 Bononcini, Giovanni Maria, 64–65, 73 Boretz, Benjamin, 259, 266n40, 268

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Boulez, Pierre, 18, 78n26, 261–64, 268 Bowles, Edmund A., 241 Brahms, Johannes, 222, 233n13 Braythwaite, William, 31–32 Broadcast Music, Inc., 259 Brody, Martin, 255n2, 256 Brook, Barry S., 242 Brown, Matthew, 219, 269 Buchler, Michael H., 176 Bukofzer, Manfred, 239n6 Cadwallader, Allen, 271 canonical transformation, 54, 113–14, 161–62, 163, 165–67, 169, 172, 185n5. canonical transformation group. See transformation group Cantor, Georg, 3n3 Capellen, Georg, 67n16 Caplin, William E., 16n13, 270 cardinal number, 41, 45–46, 89, 97, 100, 158, 167, 173, 179, 232 cardinality. See cardinal number Carnap, Rudolf, 250, 251n17 Carter, Elliott, 18, 263n35 Castrén, Marcus, 176 Chopin, Frédéric, 233n13 chord class, 116–18, 120 chord classification, 43, 115, 121, 123; Hindemith’s, 138, 140–41, 142; Klein’s, 116–18; Krenek’s, 138, 142; Weigl’s, 120–21 Chrisman, Richard, 91n10, 100–102, 122. See also equivalence; successive interval array Christensen, Thomas, 137, 219–20 CINT. See cyclic interval succession class. See equivalence class Clough, John, 91n10, 98–99, 105, 107. See also equivalence Cohn, Richard, 61, 195n16 Columbia University, 17, 18, 19, 265, 267 combinatoriality, 26, 83, 92–94, 95, 97, 143, 145n12, 205–6; hexachordal all-, 94, 95; hexachordal inversional, 92, 94, 95; tetrachordal, 95. See also Schoenberg, Arnold: use of hexachords

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index combinatorics, 43, 92 common-note function, 161–62 common-note vector, 158, 159, 161–62 communality and contextuality (Babbitt), 26, 179 complement (set operation), 49–50, 87n3, 89, 92, 93, 94, 96–97, 122, 175, 183, 207, 209, 215–16, 222, 230, 237n2; definition of, 49. See also set complex complement theorem, 97, 175 Composers’ Forum, 259 Composers in Public Schools Program, 260, 272 comprehensive musicianship, 273 computer, 28, 91, 96, 116n29, 132, 137, 268, 271, 275–76; Forte’s view on the, 248–49, 276; influence of the, on PC set theory, 96n15, 237–38, 247–50; influence of the, on the scholar’s perspective, 240–41; technology, positive reception of, by scholars, 238–41, 248. See also segmentation Computer and the Humanities, 242 Cone, Edward T., 2n2 Contemplating Music (Kerman 1985), 3, 22–23 contemporary music, 254, 274; funding of American, 259–61; rejected by American orchestras, 259 Contemporary Music Project, 272–75 contextuality. See communality and contextuality (Babbitt) contour, theory of, 182 contour class, 182–83 contour segment (cseg), 182n3 Cook, Nicholas, 25n21, 136n6, 221n3, 223–24, 225. See also segmentation Copland, Aaron, 256n23 Cornell University, 239n6, 264n37 Craft, Robert, 201, 163n35 criticism, 23, 223, 227n10, 235n15 cseg. See contour segment “cycle-of-fourths-equivalence” (Howe). See equivalence “cycle-of-fifths-equivalence” (Howe). See equivalence

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295

cyclic interval succession (CINT), 101n22. See also successive interval array cynosural set, 217 Dahlhaus, Carl, 31n1, 38–39, 82, 134n4, 137 Daniel, Oliver, 259 DARMS. See Digital Alternative Representation of Music Scores Darmstadt Summer Courses for New Music, 261n30, 263 Davis, James A., 190–91, 249n14, 250 Davis, Miles, 225 Debussy, Claude, 45, 89, 91n8, 126n36, 234; Fêtes (Nocturnes, No. 2), 194–98, 217; Prelude Brouillards, 233; Prelude La Cathédrale Engloutie, 275n51 Dedekind, Richard, 85n1 Delaere, Mark, 230 Deliège, Célestin, 4 Dello Joio, Norman, 259 Dempster, Douglas, 219, 269 “dependent” series. See twelve-tone series developing variation, 137 difference (set operation), 49–50 difference tone, 139, 142n9 difference vector. See interval-difference vector Digital Alternative Representation of Music Scores (DARMS), 244. See also Ford-Columbia (input language) directed PC interval. See pitch-interval class (PIC) Dirichlet. See Lejeune Dirichlet dualism. See harmonic dualism Doerksen, John F., 189 dominant, 53, 65, 67, 70, 205, 213; -flatninth chord, 128; -seventh chord, 113, 125, 199, 200; -thirteenth chord, 200 Dunsby, Jonathan, 2n2, 223 Eastman School of Music, 20, 22 Eggebrecht, Hans-Heinrich, 221 Eilenberg, Samuel, 86 Eimert, Herbert, 26, 57, 80–81, 116, 242, 261–62. See also all-interval series; multiplication (M)

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Élégie (Fauré), 181–83, 188 Epstein, David, 233 equal temperament, 26, 29, 33, 34n7, 45, 61, 84, 91, 228n11 equality of sets, 29, 41, 87–88, 89 equal-vector relation, 92, 98–99, 107, 131, 174. See also Z-relation equivalence, 29–30, 142, 151, 168, 177, 179, 182, 189, 269; algebraic, 84–86, 88, 92, 98n19, 99, 100, 101, 107; canonical, of PC sets (Lewin’s definition), 114; of chords, 116, 120–22; contour-, 182; “cycle-of-fourths-” (Howe), 82; “cycle-of-fifths-” (Howe), 82; informal notions of, 84–85, 86, 87, 100; of intervals, 36–39, 85, 88; of “lines” (Rahn), 110; octave-, 29–31, 36–37, 190, 199; operational, of PC sets (Howe), 99; partitional view of, 85–86, 109, 114; of PC sets (Clough’s definition), 98–99; of PC sets (Forte’s definition), 89, 92, 97–98, 103, 105, 107, 177, 206; of PC sets (Rahn’s definition), 110, 113–14, 177; of pitch sets (Morris, Rahn), 110, 123–25; sentential, 38n10, 87; set-theoretic, 86; sound and, 123–28; of successive interval-arrays (Chrisman), 100; of twelve-tone series (Babbitt), 87–88, 89. See also APIC vector; inversion (I); multiplication (M); similarity; transposition (T) equivalence class, 30, 46, 85, 86n2, 92, 100–101, 107, 109–10, 113, 123, 131, 169, 177, 222. See also “type”; chord class; contour class; contour segment (cseg); FB-class; PCINT-class; pitchclass (PC); pitch-interval class (PIC); pitch-set class (PSC); set-class Erpf, Hermann, 120 FB. See figured bass FB-class, 124, 125 Fauré, Gabriel, 181–83, 202 Ferentz, Melvin, 116n29, 242 Ferneyhough, Brian, 225 Fétis, François-Joseph, 25, 91

