Technical Bases of Nineteenth [PDF]

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78-7490 PROCTOR, Gregory Michael, 1941* TECHNICAL BASES OF NINETEENTH-CENTURY CHROMATIC TONALITY: A STUDY IN CHROMATICISM. Princeton University, Ph.D., 1978 Music

University Microfilms International f

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Ann Arbor, M ichigan 4C106

1978

GREGORY M ICHAEL PROCTOR

ALL RIGHTS RESERVED

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TECHNICAL BASES OF NINETEENTH-CENTURY CHROMATIC TO N A LITY; A STUDY IN CHROMATICISM

Gregory M ichael Proctor

A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN C AND ID AC Y FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF

MUSIC

January I978

The Principal Readers for this Dissertation were; M ilto n Babbitt J. K . Randall

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TECHNICAL BASES OF NINETEENTH-CENTURY CHROMATIC TO NALITY: A STUDY IN CHROMATICISM

Gregory M ichael Proctor

ABSTRACT

The application of a strict interpretation of Schenker's theories to classical tonal music w ith respect to the generation of chromaticism yields adequate a n a lytic results fo ra great number of tonal pieces. The adequacy of this system in handling some of the most involved chromatic w ritin g seems to suggest its a p p lic a b ility to the entire "common practice p e rio d ." But in the second quarter of the 79th century ex­ amples appear w ith increasing frequency which do not provide the same c la rity under these a n a lytic techniques as, say, 18th-century pieces do. Approaching these intractable passages from the viewpoint o f the 20th cen­ tu ry, however, leads to the positing of newer techniques of chromaticism which pro­ duce simple, coherent analyses. The application of these newer a n a lytic techniques displays consistency between these later compositions in their use o f chromaticism. These techniques thus provide a lin k between classical to n a lity and the music of the 20th century. The thesis here presented is that there is not a single "common p ractice" ex­ tending from the early 17th century through the end o f the 19th century. Rather, the era can be divided into two large, overlapping style systems, herein referred to as: classical diatonic to n a lity , and 19th-century chromatic to n a lity . The two systems are • • •

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distinguished from one another p a rticu la rly by th e ir respective conceptions of the ba­ ses of chromatic generation. Chromaticism in classical diato nic to n a lity is defined as the product o f the interaction o f d ifferent diato nic scales, w ithout reference to any particular chrom atic scale system (of which there are any number). For the d e fin itio n of the p ractice, the first part o f the dissertation contains a strict form ulation o f the operations o f tra d itio n ­ al counterpoint and o f the range of application o f Schenkerian concepts related to those operations. The specific processes by which notes foreign to the fundamental major or minor scale for each composition are generated are Schenker's formulations o f mixture and to n lc iz a tio n . These chromatic processes o f the classical tonal system adequately account for such tra d itio n a l chromatic constructs as: the diminished sev­ enth chords, N eapolitan chords, augmented sixth chords, and augmented triads. En­ harmonic "equivalence" is taken to be a no ta tio n a l, rather than a tonal, phenomenon. Chromaticism in 1 9 th -ce n tu ry chromatic to n a lity is defined in the second part of the dissertation. The basic prin cip le upon which it rests is the substitution for the diatonic scales of the equally-tem pered tw elve pitch-class collectio n as the source of a!! tonal m a te ria l. D iatonic material is then construed to be a special derivative of this underlying chromatic scale, and, insofar as diato nic matter is present, the classical procedures for its m anipulation are carried through in to the new system. But the concept of the underlying chromatic scale opens up new structural possibilities, w hich, when used in p ro life ra tio n , eventually work to undermine the specific q u a li­ ties of tonal directedness peculiar to classical to n a lity . The first new structural posiv

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s ib ility is that enharmonic equivalence as a to n a l, as w e ll as a n o ta tio n a l, phenom­ enon is e xp lo ite d . This leads, fo r example, to the m ultiple interpretation o f diminished seventh chords and the re interpretation o f German augmented sixth chords as dominant sevenths. It also provides for the symmetrical division o f a tone-space in to equal-si zed segments as measured in tempered semitones. Also presented is a new "o p e ra tio n ,11 that o f transposition. In classical to­ n a lity , in d ivid u a l harmonic voices from chord to chord are either m aintained by com­ mon tone (which includes the notion o f arpeggiation) or are displaced by step, in ac­ cordance w ith the operations o f tra d itio n a l counterpoint. The transposition operation allows for th e ir wholesale shift from one pitch level in tone-space to another, w ith ­ out reference to common tone or stepwise displacement. Numerous examples o f a ll these chrom atically tonal techniques are provided, including the use of symmetrical division as the p rinciple o f transposition. The dissertation closes w ith a discussion of the gray area where the d is tin c ­ tion between the two systems in some composition is th in . This discussion shows d ia­ tonic movement as the basis o f the p rin cip le o f transposition, and classically tonal models as analogs— but not analyses— o f 19th-century chrom atically tonal passages.

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PREFACE

The history o f Western music is ty p ic a lly divided into style periods differen­ tia te d by texture, instrum entation, preferred forms, and melodic and harmonic cliches. W ith a certain tim id ity , observations are occasionally advanced concerning d iffe r­ ences o f a more pitch-structural nature, such as m odality vs. to n a lity , strict vs. free counterpoint, to n a lity vs. a to n a lity , and so fo rth . These distinctions, though fre­ quently ap t, are mostly described in vague or general terms, and allow the joining together o f such distinct compositional styles as those of M ozart and Wagner into an era of "common p r a c t ic e . T h e i r dissim ilarities, even w ith respect to harmonic pro­ cedure, and the dissim ilarities between the Classical and Romantic periods in general, are then taken to be based on differences of aesthetic approach— poetic differences, concerned w ith th e ir relative concepts o f the place of the individual in the scheme o f things. Although such differences exist, there remains the undeniable presence of harmonic practices in Wagner that could not have been found in M ozart, regardless of th e ir source. Relevant to this is the knowledge th a t, in the course of attaining a precise explanation of his description of tonal pieces, Schenker was forced to narrow the bounds of a p p lic a b ility o f his system evermore rigorously, to the inclusion of Moz­ a rt, for example, and the (apparently reluctant) exclusion of Wagner. Schenker, in

1 W alter Piston, Harmony, 3d. ed. (New York: N orton, 1962), pp. ix ana x. VI

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developing his system, was pleased to discover fundamental procedures of pitch or­ ganization in the music o f the great classical masters, and w ith such authority behind him, fe ll in to the unexpected position o f sharing w ith the trad ition al harmony teacher the b e lie f that there were basic laws of composition which the excluded composers did not so much break as fa il to comprehend. Even i f we assume the truth of some de­ gree of fa ilu re o f comprehension, we find it harder to go along w ith Schenker who, engaged in his great w ork, was not of a mind to investigate the tenuousness of his po­ s itio n , a position which leaped from that failure of comprehension to a composer's in a b ility to w rite good music at a ll. Immutable laws o f harmony are for harmony students, not for master compo­ sers. Composers do not obey rules as much as they explore the possibilities of coher­ ence w ith a certain body of received tonal m aterial, and they do not break rules as much as they proceed to an investigation of new possibilities. If the composer pos­ sesses the innate a b ility to form materials in such a way that a listener is thereby led to a contem plation of his own mental processes as a human being, then it is largely immaterial whot paths of organization he chooses to employ in this form ation. The brain works along certain patterns consistent with larger formal patterns outside it ­ self. There is no way to a direct knowledge of these patterns. The purpose of a rt, as Schenker himself argues so consistently,

is to provide the vehicle whereby the art­

ist, endowed w ith a superior sense of the forms coincident w ith these patterns, com-

2 For example, in his "Der wahre Vortrag, " Der Tonw ille V I (Vienna: Universal, 1923), pp. 36 - 40. v ii

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munfcates in His art this awareness to the p u b lic , w hich,

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the w ork, takes pleasure in the process.

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upon true comprehension of

ine relevance o f the material w ith in which the composer works lies in the audience's a b ility to use the quasi-linguistic system to its own b e n e fit, and in this lies also the circum scription of style periods, and, more p a rtic u la rly , of tonal pro­ cedures. If the m aterial o f some system is to ta lly foreign to the listener, he is more than lik e ly incapable of receiving the form, regardless of its perfection and s ig n ifi­ cance. This assumption leads to a view of tonal history which is both evolutionary and revolutionary. O ver a large stretch of tim e, differences of procedure are evident; to these we can ascribe the characteristics of a style era. But the differences are gradu­ a lly worked out by composers so that there is extensive overlap of technique. This suggests that the processes are continuous and that a ll such style boundaries are arbi­ trary. The way in which a culture views its e lf, however, as w e ll as social pressures, tends to cause a w a ve -like pattern of change where techniques gather and remain fa ir­ ly stable for a w h ile . This is where the other style considerations mentioned at the outset can help to define an era better. Y e t, each change of procedure is a deliber­ ate choice on the part of the individual artist working from the materials which sur­ round him , and in this respect the process is revolutionary— the apparently continu­ ous wave is the sum of discrete values. If an in dividual master's extra step is incorpo­ rated in to the general paractice of his contemporaries he has contributed to the d e fi­ nition of a style era. That Liszt composed a tw elve-tone melody in the middle of the 19th century is no evidence of his influence on composers of the middle of the 20th

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century, although if might be taken as an example o f what had become conceivable in his tim e. The d e fin itio n o f tonal practices o f groups o f roughly contemporaneous com­ posers into style eras can help us grasp their works w ith greater penetration o f th e ir meaning by putting them in to th e ir tonal context, especially as th e ir "common prac­ tic e " recedes from us in tim e . Such d e fin itio n has been poorly developed in the past, w ith Schenker being the first to adequately describe the procedures of onesych style, which w ill be referred to herein as classical diatonic to n a lity . In his Structural Hearing^ Felix Salzer extended the bounds placed by Schenk­ er on his own a n a lytic system. His purpose was to demonstrate that the great part of Western music is pitch-w ise coherent according to consistently shared principles of tonal organization, regardless of specific style contexts, or even despite differences concerning the constituents o f a stable sonority in the system of each era. In particu­ la r, Salzer based his form ulation on a generalization o f the Schenkerian principle of structure and prolongation; he showed that the principle can be found in very early polyphonic music and in 20th century music, as w ell as in the intervening more fami­ lia r areas o f its a p p lica tio n . Salzer's feat is further to be admired on the grounds th a t, in performing it , he made a contribution sim ilar to that of the artists themselves who composed the music, inform ing us o f some o f the ways in which we have perceived the w orld.