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figured bass, 31, 32, 100, 121, 124, 128 Fludd, Robert, 254 Ford Foundation, 260, 272 Ford-Columbia (input language), 244, 246 Forte, Allen, 3, 4, 10, 47, 103–9, 179–80, 229, 232, 236, 237n2, 271n46; analyses of Webern’s Op. 5, no. 5, 56, 59–60, 61; computer-aided analysis project, 91, 242, 246–48, 275; educational philosophy of, 275–76; influence of positivist epistemology on, 248–49; involvement with atonal music, 60, 89, 91, 132–33, 180, 248; polemical exchange with Taruskin on The Rite of Spring, 219, 225–26, 228n11, 234; programming expertise of, 244; on Schenkerian theory, 13, 16, 273–74; score-reading program, 246–47. See also computer; equivalence; genus; integers; interval class; interval content; invariance; inversion (I); Michigan Algorithm Decoder (MAD); normal form; normal order; prime form; progenitor; segmentation; table of set-classes; set complex; similarity; similarity measures; transposition (T); Yale University; Z-relation Forte, Allen, works by: The Atonal Music of Anton Webern (1998), 60, 61; “Context and Continuity in an Atonal Work” (1963), 88–89, 90, 91; The Harmonic Organization of the Rite of Spring (1978), 23, 225–26, 227; “The Magical Kaleidoscope” (1981), 183–84, 187, 191n13; “Pitch-Class Set Analysis Today” (1985), 233–34; “Schenker’s Conception of Musical Structure” (1959), 15–16, 273; The Structure of Atonal Music (1973), 19–20, 24, 27, 38, 56, 57, 83, 91, 99, 104–5, 107, 108, 109, 126, 134–35, 136, 148, 179–80, 206–7, 209, 212, 271; “Theory of SetComplexes in Music, A” (1964), 27, 89, 91–92, 98, 104–5, 134, 144–48, 179, 206n26, 210

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index FORTRAN, 237n2 Foucault, Michel, 17n15 Fowler, David, 86n2 Friedmann, Michael L., 39–40, 78n27, 182, 228, 237–38, 249. See also segmentation Fromm, Paul, 261 Fromm Music Foundation, 261 fugue, 5, 7n8, 23, 53, 62, 63, 65, 66. See also rectus / inversus fundamental bass, 137, 205n24 fundamental structure. See Ursatz Fux, Johann Joseph, 64–65, 73 “fuzzy inversion.” See inversion: “near-” (Straus) “fuzzy” relations, 130 “fuzzy transposition.” See transposition (T): “near-” (Straus) Gadamer, Hans-Georg, 218n1, 219n2 Gagné, David, 271 Gauss, Carl Friedrich, 85n1 gender studies, 235 generalized interval system (GIS), 71 Generative Theory of Tonal Music, A (Lerdahl and Jackendoff 1983), 25, 269 genus, 27, 179, 191n13, 216–17 geometry, 55, 130, 153–54, 169 Gilbert, Steven, 13, 271 GIS. See generalized interval system Gjerdingen, Robert, 136n6 Gloag, Kenneth, 201 Godlovitch, Stan, 225 Goehr, Lydia, 137 Goelet, Francis, 260 Gouk, Penelope, 254 Gould, Samuel B., 277 Greissle, Felix, 31 “Grenze der Halbtonwelt, Die” (Klein 1925), 43, 116 Griffiths, Paul, 262 group. See transformation group group action, 50–51, 114–15 Grout, Donald J., 263 Grundgestalt. See “basic set” Guggenheim Fellowship, 257n24 Guido of Arezzo, 30

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Hába, Alois, 43 Haimo, Ethan, 228n11 Hanson, Howard, 26, 43 harmonic dualism, 67n16, 213n32, 215–16 harmonic function, 25, 122, 185, 213, 219 harmonic inversion, 121, 128. See also bass position Harmonielehre (Weigl), 120–21 harmonisches Gefälle, 143, 177 harmony, tonal, 2n2, 32, 100, 200, 232, 233; theory of, 25, 33n4, 116, 120–21, 128, 143, 203–5, 213, 251, 254n20. See also bass position; dominant; harmonic dualism; harmonic function; harmonic inversion; subdominant; tonic; tonicization Harvard University, 18, 19, 265, 267, 269n45 Hasty, Christopher, 227, 229 Hauer, Joseph Matthias, 26, 44n16, 117, 122 Haydn, Joseph, 242 Heinemann, Stephen, 78n26 Helm, Everett, 263 hemiola, 199 Hempel, Carl, 250 Hermann, Richard, 176 hexachord, 88, 92, 94–97, 102–3, 109, 122, 145n12, 161, 164; medieval, 191. See also combinatoriality; Schoenberg, Arnold hexachord theorem, 96. See also complement theorem High Fidelity, 255 Hilbert, David, 254n20 Hiller, Lejaren A., 241 Hindemith, Paul, 233, 254n20, 271, 277; discussion of interval qualities, 139, 142; on the use of chord relations, 143–44, 177. See also chord classification; harmonisches Gefälle Hollerith, Herman, 238n3 Howard, Walther, 117, 122–23 Howe, Hubert S., 81–82, 91n10, 99, 102–3, 109. See also equivalence; multiplication (M)

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Huizinga, Johan, 265n39 Hüschen, Heinrich, 251n16 Husmann, Heinrich, 139n8 IBM. See International Business Machines Corporation IML. See Intermediate Music Language inclusion, 133, 148, 150; application of the concept of, 180–81, 187–89, 193– 94, 199; and complementation, 208– 9; degree of, 158, 160; and invariance, 206; involving musical objects with plural identities, 202, 205; relation, the, as a property, 191, 194; relation, transitivity of the, 214–15;. See also set complex; subset; superset information science, 130 Institut de Recherche et de Coordination Acoustique/Musique (IRCAM), 262 integers, 18, 29, 31, 33, 37, 45, 46, 52, 69, 82, 105–6, 110, 124, 237; Babbitt’s use of, 33–34, 50, 55, 67, 69, 79n28; Forte’s use of, 34, 35, 57, 69–71, 88 intentional fallacy, 228n11 International Business Machines Corporation (IBM), 237n2, 241–42, 244 Intermediate Music Language (IML), 238 International Musicological Society (IMS), 251n16 interruption, Schenkerian, 15–16, 230 intersection (set operation), 89, 110, 193n15, 203, 205–6, 213; definition of, 49 intertextuality, 235 interval class, 36–40, 69, 190, 191n13, 227, 252; Forte’s definition of, 37–40. See also absolute pitch-interval class (APIC); pitch-interval class (PIC) interval content, 10, 43, 46–47, 52, 82–83, 132, 200, 237n2; considered as the prime distinguishing property of a PC set, 92, 96; complementary hexachords equal in terms of, 96–97; equality of, abandoned as a basis of PC set equivalence, 98, 100–101, 103, 105, 129; Lewin’s definition of, 47;

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and sound, 97, 125–26. See also APIC vector; multiplication (M); PIC vector; Z-relation interval expansion, 78–79, 128. See also multiplication (M) interval function, 46–47, 49 interval vector, 47, 92, 105, 156, 183, 227, 228, 237, 248. See also APIC vector; PIC vector interval-difference vector, 153, 173–75 “Intervallic Relations in Atonal Music” (Teitelbaum 1965), 152 Introduction to Post-Tonal Theory (Straus 1990, 2000), 19–21, 27, 40, 41, 42, 177, 237, 270, 271 invariance, 206, 252 inverse, 38–40, 47–48, 62, 67, 68, 69, 72–73, 76–77, 78, 82, 85, 88, 114, 215 inversion (I), 10, 49, 55, 57, 62–76, 82–83, 96, 122n34, 125, 126, 133, 142, 144, 148, 157, 158, 161, 162, 169, 174, 207, 215, 227, 228, 230, 237, 247, 249, 270; axis of, 62, 67–74; defined as a multiplicative operation (Howe), 82; definition of, 62, 72–73; and equivalence, 84–87, 98–101, 103, 107–9, 110, 113, 131, 206; Forte’s definition of, 69; in fugal theory, 62–67; harmonic, see harmonic inversion; and harmonic dualism, 67n16, 215–16; index of, 69n21; Lewin’s definition of, 71–72, 74n24; “near-” (Straus), 61; normal form of, 104, 105; of pitch intervals (pitch-sets), 62, 68, 72, 73–74, 112; and prime form, 105–7; in tonal music, 65–66, 70; of twelve-tone series (Babbitt), 34, 50, 51, 67, 69; of twelve-tone series (Eimert), 80–81; by voice exchange, see harmonic inversion, bass position. See also combinatoriality; rectus / inversus; “type” (Rahn): “Tn/TnI-” IRCAM. See Institut de Recherche et de Coordination Acoustique/Musique Isaacson, Eric J., 133n2, 136, 148, 157, 163, 167–69, 173–76. See also similarity measures

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index Isaacson, Leonard M., 241 Ives, Charles, 91 Jackendoff, Ray, 25, 180n2, 269, 270 Jelinek, Hanns, 57, 116n29. See also allinterval series Jeppesen, Knud, 239 Josquin Desprez, 238–39, 242 Journal of Music Theory, 15, 17, 27, 93, 98– 99, 102, 152, 210, 244, 271, 272, 273 K. See set complex Kassler, Michael, 97n16, 239, 266n40 Katz, Adele, 273 Kepler, Johannes, 254 Kerman, Joseph, 3–4, 15, 16n14, 22–23, 219, 227n10, 235, 255n21, 256, 261n31, 273n49 Kerr, Clark, 264n37 key, 32, 43, 53, 64–65, 199–200, 201, 202–4; chords as subsets of a, 203, 213. See also regions, tonal key relationships, 68, 203, 205, 213–16; Weber’s chart of, 204 Kh. See subcomplex King, Martin Luther, 75 Klein, Fritz Heinrich, 43, 116–20. See also chord classification Koch, Heinrich Christoph, 32, 52, 53, 54n4, 55 Komar, Arthur, 15n12 Kopp, David, 195n16 Korsyn, Kevin, 235 Kostka, Stefan, 19–21, 22n17 Kraehenbuehl, David, 268, 271–73 Kramer, Lawrence, 22 Krenek, Ernst, 55n5, 116n29, 254n20; discussion of interval qualities, 139, 142; Studies in Counterpoint (1940), 26, 33, 57, 80; Über neue Musik (1937), 79–80; on the use of chord relations, 143. See also chord classification Kruyt, Hugo Rudolph, 265n39 Kuhn, Thomas S., 18n16 Kurth, Ernst, 220–21, 254n20 Lansky, Paul, 40, 57