3 Felix Salzer, Structural Hearing (New York: Charles Boni, 1952); 2nd e d itio n , (New York: Dover, 1962). ix

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It was not Salzer's intention to point up the differences between eras, how ever, and this is the task that confronts us now. It is only in a distant sense that one

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can say that the seeds o f the a to n a lity o f the Second Viennese School are contained in the tonal system of the First Viennese School. The evolutionary concept requires

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that we first search for this style in the late 19th century, and other knowledge gives us a good idea o f where to lo ok. But the 19th-century innovations do take classical to n a lity as th e ir point o f departure, and the more specific we are in our search, the

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more we can discover the development of a lineage lin k in g Schubert, Berlioz, Chopin, Wagner, Liszt, Bruckner, W o lf, G rie g , and Chaikovsky to the Impressionist and Mod­ ern composers. Salzer showed that the many dialects of the Western polyphonic tradition are capable o f relation to a common generalized lin g u istic pattern; but to c la rify the his­ tory of the development from d iato nic to n a lity to the 20th century, it is useful and necessary to focus upon the specific differences of d ia le c t. The thesis here is that the Romantic style differs on the grounds of tonal tech­ nique from the Classical style perhaps even more s ig n ific a n tly than the two d iffe r with respect to the more commonly noted textural patterns and aesthetic motivations of the respective classes of composers. It is a comparison of pre-Romantic and Romantic styles from the point of view o f pitch organization as it is displayed most d ire ctly in their re la tive uses of chromaticism. The tool for such a d iffe re n tia tio n w ill be Schenker's description of diatonic

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to n a lity confined w ith in its narrowest lim its. This w ill enable us to observe the special

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ways in which Romantic music fa llin g outside of those lim its differs from classical d ia -

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tonic to n a lity . The differences define a new body of techniques w hich, in turn, form

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the basis o f a newer tonal language from which post-tonal music develops.

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The first part of this paper is a fa irly rig id description of the elements o f Schenker’ s system pertinent to our task, emphasizing those elements bearing upon chro­ maticism. Some o f the elements are defined d iffe re n tly from Schenker's formulations but they are intended to conform w ith those formulations. The second part of the paper is a description of the special techniques pe­ cu lia r to the 19th-century departure from the old system. The new system w ill be re­ ferred to as 19th- century chromatic to n a lity .

This thesis depends on the work o f many who have been my teachers, and i am deeply grateful to them. Their contribution is somewhat evident in the references in the te xt of this paper. But what cannot be so indicated is my indebtedness to the late G odfrey Winham, whose relentlessly insistent and penetrating mode of thought has had a direct impact on this w ork. Although he le ft behind many notebooks which contain his contemplations on theoretical subjects, Winham published very little in his life tim e , and nothing in the area of tonal music. Posthumous publication of some of the m aterial in his notebooks is in preparation; when that publication takes place, his mark on this work w ill be cie ar.

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TABLE OF CONTENTS PAGE ABSTR AC T..................................................................................................................

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P R E F A C E ..................................................................................................................

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PART ONE: CLASSICAL D IATO N IC TO NALITY CHAPTER I.

SCHENKER'S SYSTEM AS B A S IS ........................................................................ Level and M o t i v e ............................................ * • Tonic and Key . Harmony and Chord ............................... Scale, Consonance and Dissonance, Q u a l i t y ................................

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THE OPERATIONS OF TRADITIONAL COUNTERPOINT

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Class I: Space-defining O p e ra tio n s ............................................. Class II: Note-generating O p e r a t i o n s ...................................... Class III: Time-span-extending Operations . . . . . . The Sequence of Classes . . . . Comments on the O p e r a tio n s ............................................................... ............................................................................................... Class I Class I I ............................................................................................... Class I I I ............................................................................................... Absolute D isp lacem ent........................................................................... . Resolution, S ta b ility and I n s t a b ilit y ...................................................

13 14 16 17 19 19 22 30 33 35

CHROMATICISM IN THE D IA TO N IC SY STE M ......................................

42

The Chromatic P ro c e s s e s ..................................................................... A . M i x t u r e ........................................................................................ B. T o n ic iz a tio n ..................................................................................

43 43 65

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CHAPTER IV .

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C O M M O N CHROMATIC CHORDS IN D IA T O N IC TO N A LITY

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Diminished Seventh Chord .................................................... The N eapolitan C o m p le x ....................................... . Augmented Sixth C h ord............................................................................. Augmented T r i a d ...................................................................................

88 97 115 123

PART TWO: NINETEENTH-CENTURY CHROMATIC TO N ALITY CHAPTER I. THE CHROMATIC SCALE AS B A C K G R O U N D ......................................

131

Enharmonic E q u iv a le n c e ....................................................................... Chromatic Neighbor and Passing N o t e s ............................................. M o d a lity ..................................................

131 139 143

SYMMETRICAL D I V I S I O N S ......................................................................

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Effect o f Symmetrical Division on T o n a l i t y ....................................... Symmetrical D ivision vs. A r p e g g ia t io n .............................................

155 157

THE TRANSPOSITION O P E R A T IO N .........................................................

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The Principle o f T ra n sp o sitio n ................................................................ Transposition as an O p e r a tio n ................................................... .

165 168

EXAMPLES OF SYMMETRICAL DIVISIONS A N D T R A N S P O S ITIO N S ..........................................................

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Symmetrical Division W ithout T ra n s p o s itio n ....................................... Symmetrical D ivision W ith T ra n sp o sitio n ............................................. Linked Symmetrical D i v i s i o n ................................................................ Layering of Symmetrical Divisions . C o m p o u n d s ................................

171 181 200 205 211

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D IA TO N IC INFLUENCE O N CHROMATIC TO N A LITY

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D iatonic Basis o f the Principle of T ra n s p o s itio n ................................ Tonal Models as Analogs of Chromatic M o d e l s ................................

220 237

B IB L IO G R A P H Y ......................................................................................................

251

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PART ONE CLASSICAL D IATO N IC TO N A LITY

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CHAPTER I SCHENKER'S SYSTEM AS BASIS

We begin by asserting that Schenker's theories, as expounded in his Neue Musikaiische Theorien und Phantasien, and the periodicals: Der Tonw ille and Das Meisterwerk jm der M usik, 1 Forms the absolute basis of this description of classical diatonic to n a lity , and, w ith triv ia l exceptions, this description swallows his system whole and e n tire .2 What is om itted, for example, is any attempt to ju stify his aside that neighbor motion is derived from passing m otion,3 elsewhere contradicted and relating to some extent to Schenker's more general use of the term "passing, "4 or that the requirement o f the downward resolution of a suspension in strict counterpoint

1 Heinrich Schenker, Neue Musikaiische Theorien und Phantasien: V o l. I: Harmo­ ny, edited by Oswald Jonas (University of Chicago, 1954), a translation of: Harmonielehre (Vienna: Universal, 1906); V o l. II: Kontrapunkt. Part I (Vienna: Universal, 1910), Part II (Vienna: Universal, 1922); V o l. Ill: Der Freie Satz (Vienna: Universal, 1935), 2nd. ed. (Vienna: Universal, 1956); Der T o n w ille , ten volumes, referred to herein as: l - X Nos. V III and IX issued as one number (Vienna: Universal, 1921 24); Das Meisterwerk in der M usik, three volumes, referred to herein as I - III (Mu­ nich: Drei Masken, 1925, 1926, 1930). Translations of much o f Das Meisterwerk can be found in : Sylvan K a lib , "Thirteen Essays from the Three Yearbooks 'Das M eister­ werk in der Musik1 by Heinrich Schenker: An Annotated Translation" (Ph.D) disserta­ tio n , Northwestern U niversity, 1973), three volumes. 2 A thorough statement of Schenker's theories in English can be found in : Sonia S latin, "The Theories o f Heinrich Schenker in Perspective" (Ph.D , dissertation, Co­ lumbia U niversity, 1967). This treatise is here accepted for the most part as an accu­ rate statement of Schenker's concepts. 3 Schenker, Tonw ille V l l l / l X , 50. 4 See below , p. 30.

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is due to the prior presence of a passing note configuration, the upper boundary of which is e lid e d .

A p p lica tio n of these constructs is extremely lim ited in specific

analyses by Schenker of real compositions— w ith rare exceptions like the presenta­ tion of one instance of the la tter in his analysis of M ozart's A minor Sonata K . 3 l0 .^ Also om itted is the generation of "altered chords" (= augmented 6th type) by simul­ taneous statement of two harmonies as proposed in his H a rm o n y , ^ on the grounds that the phenomenon is more d ire c tly accounted for by other, more universal, com­ ponents of the system. Since the aspect of Schenker's theory associated w ith structural analysis is i:

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brought to its highest sophistication in Der Freie Satz, Schenker's e a rlie r writings tend to be discounted as early tentative steps toward his fu lly developed model and are thought to be safely disregarded in learning what that method is about. This ne­ gative discrim ination is especially applied to his Harmony. Schenker him self, how­ ever, as late as Der Freie Satz, regularly refers to concepts developed in Harmony, and for good reason: firs t, Harmony contains the original firm statement of the d if­ ference between the harmony on the one hand

and the actual m aterial on the sur­

face of the composition— a distinction basic to the entire further development of the system; second, it contains his only systematic reference to the origins of chro­ matic notes in a d iato nic tonal piece. Schenker never disavowed in print the basic

5 Schenker, Kontrapunkt I, pp. 340 -4 6 . 6 Schenker, Tonw ille II, 16. 7 Schenker, Harmony, 277ff.

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assertions of the early w ritin gs, c h ie fly because there is no real opposition between them and the fin a l version of his theory as it is contained in Der Freie Satz. The clear difference (aside from the development of his graphic notation) between his earlier and later writings lies in the depth to which concepts in the former are car­ ried out in the la tte r, most p a rticu la rly in the extensions upwards in the hierarchy of tonal relations of the composing-out principle to a single structural progression, the Ursatz, underlying the entire composition.

Level and M otive There are two interrelated fundamental features of Schenker's w ritings w ith ­ out which no adequate presentation of his ideas is possible. They are 1) the concept of to n a lity as a hierarchal system, and 2) the necessity for m otivic development as the associative medium for music. N either of these ideas is special to Schenker's work alone, but what marks his theory o ff from other descriptions of to n a lity is the extent to which— and the specific way in which— they are carried out, fo llo w in g Schenker's understanding of the process of composition. Thus, for the hierarchy of to n a lity to be made e x p lic it requires the representation o f the composition as a se­ ries of levels from some background to the foreground, the composition its e lf, where each level is a true variation of the prior level carried out in accordance w ith a fixed set of procedures common to a ll (where possible). It is clear in the study of harmony under any theorist that a "secondary" or "app lied" dominant, for instance, owes its existence to the prior presence of the "secondary" to n ic , the object chord. Peculiar to Schenker is the extension of the p rio rity concept to a ll chordal re la tio n -

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ships regardless of la bel.