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LaRue, Jan, 240, 242 Laudan, Larry, 18n16 Leeuw, Ton de, 4n6 Lehrbuch der Zwölftontechnik (Eimert 1950), 26, 57, 80–81 Leibowitz, René, 57, 87n5 Leichtentritt, Hugo, 234n14 Lejeune Dirichlet, Johann Peter Gustav, 85n1 Lerdahl, Fred, 23–25, 180n2, 183, 187n9, 189, 205, 269, 270. See also music cognition; segmentation Lessem, Alan P., 277 Lester, Joel, 22n17, 213n31, 252, 270 Lewin, David, 2n2, 26, 36, 46–47, 50, 71–72, 74n24, 92, 96–97, 98n18, 114, 158n20, 161, 163–65, 175, 185n5, 206, 223; critique of numerical PC labels, 35, 69; Generalized Musical Intervals and Transformations (1987), 19–20, 61; phenomenological perspective of, 198n18; “A Response to a Response” (1980), 167–69, 172. See also complement theorem; equivalence; generalized interval system (GIS); interval content; interval function; inversion (I); partial ordering; similarity measures; transformation group; transformations, theory of “Liberal Arts,” 254 Lincoln, Harry B., 242 Lippius, Johannes, 30 Liszt, Franz, 89 Lockwood, Lewis, 238–40 logic, mathematical, 38n10 logical positivism. See positivism Loquin, Anatole, 26, 33, 43, 117, 121, 124 Lord, Charles H., 135–36, 148, 153n16. See also similarity measures Louis, Rudolf, 120 Lowinsky, Edward E., 266 Luening, Otto, 259 Mac Lane, Saunders, 86 MacDowell, Edward, 265 MAD. See Michigan Algorithm Decoder Mann, Alfred, 54n4, 64n13, 64n14

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Mannes College of Music, 244 Marpurg, Friedrich Wilhelm, 54n4, 64–65, 66 Martin, Robert, 225n8 Martino, Donald, 47, 88, 92, 96, 97n17, 102–3, 259 mass education, 28, 277–78 mathematics, 10, 18, 27–28, 79, 86–87, 123, 128, 145, 151, 167–68, 237, 255, 261, 264, 277. See also absolute value, algebra, combinatorics, equivalence, geometry; group action; logic, mathematical; operation; relation; set theory, mathematical; standard deviation function; statistics; transformation group; vector Maus, Fred E., 223 McClary, Susan, 22, 23n18 McCreless, Patrick, 3, 17n15, 250, 268, 271 Mellor, D. H., 190n10 Mendel, Arthur, 238, 239n5, 266 Menke, Johannes, 116 Mersenne, Marin, 254 Messiaen, Olivier, 199–201, 202 Meyer-Eppler, Werner, 262 Michigan Algorithm Decoder (MAD), 247–48 MIR. See Music Information Retrieval “modes of limited transposition” (Messiaen), 199 mod(ulo) 7, 124 mod(ulo) 12, 29–30 Monteverdi, Claudio, 233 Morris, Robert D., 19–20, 101n22, 115, 123–25, 135, 136, 163–65, 167–68, 172, 176, 180n2, 185n5; on the audibility of PC set relations, 123, 133–34; Composition with pitch-classes (1987), 39, 61, 123; “A General Theory of Combinatoriality and the Aggregate” (1977/1978), 83. See also combinatoriality; contour, theory of; contour class; contour segment (cseg); cyclic interval succession (CINT); FB-class; interval class; multiplication (M); partial ordering; PCINT-class; pitch-class

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interval succession (PCINT); pitch set; pitch-set class (PSC) ; segment; similarity measures; Starr, Daniel “mother chord,” 116. See also all-interval series Mozart, Wolfgang Amadeus, 233 multiplication (M), 49–50, 51. 76–83, 157, 271n46; applied to twelve-tone series, 79–81; Boulez’s definition of, 78n26; defined by Eimert, 81; defined by Howe, 81; definition of, 76, 77–78; and equivalence, 110, 114–15; and interval content, 82–83; of order numbers, 79; of pitch intervals (pitch sets), 76; transforming a chromatic scale into cycles of fourths or fifths, 79–80. See also equivalence: cycle-offifths-; equivalence: cycle-of-fourths-; interval expansion; transformation group; “type” (Rahn): “Tn/TnI/M5/ M7-” Music Analysis, 225 music cognition, 269 Music Forum, 17 Music Information Retrieval (MIR), 239 “Music for the Masses” (Brody 1993), 256 Music Theory Spectrum, 17 musica ficta, 239n5, 264 Musical Quarterly, 55, 79, 256, 267 musicology, 239; new, 22–23 National Center for Bibliographic Data Processing in the Humanities, 242 National Endowment for the Arts, 260 National Science Foundation, 241 Neff, Severine, 33n5 new musicology. See musicology New York Philharmonic, 244 New York Times, 225n9, 263 New York University, 242 Newlin, Dika, 222n6 Newton, Isaac, 254 nexus set, 212–13, 214. See also set complex Nolan, Catherine, 33n4, 33n5, 43, 116n27, 121

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index normal form, 102–4, 237, 248; algorithm (Randall), 102–3, 105; Babbitt’s description of, 102. See also normal order normal order, 93, 102–4, 105, 107n23; algorithm (Forte), 104; “best” (or “optimal”), 104. See also normal form Notker Labeo, 30 null set, 45, 46, 110, 116n30, 193n14 numerical notation. See integers O King (Berio), 75–76 octatonic scale, 13, 109, 126, 194, 200– 202, 217; “collection I,” 126–27, 200; “collection II,” 126–27; “collection III,” 126–27, 200 octave equivalence. See equivalence Oettingen, Arthur Joachim von, 215–16 Olive, Joseph P., 242 Oliver, Alex, 190n10 operation, 27, 84, 85, 96, 99–100, 107, 110, 113–14, 115, 123, 128–29, 130, 157, 252, 271n46; binary, 49; as defined in PC set theory, 50–51, 76; twelve-tone, 10, 33–34, 50, 51, 67–68, 81, 82–83, 84, 96; unary, 50. See also complement; difference; group action; intersection; inversion (I); multiplication (M); retrograde; retrograde-inversion; transposition (T); union ordered PC interval. See pitch-interval class (PIC) “Our Musical Idiom” (Bacon 1917), 33, 122 overtone series, 251 Paine, John Knowles, 265 Palestrina, Giovanni Pierluigi da, 239 Parker, Horatio, 265 Parks, Richard S., 45, 179n1, 216–17, 233–34 parsimonious voice-leading, 195 partial ordering, 185 partition function, 158–62, 165–66 partition vector, 158–61 Payne, Dorothy, 19–21, 22n17

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PC. See pitch-class PCINT. See pitch-class interval succession PCINT-class, 124 PCSET (collection), 50, 84, 85, 86, 101, 109, 110, 114; definition of, 45–46 pedal, 195, 198 performance, analysis and, 220–21, 223–24 period (musical form), 15, 16 Perle, George, 4n6, 23, 57, 68n18, 70–71, 91n8, 91n9, 105, 184–85, 252, 270. See also absolute pitch-interval class; segmentation Perspectives of New Music, 17, 24, 35, 81, 83, 90, 158, 163, 167, 259n25, 261n30, 261n31 “Petrushka chord,” 200–202 PhD, 17, 266 PI. See pitch interval PIC. See pitch-interval class PIC vector, 47–48, 162n22, 168n28, 175–76, 206n25 PITCH (collection), 52, 61, 76–77, 84, 252; definition of, 45 pitch interval (PI), 9, 35–36, 37, 38, 52, 62, 68, 72–74, 75, 76, 123–24, 176; definition of, 35. See also inversion; multiplication (M); transposition (T) pitch set, 76, 110, 123–25, 176, 181, 271n46 pitch-class, definition of, 30 pitch-class interval succession (PCINT), 101n22, 124 pitch-class set (PC set), definition of, 40–41 pitch-class set complex. See set complex pitch-class set genus. See genus PITCHCLASS (collection), 50–51, 52, 57, 61, 76, 84, 93, 96, 115, 193n15, 252; definition of, 45 pitch-interval class (PIC), 36–37, 39–40, 46–48, 52, 55, 62, 72, 74, 82–83, 97, 100, 121, 124, 206n25. definition of, 36. See also absolute pitchinterval class (APIC); inversion; multiplication(M); transposition (T) pitch-set class (PSC), 123–25

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Pitch-Time Correlation (Randall), 82, 102 Pomeroy, Boyd, 195 Popular Front, The, 256n23 positivism, 23, 34–35, 249, 250, 253 Pousseur, Henri, 9 power set, 45, 191n12, 193 prime form, 105–7, 108, 109, 110, 112, 126, 177, 209, 212n29, 228, 237, 248; algorithm, 104, 105–6; of a twelvetone series, 57, 67–69, 80, 87n5, 92, 94n13 Princeton University, 9, 17, 237n2, 238, 250, 261n30, 266. See also Perspectives of New Music; Seminar in Advanced Musical Studies progenitor, 191, 193n15, 198n19, 199, 206, 215, 216; Forte’s definition of, 191n13 PSC. See pitch-set class PSET (collection), 111, 121, 123–24 Pythagoras, theorem of, 153 Rahn, John, 36, 114, 122, 123, 136, 142, 157n17, 163, 169, 172, 177, 183n4, 206n25, 268; Basic Atonal Theory (1980), 19–21, 27, 39, 44–45, 83, 109– 13, 237, 271; “Relating Sets,” (1980), 165–67; on tonal theory, 45, 113. See also equivalence; multiplication (M); similarity measures, “type” Rameau, Jean-Philippe, 30–31, 41, 53, 121, 134, 137, 254 Ramirez, Francisco O., 265 Randall, James K., 81–82, 102, 105, 238 Ratz, Erwin, 233 Ravel, Maurice, 233 rectus / inversus, 5, 67, 105 Regener, Eric, 36, 41, 97n16, 101n22, 109, 136, 144–45, 148n14, 157–61, 162–63, 165–66, 172, 238. See also common-note function; common-note vector; inclusion: degree of; partition function; partition vector regions, tonal, 205, 215 Reich, Willi, 79–80 Reicha, Anton, 203, 213n31 Reihe, Die, 35, 261n31

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relation, definition of, 84 Répertoire International de Littérature Musicale (RILM), 242 representative form (of a PC set). See prime form Réti, Rudolph, 233 retrograde, 34, 51, 55, 67, 80–81, 94, 111, 113; defined by Babbitt, 50 retrograde-inversion, 34, 51, 68, 80, 94–95; defined by Babbitt, 50 Riemann, Hugo, 25, 215–16, 254n20 RILM. See Répertoire International de Littérature Musicale. Rivera, Benito V., 31n1, 41 Robison, Tobias D., 238–39 Rockefeller Center, 275 Rockefeller Foundation, 241, 260 Rodriguez-Pereyra, Gonzalo, 190 Rogers, David W., 163, 168–69, 171 Roslavets, Nikolai Andreyevich, 91n8 Rothstein, William, 221, 273n49, 274 Roussau-Galin-Paris-Chevé method for sight-singing, 31 Rufer, Josef, 56 salience, 59, 188–89, 194, 199, 201–2, 205. See also set-class Salzer, Felix, 233, 252, 273 Sand, Ole, 276–77 Scarlatti, Domenico, 233n13 Schachter, Carl, 205n24 Schaffer, Sarah, 227 Schat, Peter, 31, 34, 35 Schenker, Heinrich, 3–4, 13–17, 32, 34, 220–22, 223–24, 230, 233, 239n4, 250n15, 252, 271, 273–74, 275n51, 276; Der Freie Satz, 14. See also interruption; Ursatz Schillinger, Joseph M., 33, 43 Schmalfeldt, Janet, 212n30 Schoedinger, Andrew B., 190n10 Schoenberg, Arnold, 9, 10, 18, 44, 55n5, 82, 83, 87n5, 89, 91, 96n14, 105, 116n28, 118, 121, 131, 135, 137, 143, 152, 180, 229, 234, 235, 239, 248, 250n15, 255n21, 264, 277; as an analyst, 137, 220–22, 233; historical awareness