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It is the same w ith m otive. Many theorists and musicographers discuss motives in a musical work, but norm ally only on what might be considered an absolute

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foreground le v e l, a feature most noticeable by the c h ie fly rhythm ic, rather than to n -

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u n til arriving at the ultim ate m otive, the U rlin ie , or basic m elodic lin e .

al constitution, o f the material involved. Schenker, on the other hand, carries motiv ic development through a ll the layers o f to n a lity , layering the motives themselves

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W ith Schenker, unless a ll pitch events in a composition participate unequivo ca lly in a hierarchy, and can therefore be found to originate at some specific lev­ el with respect to other prior events, a piece fa ils to be tonal; and unless the motives themselves are layered in the form of Zuge and underlying neighbor note configura­ tions, thereby d icta tin g the boundaries and nature of subsequent motives, a composi­ tion fails to be a masterpiece at best, and is probably not even interesting. The two basic concepts come together in that motive is the key to the generation of subsequent levels.

Tonic and Key

A t the apex of this tonal hierarchy is a single pitch-class® which is the ton­ ic note. D iatonic to n a lity names this note as the key of the composition said to lie

8 In what follow s, the distinction between pitch and p itch - class is sig nifica nt, whereas "note" is used indiscrim inately for either when the distinction is irrelevant to the discussion.

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w ith in the system. The tonic note bears w ith it its tria d , which Schenker derives

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from relations contained in the overtone series, although the v a lid ity of his system

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(or of any system) can by no means be considered to depend on this or any other

Hi em pirical ju stifica tio n of its basic assumptions.

Harmony and Chord

The operation in music of the to n ic tria d as the source of a ll subsequent e vents can not be attained unless this tria d , a pitch-class c o lle c tio n , is made speci-§

fica n d participates in change. The to n ic tria d is a harmony, and as such is an abs­ tra ctio n . M usic requires its concretization through the medium of counterpoint, a

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process em ploying the operations of tra d itio n a l counterpoint, dealt with below in Chapter l i . This accounts for Schenker's refusal to acknowledge the ultim ate to n ic triad (an abstraction) as a phenomenon representable by music notation (which is p itch -sp e cific) except by name or Roman numeral.. It also leads to a c ritic a l distin ctio n between chord and harmony,

a distinction peculiar to to n a lity .

Any simultaneously operating c o lle c tio n of pitches contitutes a chord, or at least a sim ultaneity (Zusammenklang). Chords are the result of the app lication of the operations of counterpoint. A harmony, on the other hand, is the abstrac­ tion of the pitches of some chord into a pitch-class co lle ctio n for the purpose of establishing boundaries for the subsequent generation of new pitches and chords. It is this distinction that removes common Schenkerian references such as change of

9 See, for example: S latin, "Theories of S chenker," Chapter III.

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register, Ubergreifung, U ntergreifung, voice exchange, et a l i i , from the set of operations. They are compositional procedures and are what the abstraction in to harmony accomplishes. M usical presentation o f more than one part at a t im e ^ de­ picts pitches and displays counterpoint. A harmony, as an abstraction,I ite ra lly ap­

The theory o f levels thus-carries w ith it the necessary notion o f representa­ tio n , i f the levels are to have any serious re latio n to the construction and percep­ tion of the composition. The elaboration of a contrapuntally generated prior level chord which has been abstracted in to a harmony and then rearranged to prepare the elaboration, is a representation of that prior chord now a harmony. Any chord or

^

’;

V

S

S

v

T

S

l

v

'

S

i

T

i

T

;

;

pears nowhere in the piece but is represented by pitches in the counterpoint.

11

:-

■ :

ornamental note at a subsequent level is part of a representation of the m odel.

A t the foreground le v e l, for exam ple, signs of ornamentation are used together w ith the note to which they are app lied, so that the notation depicts that which is to be represented together w ith a scheme for representing i t , but the notation does not contain the pitches to be produced lite ra lly displayed as noteheads and stems. G i­ ven a notation, Ex. I: Example 1. CO

10 "V o ice " and "p a rt" are understood here to be distinguished according to the formulations in: Arthur Komar, Theory of Suspensions (Princeton University Press, 1971), 40. ~ 11 This does not speak, of course, of events generated at some level which are unaffected at the subsequent le v e l, where "le v e l" is here used in the general sense. Pedal notes are also background events that show up on the foreground and repre­ sent themselves.

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we understand It to be a shorthand for Example 2, where the method of depiction Example 2.

I 8

leaves some lit t le submetrical room for rhythmic variance. The note c2 is understood

S? to be represented by the entire group of sixteenth notes, but the performance includes a d^ and b l in addition to two statements of c2. The two c2's in the result are not '.v:

the note undergoing prolongation; they are instances of that note. S im ila rly, with chords on a larger level (Example 3), "representation o f" and "instance o f" are to be Example 3 . M ozart: Piano Sonata K. 284, I I ; mm. 1 - 2„

|

i

‘f i t

'

~

Tt

distinguished. There are several instances of the notes of the A major triad in Example 3 but none of them in particular is the harmony — also an A major triad — undergoing prolongation, i . e . , representation. 12

12 The failure to make this distinction is what lies at the heart of the criticisms of Schenker's theory as "s ta tic ," and, more sig n ifica n tly, leads to the misconstruction of the meaning of his use of Roman numerals in analyses.

r 'i

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The graphic notation of levels is thus a notation of counterpoints as prolongational of harmonies abstracted from prior level counterpoints. The discipline of counterpoint is, th e n , the study of chord generation, and the discipline of har­ mony is the study o f layers o f counterpoint. That this distinction between harmony and chord is tra d itio n a l and w idely established can be seen from the fo llo w in g quotes. The first is byM arpurg,the fore­ most advocate of Rameau's theories in Germany in the 18th century: H a r m o n i e und A c c o r d sind wie das Gantze und ein TheiI von einander unterschieden; und wenn ein m it einem andern zu gleicher Z e it g e horter Thon ein A c c o r d heisset, so kan die H a r m o n i e durch eine Reihe aufeinander folgender Accorde, so w ie die M e I od i e durch eine Reihe aufeinander folgender eintzelnen Kliinge oder intervallen e rk la ret werden. 13 [Harmony and chord are distinguished from one another as is the whole to the part; and if simultaneous tones are called a chord, so also can a ser­ ies of successive chords be explained as a harmony, just as a melody can be described as a series o f successive sounds or intervals^ The next pair of quotes comes from a very different source— popular music of the old days: The basic harmony is the essential, and embellishments must not turn the course of the tune; . . . One need never be afraid to w rite notes in the melody that do not belong to the harmony. 14 And: Passing chords are of great im portance, both in serious and popular mu­ sic, and especially in hymn tunes, but they must never be confused w ith

13 Friedrich W ilhelm Marpurg, Der Critische Musicus an der Spree, N o . 18 (Ber­ lin , 1749), p . 147. 14 Frank Patterson, How to W rite a Good Tune (New York: Schirmer, 1925), 19.

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10

basic harmonies. A tune must belong to the basic harmonies, not to the passing chordsf the object of which is m erely to strengthen the rhythm. There are no more fa m ilia r examples o f this than the Rosary and Love Sends a L ittle G ift o f Roses. (a) Rosary-Nevin J f - f = = - - - - i - ----------------------------i -------------------

(b) Love Sends a L itt e G ift

1- fjt— |------------------- r+-&— h r--------- 1---------—

i- '- s . -

Ni l 1 -Mia---- ----------

.4 — 4 —

OK/*"'1------------¥~'------TTi— I a f 7 1f

lit

Q -A ” ---------- j i ------------■

r r

= Tonic

r

j ■■■


p ——i

a n t.

=»

m

%--

Suspensions and anticipations are norm ally discerned in context by the presence of

3 Komar, Suspensions, 39.

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at least one other voice whose behavior suggests the application o f the Class III op­ erations rather than the rhythm ic generation of the notes d ire c tly in th e ir fin a l form .4 In Example 10, the connection is taken by the simplest association, that o f having the suspension and a n ticip a tio n intervals arise out o f a stream o f parallel sixths, ra­ ther than d ire c tly in th e ir role as neighbor and passing notes w ith dis|unct attacks creating a string of a lte rn a tin g ly dissimilar intervals. Example 10. from: F-

I' B

&L

I

~0~

£

5 -6

7 -6

7-6-5

from:

from: TW

t§

i

W ii;

It II

The Sequence of Classes The note generating operations require the prior application of the space definers, and the time-span extenders require the prior application of the note gener­ ators. The conclusion that Class I is background to Class II and that Class II is prior

s!i

m to Class III does not prohibit in any way the subsequent application o f a prior class type to configurations generated from a subsequent class type at a higher le v e l. Thus,

i to ;.

a suspension may be generated and the repetition operation applied to it to create a

i i I

I

4 Komar, Suspensions, 26.

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a "re-struck" suspension. Restriking is, in fa c t, the normal way for an an ticip a tio n to be stated at the foreground le v e l. Example 11.

(Ex. 11.)

Beethoven: Piano Sonata, O p. 31, N o . 2; mm. 5 -6 .

i j O D f) J jj % § This example contains both restruck suspensions and restruck anticipatio ns. The first note in the right hand of each of the first three beats is a suspension. The g j^ is an anticipation of the seventh degree of A major altered from the fourth degree of D minor, and acting as a member of a neighboring leading tone chord to the A major tria d . The sixteenth note a^ in the second measure o f the example is an an ticip a tio n of the a l of the chord of resolution. (Ex. 12.) Example 12.

rIV

r v-

The foreground is derived from this model by the application at s till lo w e rje ve ls of (A ft-

t': &

the Class III operations and then subsequently, the Class I operation o f re p e titio n . Any operation may occur at any level provided the proper sequence of classes is fo l­ lowed. This results in the prevention of the appearance of Class II operations on the pre-Ursatz le ve l, and the appearance o f Class III operations on the Ursatz le v e l.

K! 8 P ¥ I

(The pre-Ursatz level is, of course, the level of the conception of the to n ic harmony.)