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index of, 70, 94n13, 137; interval notation of, 56–57; PC-set awareness of, 97, 228n11; use of hexachords, 92, 94, 97. See also “basic set”; “basic shape”; developing variation; set; regions, tonal; “strong” and “superstrong” progressions; twelve-tone series Schoenberg, Arnold, works by: Book of the Hanging Gardens, Op. 15, no. 11, 237; “Composition with Twelve Tones” (1950), 44, 56–57, 92, 222n5; Four Orchestral Songs, Op. 22, 222n5, 233; Fundamentals of Musical Composition (published 1967), 180n2; Modern Psalm, Op. 50c, 95; Moses und Aron, 94; Piano Piece, Op. 11, no. 1, 41, 50, 183–87, 188, 194, 199; Piano Piece, Op. 23, no. 3, 4–7, 8, 9, 10, 13, 23, 70–71, 74; Piano Piece, Op. 33a, 95, 96, 111, 112, 113, 123, see also segmentation; Six Little Piano Pieces, Op. 19, 88–89, 90 (no. 2), see also segmentation (no. 6); String Quartet No. 3, Op. 30, 252–53, 270; String Quartet No. 4, Op. 37, 95, 102–3; Structural Functions of Harmony (published 1954), 198n17, 205, 214–15; Suite for Piano, Op. 25, 54 (Prelude), 41, 42, 270 (Gavotte); Theory of Harmony (1911), 187n7; Variations for Orchestra, Op. 31, 68, 88, 94; Wind Quintet, Op. 26, 67–68, 94n13 Schott Music, 120n33, 261 Schuller, Gunther, 259 Schumann, Robert, 14–16, 200n21, 274 science, history and, 218 science, music theory and, 23–24, 34–35, 102, 250–56, 264, 265–71, 274, 277– 78. See also Babbitt, Milton; university Scriabin, Alexander N. 45, 58, 89, 91n8, 135, 136 Scruton, Roger, 223 Seeger, Charles, 239n6 segment, Morris’s definition of, 180n2 segmentation, 23–25, 61, 70–71, 185, 189, 226–27, 228; ambiguity in, 181, 229; Benjamin’s criticism of Forte’s approach to, 24n20; computerized,

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244–47; Cook’s, of Schoenberg’s Piano Piece, Op. 19, no. 6, 229–32; Lerdahl’s view on, 23–24, 180n2, 183, 269; Perle’s, of the first measures of Schoenberg’s Piano Piece, Opus 11, no. 1, 184, 185; by sequential imbrication (Friedmann, Forte), 237–38, 249; Straus’s view on, 228, 229, 249 Seminar in Advanced Musical Studies, 261n30 Seminar on Comprehensive Musicianship, 275, 277 sentential calculus. See logic, mathematical Serial Composition and Atonality (Perle 1962), 19, 57–59, 91n8, 91n9, 184, 252 Sessions, Roger, 257, 273 set, definition of, 29 set complex, 27, 99n20, 179–80, 207–17, 230, 237n2, 247, 249, 271; analysis, 209, 211–12; evolution of the, 209; reference pair of a, 209, 212; tonal theory through the lens of the, 213– 16. See also subcomplex set theory, mathematical, 3n3, 29, 40, 89 set-class, 92, 100–101, 109, 207–8, 216, 226, 229–30, 237, 247, 216, 226, 229–30; identification, 237, 247; representing a, 103–5, 107, 109; salience, 188–89, 191, 194; and transformation group, 114–15. See also equivalence class; table of set-classes Shreffler, Anne C., 267n42 similarity, 179, 180, 227, 247, 249, 269; aural experience of, 131, 133–34, 136n6, 177; equivalence and, 130, 177, 182; from the composer’s perspective, 136, 177; as a focus of the study of musical structure and style, 132, 135; Forte’s introduction of PC set, 131, 173; maximal (first-order), see similarity measures: R1 (Forte); maximal (second-order), see similarity measures: R2 (Forte); minimal, see similarity measures: R0 (Forte); origins of the concept of, 130; and the pedagogical tradition of music analysis, 133, 134; “strong, “superstrong,” and “weak” (Lewin), 161–62

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similarity measures, 132, 133, 134, 136, 179, 222 ; ASIM (Morris), 164–65, 172, 179; ATMEMB (Rahn), 167, 169, 172, 179; EMB (Lewin), 165, 167; IcVSIM (Isaacson), 136, 173– 76; PM (Morris), 176; R0 (Forte), 134, 135, 136, 147–49, 150, 152, 158, 172, 247; R1 (Forte), 134, 135, 136, 145–47, 148, 150, 154, 155, 158, 160, 172, 247; R2 (Forte), 134, 136, 145–47, 148, 150–51, 154, 158, 172, 247; Rp (Forte), 134, 135, 136, 148, 157–58; RECREL (Castrén), 176; REL2 (Lewin), 167–69, 171, 172, 173, 179; S.i., (Teitelbaum), 132, 152–57, 162, 164, 165, 169, 172, 173–74, 175; sf (Lord), 135–36; SIM (Morris), 133–34, 163–65, 167; TMEMB (Rahn), 166–67, 183n4. See also APIC vector; common-note function; common-note vector; inclusion: degree of; partition function; partition vector Simms, Bryan R., 4n6, 7n8 Skizze einer neuen Methode der Harmonielehre (Riemann 1880), 215 Small, Christopher, 225n8 Smith, P. H., 239 Smith, Russel F. W., 266 Snarrenberg, Robert, 221, 273n49, 274 SNOBOL, 244, 246 Society for Music Theory (SMT), 17 “Some Combinational Properties of Pitch Structures” (Howe 1965), 81, 99, 102–3 Sontag, Susan, 236n1 “sound,” 82, 113, 125–26, 128, 217. See also equivalence; interval content source set, 96 “Source Set and Its Aggregate Formations, The” (Martino 1961), 47, 97n17, 102–3 standard deviation function, 173 Starr, Daniel, 83. See also Morris, Robert D. statistics, 130, 165, 167, 173 Stein, Erwin, 4n6, 31, 55n5, 117, 118, 120–21 Stein, Leon, 180n2