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19

Comments on the Operations

Class I The pair o f operations making up Class I are not conceived of by Schenker as of the same sort o f contrapuntal a c tiv ity as those o f the other two classes. This is due to the notion of representation, where the analysis o f a counterpoint stands for the discovery of a model progression, a prior level counterpoint, where elements of the prior level may be instanced in the subsequent level but are not id e n tica l w ith those elements of the prior le v e l. The presentation here seems to suggest that given, for example, a passing note fig u re , the boundary notes are an arpeggiation o f the members o f a chord at the prior level and are id entical w ith those members. This is not s tric tly indended. Arpeggiation as w e ll as repetitio n are meant here to derive from abstraction o f the chord— the harmony. An analysis which includes the reduc­ tion out o f the counterpoint of a passing or neighbor note (leaving the boundaries standing) indicates an extra stage not normally notated d ire c tly in Schenker's graphs. E i

P

Indeed, Schenker does not observe the existence o f our Class I as preparation for

I Class II operations. He sees neighbor and passing note figures as a ll together derived & fy-

I

immediately as linear expressions of chordal facts. This is to say that the repetitio n component in neighbor figures and the arpeggiation component in passing figures come together w ith rhe passing or neighbor notes themselves, and are inseparable from them. G iven the scheme here presented, nevertheless, once the arpeggiation

Si:

in a passing fig u re , for instance, is discovered and the notes are brought together in a chord, these boundary notes may or may not be id entical in register w ith those

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of the model progression. If they are id e n tic a l, then they d ire c tly state some aspect of the model; i f they are not id e n tic a l, the model must be shown at another graphic stage. Contrapuntal elaborations o f some prior level chord frequently introduce ad­ ditional registers and therefore more voices than those in the model, sometimes few­ er. To fa il to ascertain the actual voices o f the model may be consistent w ith anal­ ysis by reduction through the classes o f operations and thus consistent w ith a basic element o f Schenker's system, but w ill fa il in the sp irit o f representation through the stage o f abstraction,and frequently cover up the passing and neighbor note mo­ tives fundamental to the compositional process as described by Schenker. It may fa il furthermore— by the constant retention in more background level representations of

p ggl

i

I

a ll the voices generated by contrapuntal elaboration— to demonstrate the common use of passing note motions to a tta in registers and chord positions not present in the model. It thereby elim inates the possibility of precise depiction o f the composition­ al procedures of octave transfer, voice exchange, and so on.

i

«?.>«

A simple instance of the misleading effect o f the lite ra l application o f the sequence of classes, w ithout the intervening consideration o f the original form of

i the harmony elaborated, is found in the reductive analysis of the common technique 15 r: fi m

of generating a chord in one harmonic position at one level and then forming from it new positions a t subsequent levels w ith an eye toward the m otive. Let us gener­

1 h

ate at some level a passing note figure where the passing notes are accompanied by a neighbor (Ex. 13): Example 13.

i

I

i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

21

and then compose out the figure, adding a roof fo the dominant ^ by way of inten­ sification of the motive o f a third (Ex. 14): Example 14.

An analysis that reduces out the passing notes w ith in the dominant harmony might easily result in a graph w ith more voices than the model and w ith a root position dominant seventh. (Ex. 15.)

from:

b. 3 fc fc = = ± = t= =

A lth tv’-!

1

from :(?)

.i

Example 15.

b j

;-2 su iti in a rational source for the composed-out passage it does not

represent the model employed, and has at least one more voice at some stage o f the analysis than is necessary. What it does show is the harmony of to n ic and dominant with a ll the voices used, but not the chords o f the model counterpoint. Thus, a l­ though this notation is inform ative in some ways, p a rticu la rly as a pedagogical to o l, it counters the Schenkerian position that counterpoints, not harmonies, are graphed. This is related to the question of the U rlinie as background (see below: Absolute Dis­ placement, p. 33 .

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Time-span. The notion of "tim e-span" was employed in the description o f the operations. This conforms w ith Komar's theory o f background rhythm .^ The no­ tion of time-span is not necessary in this description as lite ra lly as it is used in Ko­ mar. It may be taken rather to indicate simply that notes are generated in vertical alignment and displacement sequence w ith respect to one another, and that levels elaboration can never disturb that sequence at its level o f generation. Schenker's position on this is that rhythm and meter are absent at remote levels of structure a l­ though sequential order is real, and that meter and rhythm gradually develop by the implications o f stress induced by the contrapuntal elaborations. He says: M it den spateren Schichten wandelt sich entsprechend auch der Rhythmus, bis er, noch immer im Kontrapunkt verankert, durch hinzutreten des Metrums seine letzfe Vordergrundfassung e rh a lt. 6 [A t later levels, the rhythm also develops correspondingly u n til, a l­ ways anchored in the counterpoint, it achieves its foreground shape w ith the emergence o f the meter.] Strict adherence to Komar's theory, however, in no way disturbs the essential fea­ tures of Schenker's theories and, as he has shown, can be greatly inform ative.

Class II The influence of harmony allows the possibility o f altering the pure forms of the Class II operations without creating uncertainty as to their true function. This point, in fa ct, constitutes much of the original contribution o f Schenker's Kontra­ punkt.

5 Komar, Suspensions, esp. Chapter III. 6 Schenker, Der Freie Satz, 65. (Page numbers are from the 1956 e d itio n .) $ I S it

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Either boundary note o f a neighbor note figure can be suppressed, resulting in an "incom plete" neighbor note fig u re . (Ex. 16.) Example 16. In ct n . 1 frv ¥ -------

9

X—

-

—

^9-------—

©— y—

| £ BJ, £

P-----:

■»-

%---

ns Thus (Ex. 23): Example 23.

The c^ and a^ are indeed members o f the same chord at the level of generation of the a ^, but not at the time or at the level of the first appearance o f the c ^ . The ar­ peggiation involved in the generation o f the passing note can be represented by the labeling of the voices; as "p " and "q ;" p :(c ^ )-q :(g ^ , a^) . Arpeggiation refers to "p " and "q " and not to the notes they contain. Another possibility is the displacement of both notes of the original chord members in the arpeggiated voices so that the c^ and a i are not members o f the same chord at any level (Ex. 24): Example 24.

The arpeggiation is of p and q where each contains two elements; p :(c ^ , d^) - q:(g ^,

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aV

Schenker refers to these other possibilities when he writes:

Der freie Satz fugt den vom strengen Satz ubernommenen Zugen im Terzund Quartraum prolongierend nun auch noch Q u in t- und Sextzuge (Sextzug = Terzzug im Umkehrung), Sept- und Nonenzug fur Sekundschritte, Hoher- und TEeferlegungen in Oktavzugen hinzu. Die F u llu rg a lle r d ie ser Zuge geschieht m ittels Durchgangen, die nun einen freieren Satz fiihren. 11 [Free composition adds to the passing note motions carried over from strict counterpoint which prolong the spaces of third and fourth, those o f the fifth and sixth (sixth lin e = third line in inversion), seventh and ninth lines derived from stepwise m otion, and octave lines derived from trans­ fer o f register up or down. The f illin g - in of a ll these lines is accomp­ lished by means o f passing notes which then lead to a freer composition­ al technique.] The seventh and ninth lines as transformations of stepwise motions would have had to be generated by linear operations prior to the introduction of the passing notes, where one boundary o f the passing note figure might w ell originate as a displace­ ment of the other. Schenker expands on this issue somewhat in Per Freie Satz where he distinguishes between the interval of a second composed-out as a seventh that 19 does not make a li n e , 1 and a composed-out seventh which might also have arisen from step motion but which is a true Zug by virtue of its a rticu la tio n in to (harmonic a lly related) segments.

1o

This latter distinction seems in effect to depend on the

difference between two octave positions of essentially one voice expressed as a se­ venth (pseudo-Zug), and an octave doubling of some note by what are essentially two different voices (true Z ug).

11 Schenker, M eisterwerk, II, 26. 12 Schenker, Der Freie Satz, 120. 13 Ib id ., 124.

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29

Under normal circumstances then, a figure such as (Ex. 25): Example 25.

1

can be analyzed in one o f three ways; 1)

the boundaries o f the figure are members o f the same harmony (Ex. 26):

Example 26.

pi F major:

2)

I

the boundaries are not members o f the same harmony, but at least one o f them

is a displacement o f a voice o f a background harmony (Ex. 27): Example 27.

i A minor:

3)

£

I

the second note of the figure is a neighbor to the firs t, and the th ird note is a

neighbor to the second (Ex. 28): Example 28.

a

t

I 4----- «*-

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"Durchgang" vs. passing note. Schenker, throughout his w ritin g s, uses the term " Durchgang" (passing note) in both a general and particular sense. The gener­ al use is exem plified in the fo llo w in g comment by Schenker: Die Kansonanz lebt in D reiklang, die Dissonanz in Durchgang. Vom Dreiklang und Durchgang stammen a lle Erscheinungen des Tonlebens: Der Dreiklang kann zur Stufe werden, der Durchgang sich als N eben-, Wechselnote, Vorausnahme, als dissonante Synkope und Sept des Vierklanges abwandeln. 14 [Consonance lives in the tria d , dissonance in the passing note. From triad and passing motion grow a ll the phenomena o f tonal life : the triad can become a harmony, w h ile passing motion can be found in the guise o f neighbor or changing notes [accented passing tones], a n tic i­ pations, as dissonant syncope and as seventh o f the seventh chord .1 This use of the term is sim ilar to that in the fo lk expression: "Schenkerites care on­ ly that the first and last notes are in tune; everything else is "passing. 11 But Schen­ ker also uses the term " Durchgang11 in the present sense o f "passing:" A

Auf Grund odei Fuhlungnahme mitdem Ursatz b le ib t die erste 2 dem Gesetz des Durchgangs im Terzraum treu und begibt sich dadurch des Charakters einer tieferen Nebennote von vornherein: D u r c h g a n g u n d N e b e n n o t e sind ganz v e r s c h i e d e n e B e g r i f f e . 15 A

[On the basis o f, or in sympathy w ith , the Ursatz, the first 2 re­ mains true to the law of passing motion in the space o f a th ird , and thus rejects at the outset the a ttrib u tio n to it of the character o f a lower neighbor note: passing and neighbor note are to ta lly different concepts/ ]

Class III Suspensions toward the background. In Der Freie Satz, the issue of the sus­ pension operation is raised in Part i l l , Background. This has led to the assertion that

14 Schenker, T onw ille I I , 3 . 15 Schenker, Der Freie Satz, pu 72, in the discussion o f in terruption .

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31

suspensions can only exist on the foreground, where "foreground" stands for one o f three levels. With the exception o f the background, however, the terms "m iddleground" and "foreground" are conceived of as including sets o f levels, each set in­ cluding types of compositional procedures appropriate to its e lf. Thus, Schenker's first level o f middleground contains procedures that most p a rticu la rly influence the large-scale form o f the composition, especially the ascending and descending lines of the first order and techniques that articulate the structural line into segments. The general concept o f a suspension as prim arily a rhythmic event is naturally to be found toward the foreground in keeping w ith Schenker's view o f the emergence of meter and rhythm toward the fo re g ro u n d .^ But a suspension does not require a spe­ c ific metric association to be generated, although it may need such an association to be cle a rly expressed. It simply requires that a note generated w ith other notes in a chord be shifted in alignment by another preceeding note w ith which it forms a l i ­ near p a ir. The suspension its e lf is thus stressed m etrica lly w ith respect to its resolu­ tion— which was ru liuvcf occupied its place in the alignment— and is further empha­ sized by its lik e ly (but not necessary) presence simultaneous w ith other notes back­ ground to i t ,

but it need not be so stressed w ith respect to the foreground meter.

That so many suspensions do occur on the foreground downbeat does a llo w — to be sure— the possibility o f suspending entire chords. In any case the concept o f the se­ quence of classes is in essential agreement w ith Schenker's system, and allows sus­ pensions as soon as a linear pair is generated by a Class II operation. The circum -

16 See above, p. 22.

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32

stance where a suspension can only occur on the foreground holds only where there are but three levels a ll together, an u n lik e ly situation for any composition worth a n a ly z in g .^ . "Suspensions out o f the harmony. Suspensions, in addition to their lite ra l re­ tention o f a previous pitch,m ay be extended under the influence of harmony to en­ compass the retention o f a note stated in a different octave from the suspension i t ­ m

self, i . e . , it becomes a suspension of a pitch-class rather than a suspension o f a p itch. (Ex. 29).

Example 29. Beethoven: Piano Sonata, O p . 31, N o . 2, II; mm. 87-88.

The f2 of the second measure of the example, a suspension, is prepared an octave lower in the previous measure. This device is used w ith care, of course. Here it is emphasized as a suspension configuration by virtue of an environment of downbeat suspensions, exem plified by the suspension of the c^ in the first measure of the ex­ ample.

17 Criticism of Komar's book (op. c it . ) on this basis— that suspensions cannot oc­ cur except on the foreground— is at best re a lly a criticism of Schenker's construc­ tion of sets of levels. Komar defines as many levels as necessary to arrive at a spe­ c ific musical event w ithout classifying them . It does not demonstrate any incompat­ ib ility between Komar's theory and Schenker's.

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Absolute Displacement

As stated above, ^

graphic notation is a notation of counterpoints. The Ur­

satz its e lf is such a counterpoint and is the composing-out o f the to n ic harmony ov­ er the span of the entire piece. This suggests that the to n ic harmony, ly in g behind the firs t, i . e . , background, graphic le v e l, can o n ly be indicated by name and not by notation. It then leads to the notion o f absolute displacement, for the Ursatz is a counterpoint that underlies the composition as an e n tity occupying time and un­ dergoing change. "Der H i n t e r g r u n d in der Musik w ird durch einen kontrapunktischen Satz vorgestellt, von m ir U r s a t z

b e n a n n t."*^ [The background in music

is represented by a contrapuntal setting which I ca ll Ursatz J] This is to say that the -fundamental m elodic lin e , the U rlin ie , is a passing note fig u re , its e lf derived from arpeggiation, and the bass associated w ith it is an arpeggiation as w e ll, of the root and fifth o f the tonic tria d . S p e cifica lly: Die U r l i n i e weist gemass der Brechung, von der sie stammt, einen T e r z - , Q u i n t - oder O k t a v r a u m auf und fu llt diese Raume mit Durchgangen. Ein Urlinie-Raum muss zumindest einen Terzzug enthalten; ein Sekundschritt ist als U rlin ie undenkbar. Also ist der Durchgang der U rlin ie d e r e r s t e D u r c h g a n g uberhaupt, und gerade der ihm vom ^rengen Satz her anhaftende Zw ang, in derselben Richtung fortzuschreiten, in der erbegonnen h a t, bedeutet Z u sammenhang, macht ihn zum Anfang alien Zusammenhanges in einer musikalischen Komposition. 20

18 Page 9. 19 Schenker, Der Freie Satz, 27. 20 Ib id ., p .41.

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[in accordance w ith arpeggiation from which it derives, the U rlin ie sets up a th ird , f if t h , or octave space and fills in these spaces w ith passing notes. An U rlin ie space must contain at ieast a third lin e ; a progression of a second is unthinkable as an U rlin ie . The passing note in the U rlin ie is thus the first o f al] passing notes, and just as the constraint le ft over to it from strict counterpoint— to con­ tinue in the same direction in which it has begun— signifies connection, so does this make it the source of a ll connections in a musical composi­ tio n .] We have suggested that a passing motion always indicates the prior presence of the boundary members o f the fig u re , and that an a n a lytic graph can then show these boundaries either as members of the same chord or as displacements of simultaneous­ ly generated voice representatives at some prior le v e l. A ll displacements o f passing & note boundaries at other than the Ursatz level are therefore non-absolute, and a l­ low graphic representation of these boundaries at higher levels. (The same holds, of I i i*' p

course, for simple u n fille d -in arpeggiations.) But to Schenker, musical structure at • i the level o f the ultim ate tonic does not occupy time and participate in change. This leads us to declare the displacements in the U rlin ie as absolute displacements. When the structural m elodic line moves from 3 to 2f

that displacement ends the 3 as a

contrapuntal event in that register for the rest o f the piece. (The interruption form, construed as a repetitio n of the opening structural movement, does not contradict this

a s s e r t i o n . )21

Schenker confirms this when he says: "ich wiederhole also: der

21 This concept of absolute displacement, together w ith Schenker's practice o f mi nim izing the number of voices in the model prior level graph, constitutes the chief difference between Schenker's and Komar's references to the background, respect­ iv e ly . The directio n of analysis suggested by each approach, however, increases their differences in that Schenker's method more d ire c tly encourages the develop­ ment of rr>otivic analysis in the discovery of U rlin ie —lik e Zuge at every le v e l.

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Tonraum ist nur horizontal zu verstehen.

Q repeat, therefore: the tone-space is

always to be understood only h o riz o n ta lly !]

Resolution, S ta b ility and In sta b ility

"Resolution" is norm ally defined as the movement of a dissonant interval to a consonance, stepwise. This d e fin itio n does not accord fu lly w ith the levels con­

I cept o f to n a lity . Rather, levels relegate the question of resolution to the sphere o f i

relative s ta b ility and in s ta b ility . As Randall says: The realization that such tra d itio n a l musical concepts as the relative con­ sonance and dissonance o f intervals are irrelevant to the analysis of 12tone musichas led composersand musical analysts to re-examine the func­ tion o f these concepts in th e ir trad ition al context. What we have sought to define is some ahistorical sense in which these concepts may be said both to cohere and to cohere in such a way as to shed lig h t on the la ye rstructure peculiar to music in the tonal system. 23 Komar's own re-evaluation led him to classify the consonant intervals according to their degrees o f consonance and then to use this classification as a clue to the rela­ tive p rio rity of the members of one chord over those o f

a n o th e r.

^4 But Komar recog­

nizes the contextual nature o f p rio rity fudgements, which also must consider other aspects besides consonance and dissonance. The determination w ith respect to re la -

22 Schenker, Der Freie Satz, 44. 23 J .K . Randall, "Two Lectures to Scientists: 1-Theories of Musical Structure as a Source for Problems in Psycho-acoustical Research," in Perspectives on Contemp­ orary Music Theory, edited by Benjamin Boretz and Edward T. Cone, ""[New York: N orton, 19/7! ) , p . 121. 24 Komar, Suspensions, p. 28ff.

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five consonance is then only a clue and nota rule o f levels determ ination. We shall distinguish here between consonance and dissonance on the one hand, and s ta b ility and in sta b ility on the other, as separable references. When a linear note is generated it becomes, by d e fin itio n , unstable w ith re­ spect to the note or notes from which it was generated. The source note is stable w ith respect to the linear note which it generates, regardless of the course of subsequent elaborations o f e ith e r. A ll notes of the piece have a time-span during which they operate, regardless o f whether or not one takes the notion of time-span to refer to specific measurable durations or— especially at more remote levels of structure— merely to sequence and sim ultaneity of alignm ent. The time-spans are themselves layered. Since a ll the notes o f some composition derive u ltim a te ly from a seiies of operations on the prior layers, at each layer newly derived notes have a time-span drawn from the prior level notes— carved out of whatwas there before, so to speak. A t each le v e l, a note is said to resolve when its time-span is concluded and it completes a motion to a source note. This requires the presence of the conclud­ ing source note, and excludes the echappee figure where the second source note is suppressed. The true echappee note w ith no understood resolution, either in the har­ mony or in another register, and which is not retrieved after an interruption and car­ ried forward to a prior level note, never resolves. But no note may be said to re­ solve to a note which is foreground w ith respect to its e lf. Two members of a linear pair arising on the same level have no p rio rity relationship between them.

In the

cases where, for example, the interval of a fourth is established by arpeggiation and two passing notes are interpolated, the passing notes areon the same le v e l. The sec-

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37

ond passing note is neither generated from the first (with respect to level) nor can the first be said to resolve to the second, even though its time-span is concluded at its displacement. A passing motion may perhaps be said to resolve upon the arrival of the second boundary note but not in dividual passing notes, except for the last. A displacement relation*—and consequently, a linea r pair— can be made o f two notes neither of which is prior to the other and w hich need in volve no invocation of the concept o f resolution. In the case of suspensions, the question o f resolution is obscured unless the suspension configuration is separated in to its parts. As defined above, ^

a suspen­

sion arises when the first note o f a linear pair is extended in to the time-span pro­ per to the second note o f the linea r p a ir. The suspension proper is only that part of the total duration o f the suspended note which occupies the prior time-span o f the second note. The suspension is then said to resolve when the second note of the lin ­ ear pair displaces the suspended portion of the previous note of the p a ir. This estab­ lishment o f both the fa c t of resolution and the tim e-p oin t o f the resolution of the suspension does n o t, however, provide any inform ation about the resolution charac­ teristics of the original linear p a ir. Let us ta k e , for instance, a C chord composedout w ith passing notes (Ex. 30): Example 30.

25 Page 16.

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Both c|2 and f l are passing notes on the same level and are foreground to the C chord members bounding them. They resolve as passing notes simultaneously to the notes which fo llo w and displace them. The soprano note, e2, as a generator, cannot be said to resolve to d2 in any sense; as a p rim itive it admits of no resolution. If now, however, the e2 is suspended in to part of the tim e-sp an o rig in a lly occupied by d2, then the suspension its e lf is foreground to the d2. The suspended portion of the e2 therefore resolves to d2 ( Example 31) : Example 3 ] . I

°

U

m a) e2 as suspension resolves; b) d2 as passing note resolves. The original e2 of the example does not resolve. If the d2 is likew ise suspended into part of the prior time-span of the fin a l c2, then the d2 as passing note resolves to c2 as shown in Example 31 (b), w h ile the suspension of d2 resolves to the c2 at its tim e wise displaced appearance (Example 32) : Example 32.