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Steinecke, Wolfgang, 263n34 Stockhausen, Karlheinz, 34–35, 261–62, 263 Storm van Leeuwen, W., 265n39 Straus, Joseph N., 27, 40, 41–42, 61–62, 125, 137, 177, 228–29, 235, 237, 249, 267, 270, 271; and the “myth of serial tyranny,” 267. See also inversion; segmentation; transposition (T) Stravinsky, Igor, 18, 44–45, 180, 233, 235, 257, 263n35; Agon (“Pas de deux”), 8–13, 24, 194, 199; PC-set awareness of, 228n11; Petrushka, 201; Symphonies of Wind Instruments, 126– 28, 200; The Rite of Spring, 212n30, 225–26, 228. See also octatonic scale; “Petrushka chord” Street, Alan, 4 “strong” and “superstrong” progressions (Schoenberg), 198n17, 205n24 “strong,” superstrong,” and “weak” similarity (Lewin). See similarity subcomplex, 208–12 subdominant, 213, 215 subset, 49, 88–89, 165, 176, 179, 181; common, 148, 157–58, 161, 166–67, 168; content, similarity measures based on, 148, 157–163, 165, 168, 172, 176; definition of, 45; invariant, 206; nomadic, 199, 216; proper, 45, 207; settled, 199, 206, 216. See also inclusion; key; set complex subsets, family of, 191–94, 199, 207, 209, 214–16 successive interval array, 100–102, 103, 122. See also cyclic interval succession (CINT) superset, 179, 181; definition of, 45; proper, 45, 207. See also inclusion; set complex supersets, family of, 191, 194, 199, 200, 209, 214, 216 tablature, 31, 32 table of set-classes, 93, 99, 103, 108 Tanglewood Festival of Contemporary Music, 261n30 Tarski, Alfred, 38n10

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index Taruskin, Richard, 23, 126n37, 127, 200, 219–20, 227, 234, 269; polemical exchange with Forte on The Rite of Spring, 219, 225–26, 228n11, 234 Tawa, Nicholas, 259 Teitelbaum, Richard, 91n10, 131–32, 135, 137, 152, 165, 172, 173, 176. See also similarity measures Thuille, Ludwig, 120 “Tn-type.” See “type” “Tn/TnI-type.” See “type” “Tn/TnI/M5/M7-type.” See “type” Toch, Ernst, 277 Tomlinson, Gary, 22, 23n18, 219 Tonal Harmony (Kostka and Payne 1984), 19–21, 22n17 tonic, 53, 65, 67, 69, 70, 200, 205, 213– 14, 215, 233 tonicization, 203 Traîté de l’harmonie (Rameau). See Treatise on Harmony transformation group, 51, 82n32, 114; canonical, 113, 115, 162; the concept of a, invoked by Lewin, 114 transformations, theory of, 26 translation, 55 transposition (T), 10, 49, 59–61, 78n26, 82–83, 87n4; definition of, 52; different labels for, 56–57; and equivalence, 85–86, 98–101, 110, 126, 128; Forte’s definition of, 57; in fugal theory, 53–54; as an inappropriate equivalent of Versetzung, 54n4; index of, 52, 69n21, 85, 99; Koch’s definition of, 52–53; “modal” (Friedmann), 78n27; of twelve-tone series (Babbitt), 34, 50, 51, 55–56; “near-” (Straus), 61–62; of pitch intervals (pitch sets), 52. See also invariance; “modes of limited transposition” (Messiaen); partition function; transformation group. transposition number. See transposition(T): index of Treatise on Harmony (Rameau 1722), 41, 53, 137 Treitler, Leo, 219, 223, 235n15 “Tristan” chord, 113, 125

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trope, 122 twelve-tone operations. See operation twelve-tone serialism, 9, 26, 33, 37, 83, 113, 143, 228; European versus American, 56, 225, 261n31. Milton; PC set theory influenced by, 31, 43–44, 60, 105, 206, 225; and scientism, 267. See also Babbitt, Milton twelve-tone series, 26, 37, 54–55; Dahlhaus’s definition of, 38–39, 82; “dependent,” 87n4; forms, hierarchy of, 67–68, 87n5; and octave equivalence, 31; and PC set, 10, 44, 59, 60–61; of Schoenberg’s Piano Piece, Op. 33a, 95, 96; of Schoenberg’s Third String Quartet, op. 30, 252–53; of Schoenberg’s Variations for Orchestra, Op. 31, 68, 88, 94; of Webern’s String Quartet, Op. 28, 88; of Webern’s Variations for Orchestra, Op. 30, 9, 10, 12–13. See also all-interval series; combinatoriality; equivalence; integers; inversion; multiplication; operation; prime form; retrograde; retrograde-inversion; transposition (T) “type” (Rahn), 109–10, 111, 112, 114, 123; “Tn-,” 122; “Tn/TnI-,” 237. See also Equivalence class union (set operation), 89, 193n15, 207; definition of, 49 Universal Edition, 261 universal set, 45, 57, 69, 88, 193n14 universals, 190 university, 9, 17n15, 239n6, 268n44; Babbitt’s view of the, 258–59, 264, 268; Boulez’s view of the, 263–64, 268; employment of composers by the American, 265–67; history of the American, 264–65; as a home for the creative arts, 265–66; teaching, Stravinsky’s aversion to, 263n35; types, 18. See also Columbia University; Cornell University; Harvard University; New York University; PhD; Princeton University; University of Wisconsin; Yale University

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University of Wisconsin, 20, 22, 264n36, 264n37 Unterweisung im Tonsatz (Hindemith), 138 Ursatz, 14, 221, 252, 273 Van den Toorn, Pieter C., 221n4, 45, 126, 200 Varèse, Edgar, 34, 257 vector, 47, 153. See also APIC vector; common-note vector; interval-difference vector; partition vector, PIC vector Vicentino, Nicola, 62, 63 Vincent, Heinrich, 33 Vogler, Georg Johann, 203 Waerden, B. L. van der, 86 Wagner, Richard, 124; Tristan und Isolde (prelude), 111–13. See also “Tristan chord” Wason, Robert W., 33n4, 205n24, 272 Weber, Gottfried, 203, 212. See also key relationships Weber, William, 227n10, 134 Webern, Anton, 2, 18, 34, 91, 94, 131, 152, 239, 248; Bagatelles for String