T

9

(k \

o

z r 3

A 4/

*-0 3 ---------4 j

-----------

a) d2 as suspension resolves. Questions of resolution are taken separately according to respective levels of gen­ eration.

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If is sim ilar w ifh passing nofe motion w ith in a fourth where the two passing notes are on the same level of generation (Example 33) : Example 33.

P

=

1 (a)

(b)

a) no resolution; b) "resolution" of passing m otion. A gain, although g^ as a boundary note can never be said to resolve to the a^ that i t generates, i f the g^ is suspended into the prior time-span o f the a^ the suspended portion o f the g^ resolves to a^ at the latter's point o f appearance. And i f the a^ is its e lf suspended in to the prior time-span proper to b ^ , it resolves to the b^ at the latter's point of appearance, despite the absence of any resolution relatio n between them at any prior le v e l. (Ex. 34.) Example 34. (a)

i

(b)

a

a) g^ as suspension resolves to a^; b) a^ as suspension resolves to b l . The sequence o f classes determines re la tive p rio rity and therefore resolution. But the vertical aspect of music concerns its e lf w ifh triads as a common ob­ ject o f contrapuntal generation in tonal music. Such triads are not only suitable on the grounds of th e ir in te rv a llic sim ila rity to the to n ic , but— more im portantly— by

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40

virtue o f that sim ila rity are capable o f their own elaboration analogous to that of the to n ic . Since triads are em inently capable of prolongation and therefore are lia ­ ble to be found toward the background of any diatonic progression, and since the triad defines consonance, it follows that consonant intervals are more lik e ly to be found on remote levels and dissonant intervals on foreground levels. But it is by no means necessary that a ll consonant intervals are background to a ll dissonant inter­ vals. Schenker's concept of the leading voice in a progression suggests that the pasA sing note w ith in a tonic harmony (2) is first generated as a Jissonance and only af­ terward accompanied by the arpeggiation and neighbor note which make it part of a consonant dominant tria d . Consonance of intervals is a primary clue in levels anal­ ysis but other c rite ria must be considered. At re la tiv e ly foreground levels w ith re­ spect to the entire structure, dissonant chords may not only appear as derived from prior consonant chords, but may be prolonged on th e ir own, and prolonged in such away as to generate intervals more consonant than those in the source chord. (Ex. 35.) Example 35.

i v»' I Here a diminished seventh chord is generated as a co lle ctio n o f neighbor notes, is arpeggiated, and the spaces fille d w ith passing notes. Each set o f passing notes forms a chord containing pitches of the chord of resolution. The passing notes certa inly form more consonant intervals than do the members o f the diminished seventh chord, but the sequence of generation declares them to be passing notes, and an analysis

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of the foreground would most lik e ly select that solution over the immediate resolu­ tions of m ultiple diminished seventh chords, i f only by virtue o f the harmonic rhy­ thm strongly suggested by the second measure. (The s im ila rity crite rio n here has less to do w ith in te rv a llic sim ila rity than w ith rate-of-change s im ila rity .) A ll notes are unstable w ith respect to their source. It follow s that at the U rsatz level the dominant chord is unstable w ith respect to the tonic which generates i t . The only absolutely stable chord is the tonic harmony in its background status. Appearances of to n ic chords in some other role than as background stand themselves in need of resolution. A t many stages of analysis, s ta b ility and in s ta b ility corres­

:|:»w

pond to the presence of consonant and dissonant intervals respectively, but not at a ll. C ontrapuntally, s ta b ility and in s ta b ility correspond exactly to source and deri­ vation. A consonant chord is capable o f generating many more levels o f expansion than is a dissonant chord, but both w ill resolve ultim a te ly as both are unstable w ith respect to some prior event. The distinction is best preserved by referring to the s ta b ility o f individual notes w hile consonance and dissonance refer to the kinds o f in te rv a llic re la tio n scre ated by those notes. Consonance and dissonance are absolute qualifier-; o f chords in the tonal system, and provide inform ation as to the potential for fo n iciza tio n ; sta­ b ility and in s ta b ility are relative qualifies of single notes- "and by extension, of chords— and provide inform ation as to p rio rity in the tonal hierarchy.

I

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CHAPTER II! CHROMATICISM IN THE D IATO N IC SYSTEM

our a tte ntion, there regularly appear notes not contained in the diatonic scale pro­ per to the to n ic o f the piece. A ll such notes are here called "c h ro m a tic ." The chro­ maticism of the d iato nic tonal system is contained w ith in very c le a rly defined lim its which preserve the essential diatonicism of the system. As Schenker says: "Chroma­ tic change is an element which does not destroy the diato nic system but which rather emphasizes and confirms i t . " ^ In the diatonic system, the location of the h a lf steps w ith in the scale w ith respect to the tonic e ffe c tiv e ly defines the scale.

In this sense, h a lf steps provide

more information than do whole steps as to the location of the to n ic and the charac­

I

■

II

I

ll....ry..i..

i.Ji. j -:;.J L J I!lL„!llL'.Jl'JIMieuilU>WliaBaMiailW M I

In the vast m ajority o f compositions in the classical tonal system that merit

|

ter of the mode. The introduction o f a chromatic note necessarily creates a h a lf step

respect to the notes which might be combined w ith i t contrapun tally. Both of these benefits are ty p ic a lly exploited by the composer in the application of chromaticism

—

.—

— —

I .

where none existed at a prior le v e l. It can further provide a new chord color w ith

w ith emphasis now on one, now on the other of these characteristics.

1 Schenker, Harmony, 288.

;

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43

The Chromatic Processes

There are but two ways in the classical diato nic tonal system in which chro­ matic notes may arise: m ixture and to n ic iz a tio n . These w ill be referred to herein as the chromatic processes of d iato nic to n a lity . Both o f these processes invoke diato nic antecedents. The chromatic notes derived through these processes do not require the inclusion of concepts lik e a prior chromatic scale, or an equally tempered tw e lv e note c o lle c tio n . By virtue of their d ia to n ic o rig in , chromatic notes introduced by these processes have no theoretical reference to the equal-tempered pitch-class col­ le c tio n . W ith one exception, enharm onically equivalent notes refer to different pitch­ es. Both performance and notation reinforce this diatonic source. This is the funda­ mental distinction between the chromatic practice o f d ia to n ic to n a lity and that o f the chromatic tonal system.

A . M ixture M ixture as a theoretical construct is apparently first treated systematically by Schenker, and is fu lly exposed in his in his Structural

H e a r in g . ^

H a rm o n y .

^ Salzer treats of mixture as w ell

The concept o f m ixture has since become a widespread

tool of analysis, but one employed perhaps too infrequently where more general terms lik e "chrom atic" its e lf have been favored. Schenker cle a rly considered the process

2 Schenker, Harmony, p. 84ff. John V in ce n t, in The Diatonic Modes in Modern Music (C a lifo rn ia , l£ 5 l) , contains a com pletely independent discovery an3 presen­ tation of the mixture concept. Vincent seems never to have heard o f Schenker. 3 Salzer, Structural H earing, pp. 1 7 7 - 81.

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of mixture to be a purely d ia to n ic phenomenon ly in g outside the realm of chromati­ cism altogether (at least in Harmony)/ considering as chromatic processes only fo nicizatio n and a lte ra tio n .^ - His attitude reflects the strict d ia to n ic o rigin of the concept, but since m odally non-original notes are produced by mixture i t falls un­ der our d e fin itio n as a source o f chrom atic notes and w ill be so considered here. M ixture sim ply allows the interchange o f scale degrees between different parallel diato nic scale degrees of opposite mode, w ith out alteration o f the function of the mixed scale degree. If is thus also called "p arallel mode m ix tu re ," although this term is overburdened in that non-parallel mode m ixture is not entertained. Were it to be so entertained— as i t is by V incent— it could account for every chromatic event in any piece w ith out offe ring any other inform ation about the structural im­ plications of the source and tendencies of the chrom atic notes.

(If could even be

extended to account for a ll the notes in any piece in any style . A concept so gener­ al as to be u n ive rsallyapp lica ble is em inently employable but equ ally uninform ative.) The single commonest example of m ixture is in a minor mode environment as the minor seventh degree is replaced by the seventh degree from major to effect a more sp e c ific a lly directed close on the to n ic note, and to avoid the overpowering suggestion of a real tonic on the re la tiv e m ajor. This procedure is so fundamental that i f has led to the hypothesis of a "harmonic minor s c a le ," which is no scale at a ll under the sca le -fria d associative conditions o f consonance and adjacency, in

4 "A lte ra tio n ,11 in lig h t o f mixture and to n ic iz a tio n , turns out to be a superflu­ ous concept and is omitted here. The appearance o f altered chords can be account­ ed for com pletely by mixture and fo n ic iz a tio n .

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that i t contoinsa harmonic interval between sixth and seventh steps; i t is, after a ll, the harmonic m inor. When the seventh degree has thus been absorbed from major and a linear in te g rity is to be maintained in a scalar movement to that seventh degree in ascent, the sixth degree from major must likewise be taken over. This results in the "melodic minor scale" which does f u lf ill the adjacency condition but fails that o f consonance w ith respect to the tonic tria d , and is thus re la tiv e ly in h ib itin g in the establishment o f secondary chords for prolongation. The typ ica l representation of the minor scale as a thre e-fold e n tity: natural, m elodic, and harmonic, is unnecessari­ ly com plicated.