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Quartet, Op. 9, no. 5, 211–12, 214; Five Movements for String Quartet, Op. 5, 135 (general), 91, 131, 144–45, 234 (no. 4), 56, 58, 59–60 (no. 5); Six Pieces for Orchestra, Op. 6, no. 3, 71–72, 74; Three Pieces for Cello and Piano, Op. 11, 91 (general), 245, 246 (no. 1); Three Pieces for Violin and Piano, Op. 7, no. 1, 243, 244; Variations for Piano, Op. 27, 68–69, 70. See also twelve-tone series Weigl, Bruno, 116n27, 117, 120–22, 123, 124. See also chord classification Whittall, Arnold, 2n2 Wilcox, Howard J., 97n16 Winham, Godfrey, 81–82, 238, 266n40 Wittlich, Gary, 183 Yale University, 3, 17, 237n2, 265, 267, 268n44. See also Journal of Music Theory Zarlino, Gioseffo, 62, 63, 64 Ziehn, Bernard, 67n16, 91, 120 Z-relation, 83, 98, 99, 103, 104, 107, 109, 125, 144, 162, 174, 237

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Eastman Studies in Music The Poetic Debussy: A Collection of His Song Texts and Selected Letters (Revised Second Edition) Edited by Margaret G. Cobb Concert Music, Rock, and Jazz since 1945: Essays and Analytical Studies Edited by Elizabeth West Marvin and Richard Hermann Music and the Occult: French Musical Philosophies, 1750–1950 Joscelyn Godwin “Wanderjahre of a Revolutionist” and Other Essays on American Music Arthur Farwell, edited by Thomas Stoner French Organ Music from the Revolution to Franck and Widor Edited by Lawrence Archbold and William J. Peterson Musical Creativity in Twentieth-Century China: Abing, His Music, and Its Changing Meanings (includes CD) Jonathan P. J. Stock

The Gardano Music Printing Firms, 1569–1611 Richard J. Agee “The Broadway Sound”: The Autobiography and Selected Essays of Robert Russell Bennett Edited by George J. Ferencz Theories of Fugue from the Age of Josquin to the Age of Bach Paul Mark Walker The Chansons of Orlando di Lasso and Their Protestant Listeners: Music, Piety, and Print in Sixteenth-Century France Richard Freedman Berlioz’s Semi-Operas: Roméo et Juliette and La damnation de Faust Daniel Albright The Gamelan Digul and the Prison-Camp Musician Who Built It: An Australian Link with the Indonesian Revolution (includes CD) Margaret J. Kartomi

Elliott Carter: Collected Essays and Lectures, 1937–1995 Edited by Jonathan W. Bernard

“The Music of American Folk Song” and Selected Other Writings on American Folk Music Ruth Crawford Seeger, edited by Larry Polansky and Judith Tick

Music Theory in Concept and Practice Edited by James M. Baker, David W. Beach, and Jonathan W. Bernard

Portrait of Percy Grainger Edited by Malcolm Gillies and David Pear

Music and Musicians in the Escorial Liturgy under the Habsburgs, 1563–1700 Michael J. Noone

Berlioz: Past, Present, Future Edited by Peter Bloom

Analyzing Wagner’s Operas: Alfred Lorenz and German Nationalist Ideology Stephen McClatchie

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The Musical Madhouse (Les Grotesques de la musique) Hector Berlioz Translated and edited by Alastair Bruce Introduction by Hugh Macdonald

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Letters I Never Mailed: Clues to a Life Alec Wilder Annotated by David Demsey Foreword by Marian McPartland Wagner and Wagnerism in NineteenthCentury Sweden, Finland, and the Baltic Provinces: Reception, Enthusiasm, Cult Hannu Salmi Bach’s Changing World: Voices in the Community Edited by Carol K. Baron CageTalk: Dialogues with and about John Cage Edited by Peter Dickinson European Music and Musicians in New York City, 1840–1900 Edited by John Graziano Schubert in the European Imagination, Volume 1: The Romantic and Victorian Eras Scott Messing Opera and Ideology in Prague: Polemics and Practice at the National Theater, 1900–1938 Brian S. Locke Ruth Crawford Seeger’s Worlds: Innovation and Tradition in TwentiethCentury American Music Edited by Ray Allen and Ellie M. Hisama Schubert in the European Imagination, Volume 2: Fin-de-Siècle Vienna Scott Messing Mendelssohn, Goethe, and the Walpurgis Night: The Heathen Muse in European Culture, 1700–1850 John Michael Cooper Dieterich Buxtehude: Organist in Lübeck (includes CD) Kerala J. Snyder Musicking Shakespeare: A Conflict of Theatres Daniel Albright

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Beethoven’s Century: Essays on Composers and Themes Hugh Macdonald Composing for Japanese Instruments (includes 2 CDs) Minoru Miki Translated by Marty Regan Edited by Philip Flavin Variations on the Canon: Essays on Music from Bach to Boulez in Honor of Charles Rosen on His Eightieth Birthday Edited by Robert Curry, David Gable, and Robert L. Marshall Wagner and Venice John W. Barker Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts Michiel Schuijer

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For the past forty years, pitch-class set theory has served as a frame of reference for the study of atonal music, through the efforts of Allen Forte, Milton Babbitt, and others. It has also been the subject of sometimes furious debates between music theorists and historically oriented musicologists, debates that only helped heighten its profile. Today, as oppositions have become less clear-cut, and other analytical approaches to music are gaining prominence, the time has come for a history of pitch-class set theory, its dissemination, and its role in the reception of the music of Schoenberg, Stravinsky, and other modernist composers. Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts combines thorough discussions of musical concepts with an engaging historical narrative. Pitch-class theory is treated here as part of the musical and cultural landscape of the United States. The theory’s remarkable rise to authority is related to the impact of the computer on the study of music in the 1960s, and to the American university in its double role as protector of high culture and provider of mass education. Michiel Schuijer teaches at the Conservatory of Amsterdam and the University of Amsterdam. His research focuses on topics at the interface between music theory and historical musicology. “A penetrating study of pitch-class set theory, its mathematical foundations, and the context in which it was developed.” —Mark Delaere, University of Leuven (Belgium) “Schuijer’s book brilliantly situates pitch-class set theory—our dominant mode for analyzing atonal music—in its historical, social, and cultural contexts: this is a tour de force of intellectual history. What is more, it also provides a magisterial if, at times, pointedly critical summary of the theory itself. A deeply thought evaluation of a theoretical approach that has largely escaped critical scrutiny, despite its dominant position in contemporary North American music theory.” —Joseph N. Straus, Distinguished Professor, Graduate Center, City University of New York

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