It suggests that the minor scale has no other particular character­

istic than the minor third above the to n ic , or at least that scale degrees six and se­ en have a lte rn a tive , equal bipartite id e n titie s. This la tte r concept, to be sure, is not w ithout precedent in Western theory, in that M edieval treatises consider the gamut to contain alternative pitch-class constituents for the note B. Such a concept had the practical effect of allow ing the avoidance of a ll dissonant linea r intervals except the seconds, the intervals of displacement, and had the secondary byproduct of economy of mode types, in that major and minor scales may w ell have been un­ derstood to be contained w ith in the Lydian and Dorian respectively as alternative forms. But i f we begin our modern system w ith two scales deeply rooted in the to n ic tria d , major and minor, which contrast as much as possible w hile s till fu lfillin g the criteria o f adjacency and consonance, then the process o f mixture provides us d i­ re ctly w ith the aforementioned alternative minor scales, together w ith a few other possibilities (including the appearance in a major mode environment o f the minor sixth degree, a common enough occurrence which has nonetheless fa ile d to occa­

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sion the hypothesis of a "harmonic m ajor" scale. There are more specific problems w ith the assertion of alterna tive minor scale forms. If the notion o f "scale" arises from the concept of the development o f dis­ placement notes, then the very term "harmonic m inor" would seem to suggest that melodic considerations are not e n tire ly at issue in its construction and that there is more involved in a scale than melodic displacement. The use o f the natural minor sixth and mixed seventh degrees in conjunction has precisely the effe ct of separa­ ting them from each other and defining two voices in the most direct terms, each with its own m elodic displacements operating simultaneously w ith the other. Here, the sixth degree would arise from the fifth degree alone, and the seventh from the tonic alone. The arrival at this as a scale would have to originate in the defining of a scale as a pitch-class co lle ctio n that might ty p ic a lly be associated w ith a tonic minor tria d . But i f that is the scale-generating process, where does one stop? How can other common chromatic events be overlooked? If the harmonic minor scale is the proper set o f notes belonging to a minor to n ic , there is then no way to account for the natural minor seventh when it appears. The hypothesis o f the bifurcated scale which is "m elod ic" in ascent and "n a tu ra l" in descent (aside from avoiding the is­ sue of what then is meant by "m elodic") would seem to account for the natural sev­ enth degree. But this complex accounts neither for: 1) the appearance of the afore­ mentioned augmented second in a passage moving in either direction; nor 2) for the choice of notes in re g is tra lly separate sixth and seventh degrees in successive chords i f no clear ascent or descent connection between them is im plied; nor 3) for the sim­ ple 8 -7 -8 succession where both descending and ascending directions are taken suc-

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cessively; nor

4

) for the appearance o f the m elodic form in descending passages and

the natural form in ascending passages. The addition o f the harmonic form to this bifurcated m elodic complex is an attempt to deal w ith some of these questions, but the addition contributes no new notes to the complex— they are just d iffe re n t com­ binations of the same notes— and would seem to be a h ig h ly in e ffic ie n t solution to a problem w h ich , after a ll, results simply from having no crite rio n for the estab­ lishment of a scale. Harmonic problems also are created by a m u ltip lic ity o f scales fo r a single mode. Both harmonic and m eiodic minor scalesseem to d e cla re the triad on the th ird scale degree as augmented, and statements to that effect recur in harmony b o o b nearly w ithout exception, w ith never an example from real music to support the assertion. The appearance of the m ajor triad on the third scale degree in minor is then j |

in every case a change of ke y, extending even to such a simple chord sequence as:

; |

l- lll- I V - V - l.

A g a in , since the

dominant chord is construed to be always a major

tria d , the prolongation o f that harmony by to n iciza tio n would require extensive a l­ teration of the original to n ic scale, to its virtual o b lite ra tio n . Such a circumstance would set the minor mode greatly at odds w ith the major in the consequences o f a \

move to the nearest fifth above.

Fortunately, it doesn't happen. The to n ic iz a tio n of

the dominant to any sig nifica nt extent in minor involves the dominant in its natural minor form, a llo w in g for the duplication of m otivic m aterial at the upper fifth w ith the least possible damage to the scale of the underlying to n ic . Such uses of the e x |

tended dominant in minor requires the adherents of m ultiple minorscales to also ad­ here either to the concept o f modulation in its most absolute sense, or to accept the

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introduction o f m ixture in some sense anyway. A p p lication of the mixture concept at the outset denies that any o f these problems are real musical ones, merely theoretical ones. It is worthw hile to note Bach's attitude toward the id entity of the minor scale. According to S p itta: . . . Bach recognized only two modes, the Ionian w ith the greater and the A eolian w ith the lesser th ird . . . Although the modern two mode sys­ tem is firm ly established in Bach, he yet keeps up a close connection w ith the system o f six modes, by simply taking the A eolian for the m inor. The­ o re tic a lly there existed for him one and the same scale Tor ascending and descending passages a lik e , w hile Rameau, and Kirnberger, used the scale a b c d e f * g # a , i n ascending, and Lingke, in order to get over the ano­ maly of having two d ifferent scales, made up the scale a b c d e f g # a , for both kinds o f movement. 5 Spitta appends a footnote to this comment which gives the reference: Lingke, Die Sitze der M usikalischen Haupt-Satze, pg. I 6 f f ,

and goes on to say in the note:

"Lingke la id this scale, invented by him , before the Society for Musical Sciences in Leipzig in 1744, and a ll the members approved of i t . . . Bach had not become a member at that tim e ." Needless to say, Bach did employ chromatic notes in m inor, analyzable as products of m ixture in cases of the generation of harmonic and melo­ dic minor scale forms.

Examples o f common mixtures. M ixture in the major mode most commonly in ­ volves the introduction o f the minor mode's fla t sixth degree. This is the generation of the "borrowed" or "m utated" subdominant. (Examples 36 & 37.)

5 Philipp S pitta, Johann Sebastian Bach, trans. by Bell and M aitland (New York: Dover, 1952), V oI . T n T p * T30T

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49

Example 36. Beethoven: Der Liebende; mm. 22-23.

£

JBL

ist

m if

d ir

ge-

schehn?

das

So

i

£

*

F=3 S

3

D:

V

IV 6 (m aj.)

(m in.)

Example 37. Schubert: "Morgengruss," N o .

f -sto rt

r

f

8

from Die Schone M u lle rin ; mm. 14-15.

T p H fT

dich denn mein S lick so sehr?

i H So

i muss

1 1 t o

s

s

i f

C:

IV6 (m in.)

V

This fla t sixth degree presses downward by h a lf step to the fifth , the characteristic linear relatio n of m inor, corresponding somewhat to the leading tone in major. Another pattern that includes the fla t third as w e ll as the fla t sixth is read­ ily found in places where the dominant harmony is in the process of prolongation and where the mixture notes emphasize the root and fifth of the dominant triad by h a lf

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50

step movement. (Ex. 3 8 .) Example 38. M ozart: Piano Sonata/ K . 333, I; mm. 89-90.

f: V '

The dl^ and g ^ display mixture w ith Bb m inor, the third and sixth scale degrees re­ spectively.

Mixtures in volving the third o f a scale are less common than those of

the sixth. O f the two modes, the m ixture of the minor third in to a major environ­ ment is more lik e ly than the reverse. This can be accounted for by the dual mixture reference, in that the fla t third o f the underlying to n ic coincides w ith the fla t sixth of the dominant, as in the above example. Example 39 is sim ilar: Example 39. Beethoven: Piano Sonata, O p . 7, I; mm. 37-39.

I

(P i

Bb:

i

l

l

b6 -4

5 3

Other examples o f third mixture sim ilar to the above appear where a possi­ ble am biguity as to the true resolution of the dominant to the tonic is eschewed, no-

±

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51

tably in transition sections of sonata expositions. An example is from Haydn's "La Reine" symphony (Ex. 40): Example 40. Haydn: Symphony N o . 85, "La R eine," I; mm. 52-79 (reduced).

,

J .

f n b

ii f


------------* z f --------

I -----------*

• § —j j E -----------------

■x ------------------- X

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In this analysis measures 1 - 13 are derived from an in te rru p tio n -like movement from tonic to dominant w ith middleground linear melodic displacement, but the comple­ mentary segment of the interruption does not appear. In its place is a transition to the second theme in E m ajor. The C minor triad arpeggiated in measure 12 is an instance of the suspension of the to n ic chord into the dominant harmony (E x. 71, b ) . The ge­ neration of the passing notes in the bass and a lto are shown in Ex. 71, c . Owing to the space problem, the third voice holds back one chord and its representative, G , takes the opportunity to rise to the soprano so that the conclusion of its passing mo­ tion w ill be on the e ^ , now e|> 1 , which the original top part le ft for another voice. Example 71 d shows the generation of the appoggiature to the C minor ^ suspension chord. This

chord is of special interest in that it feigns the arrival harmony. But

the fifth of the a rrival harmony, d ^, can not be found in this chord— for excellent reason: the voice u ltim a te ly containing the d^ has already (w ith respect to levels of

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87

priority) been displaced by the suspension e lt 1 which is its e lf now retarded by the sus­ pension of the passing f 1. The d 1, in other words, is present in the f 1 and this defect­ ive V 7 chord has no fifth at a l l. Indeed, as shown in graph a of Example 71, the goal V harmony is a simple dominant tria d , not a seventh chord (in keeping w ith the form of the first dominant in an interruption structure which shall not conclude its motion and which therefore has no use for the seventh). Each of the two passing chords is then prolonged via a simple progression normal to its key, shown in Example 71 e . This is the origin of the daring Bb chord; equally daring iswhat it demonstrates about the first phrase: that the tonic C major harmony has been altered to the subdominant of G du­ ring the course o f its statement. (This analysis is a variant of one suggested by God­ frey Winham. In his version only the first C in the bass represents the overall tonic harmony; a ll the rest of the repeated C chords stand as the IV of G . ) The downward tending half steps created by these prolonging progressions is maintained in the F chord by the simple device o f m ixture, creating a

6

- 5 pressure in contrast to the 4 - 3

pressure of the preceding chromatic notes derived through to n ic iz a tio n . Example 71 f indicates how the model progression is strevched out over four octaves, where the highest octave gives emphasis to the E vo C passing motion and then to the appoggia— tura chord of measures 9-1 1 . There is thus no need to invoke in this passage an underlying chromatic scale, and i f one is invoked, it would seem to provide rather less information than the diato­ nic description, which doesn't forbid the analyst to notice and delight V n th e h a lf step movement in the bass, after a ll.

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CHAPTER IV C O M M O N CHROMATIC CHORDS IN D IA T O N IC TO N A LITY *

In a group o f classically tonal compositions one is lik e ly to encounter a set of recurring chord types not indigenous to the governing diatonic scale. These are by definition chromatic chords. In harmony teaching they are normally treated as speci­ al cases, part of the baggage that comes w ith to n a lity , and in some cases—“ in parti­ cular the diminished seventh chords— are taken to be fundamental to some bifurcated scale system. Among these chords are the diminished seventh, augmented tria d , aug­ mented sixth, and N eapolitan. In accordance w ith the d e fin itio n of the scales and chromatic processes, these chords are a ll examples o f mixtures and to n iciza tio n s.

Diminished Seventh Chord

The diminished seventh chord in its commonest appearances is the direct re­ sult of the most immediate applications of m ixture. In the diato nic scales, the place­ ment of the h a lf steps creates a single im perfect fo u rth /fifth inversionai pair when the tendency notes are taken at once. In major this interval conveniently represents a dis­ placement of two tonic triad notes, one o f which is the tonic itse lf ^Ex • 7 2 ) .

C maj:

7

8

4

3

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89

In minor, this interval native to the pure minor scale also resolves to tw o members of the tonic tria d , but the to n ic note is not one o f them (E x. 7 3 ) : Example 73.

6

5

---------------------------------------------------------

t

—

1

# 2

3

The identity of this re la tio n to that o f the minor mode's re la tiv e major creates a con­ flic t which is intensified by the im p lic it strengthening o f the third degree of the m inot scale, where emphasis on the fifth instead would be preferable in preserving the identity of the m inor. The normal m ixture of the major scale's seventh degree in to mi­ nor resolves this c o n flic t, in introducing in to minor the same im perfect fo u rth /fifth pair possessed by its parallel m ajor. That this la tte r is, however, a borrowed interval is clearly indicated by the lack of h a lf step motion to the th ird degree (E x. 7 4 ) : Example 74.

7

4

8

3 (5?)

By combining both of these strong tendency-note intervals in to one chord we achieve the diminished seventh chord, a configuration thus remarkable for its h a lf srep move­ ment to a ll the notes of the tonic tria d

(w h ile the neutral whole step-related fourth

degree moves equally to the fifth or th ird , or b o th ), (E x. 7 5 ) .

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90

Example 75.

Although this configuration is lik e lie r in m inor, by virtue of the necessary seventh scale degree mixture under the conditions described above, I

the easy mixture o f the

minor mode's downward tending sixth degree in to major v /ill produce the same pheno­ menon (Ex. 76 ) : Example 76.

Here the second scale degree becomes the neutral step as a non-original tendency note in the basic mode and under the appropriate voice-leading conditions can move free­ ly up or down. The diminished sevenrh chord anatom ically described as V ||^ is thus a mix­ ture chord w ith stepwise displacement in a ll voices at its resolution. The diminished seventh chord can be applied to any tria d in the key capable of being a to n ic . It is then derived as a to n iciza tio n chord w ith appropriate mixtures. The stepwise displace­ ment in a ll voicesto the chord of resolution brings the V ll^ in to close association w ith the upper members of the dominant chord. An early resolution of the sixth to the fifth

1 Pg. 5 5 ff.

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scale degree transforms if exactly in to a dominant seventh chord. For this reason i t is often treated as an incomplete V ^ , w ith the root suppressed. Although the V ll^ is ta x o onomically distinct from the V , this nomenclature is acceptable insofar as, at a re­ la tively foreground le v e l, the two chords are interchangeable; but the incomplete dominant can never be used at a point o f true cadence which by d e fin itio n contains the root of the dominant as support for the second scale degree, the note o f melodic cadence. The two chords are perhaps best kept distinct w h ile the awareness o f th e ir similar melodic function is m aintained.

Common-tone diminished seventh. One often finds a species of diminished seventh chord not bearing rhe stepwise relation in a ll voices to the chord o f resolu­ tion. This occurs when one o f the members of the diminished seventh chord is a con.mon tone between the diminished seventh chord and the chord of resolution (Ex. 77): Example 77. Schumann: Papillons, N o . 10; mm. 25 - 27.

I

rrr

i

« * f "' f

TTT

m

The chord in the second measure of this example is a common-tone diminished seventh chord. It results from the m icrotonicization of the third and fifth o f the object chord with an added whole step neighbor to the fifth to f i l l out the sonority; the common tone is the root of the object chord (Ex. 7 8 ):

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92

Example 78.

C m a |:

I

Although examples of this fille d -o u t form of the common-tone diminished seventh are widespread in the Romantic era, sim ilar examples from the 18th century are rare. The chord is most often met w ith there in incomplete form, w ith the whole step neighbor omitted (Examples 79, 8 0 ): Example 79.

M o za rt: Piano Sonata, K .3 8 3 , I II ; mm. 78-79, a.

Example 80,

M o za rt: Piano Sonata, K .5 o 3 , I ; mm. 95-96, a.

Mr

*

.

*

=s=

b.

=*=E

------------F: V

A

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Despite the complete consistency o f the fille d -o u t version o f the common-tone dim i­ nished seventh w ith diato nic tonal p ra ctice , the preference for the incomplete form among Classical composers is due perhaps to th e ir eschewal of superfluities and their avoidance of heavily weighted textures for th e ir purely co lo ristic q u a litie s . Piston, for example, discovers such a complete version 2

in Haydn, but his example occurs

within a prolongation of a \ P chord, so that there remains a h a lf step movement in all voices (E x. 8 1 ): Example 81. Haydn: String Q u a rte t, O p . 76, N o . 4 .

u T

it Another example of this chord in the prolongation of the V ^ , on the other hand, al­ though including the missing " f ifth " of the chord omits the "ro o t" instead (E x. 8 2 ): Example 82. Haydn: Piano Sonata N o . 39, Hob. X V l/2 4 , I I I ; mm. 69-70 .

D: I

,7 V

2 Piston, Harmony, p . 282. Piston Names this chord a V l^ .

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94

In Example 82, the normal spelling for the complete version o f the chord is : a f - o f F $ -A ; the

is om itted in that there is no place for it in the motive and the compo­

ser w ill not indulge in harmonic overstatement. Example 83 also shows the commontone diminished seventh chord w ithout its "ro o t" (here C x ), in a sim ilar in te n sifica tion b ym icrotonicization o f a passing motion between a V Example 83. C .P .E . Bach: Sonata N o .

E m in , : \ / 7 I

V7

1

7

7 and V II

9 (or w ith in a V ) :

fu r Kenner und Liebhaber, I ; mm. 33^-6.

(VII7) I

Although the common-tone diminished seventh chord is usually spelled as in the preceding examples — w ith the h a lf step neighbors notated as leading notes and thus as chromatic notes by m icrotonicization — the h a lf step to the th ird of the chord of resolution may also be derived through mixture w ith the minor th ird of the scale of the harmony. This is the case in the opening o f Brahms' Third Symphony, where the minor third is m o tivic (F —A(k) ■ F : fre? aber frShlich) and is further exploited immedi­ ately after the statement o f the motto (E x. 8 4 ):

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95

Example 84. Brahms: Symphony N o . 3 , O p . 90, I ; mm. 1 -3.

★ A similar example is the opening of Schubert's C major String Q u in te t, Example 85. The mixture interpretation is intensified in this example by the composing-out o f the minor third by the first v io lin . Example 85. Schubert: String Q u in te t, O p . 163, I ; mm.

1- 6

.

Although the two preceding examples spell the common-tone diminished se­ venth chord as id entical to the secondary V i r of V , the two phenomena are not to be

6

confused. The interpolation of a cadential 4 into a dominant harmony has wrecked many a harmony teacher. Even though the chord succession is the same as in the com­ mon-tone diminished seventh case, the harmonic succession is com pletely d iffe re n t; Example

86

shows the model progression and its sequence of generation. Example 87

shows the progression in co n text:

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96

Example. 8 6 .

Example 87. M o za rt: Piano Sonata, K .2 8 4 , I ; mm. 2 2 - 6 .

m 6

V

Tfi'e confusion of these types is the result of attempting to apply Roman numerals to linear chromatic chords w h ile fa ilin g to perceive the distinction between the to n ic ization of a chord and the m icrotonicization of members o f some chord. Roman numer­ als are especially misleading in the case o f common-tone diminished seventh chords. Their spelling is w h olly dependent on the specific m icrotonicization ( or m icro to n ici­ zation plus m ixture) involved, so that enharm onically equivalent diminished seventh chords refer to com pletely different chords of resolution, just as they do in the form of secondary V I^ chords. This is not to sugg it that a composer may not take advan­ tage of the am biguity by a lte ra te ly employing each potential meaning o f the dim i­ nished seventh chord. Compare, fo r instance, m. 5 4 ff. o f the Schumann example c it­ ed above (O p .

2

, N o . 10) w ith Example 77.

The common-tone diminished seventh is but the most discussed member of a large class of pedal-point chords, both chromatic and d ia to n ic .

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97

The N eapolitan Complex

When in a minor mode environment the second scale degree of the underly­ ing tonic is lowered a chromatic h a lf step, the relatio n to the to n ic note of this fla t second degree is frequently referred to as the

N eapolitan re la tio n , " the triad (a

consonant II chord) of which it is the root is called the "N e apolitan c h o rd ," and the first inversion o f this chord (the II6 ) is known as the " N eapolitan sixth " chord. The geographical reference presumably alludes to the phenomenon's being a favored de­ vice of mid-Baroque opera composers led by Alessandro S c a rla tti. It also avoids the issue of what source in diatonicism this scale alteratio n could have, since it has long been known among theorists that to give an event a nameclouas the issue w ith an ap­ parent explanation su fficie n t for at least a generation, leaving one's students to deal with the problem. Schenker^ uses the term "Phrygian II" for the chords of this com­ plex, and he treats the relatio n as the survival of a Phrygian element in modern mu­ sic where a consonant II chord is required for the purpose of m o tivic elaboration. He says: "For the major tria d resulting from this fla ttin g is able to comprehend the m otive, and at least a temporary v ic to ry has been won by the motive over the system. ^ This is to say that the Phrygian II is a fact of tonal compositions and that it is extra-sys­ tem atic. Salzer makes i t systematic by allow ing the possibility of m ixture w ith the parallel Phrygian m ode.5

The attractiveness of this assertion is that the Phrygian

3 Schenker, Harmony, p. 109. 4 Ib id ., p. 110. 5 Salzer, Structural H earing, p . 177.

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98

mode is w e ll suited to stand fo r the minor mode; in terms o f contrast w ith the major it has the contrary qua lities (m inor vs. m ajor) fo r the im perfect intervals formed by its scale members w ith the to n ic note. It loses on the grounds o f the consonance condi­ tion w ith non-adjacent tria d members by its lack o f a perfect fifth for the dominant note and therefore o f a consonant dominant chord. It is thus useless for prolongation by means o f this otherwise sig n ifica n t fifth re la tio n — and w hich causes i t to possess a unique cadence formula in Renaissance music — although preferable to minor fo r the m otivic uses of its supertonic tria d . The suggestion of a strongly contrasting mode to major for the attainm ent o f consonance on one m o tiv ic a lly useful scale degree, the II, and a consonant fifth -re la te d tria d on the dominant at the same tim e lends strength to the idea that the one mode can be changed to the other for each o f these purposes in turn. The only other m ixture method fo r gaining a consonant tria d on the second de­ gree of the minor scale is to raise the fifth o f the supertonic tria d . This could then be understood as m ixture of minor w ith major (with Dorian as the result), but it produces the major sixth scale degree which is fa r less d ire ctio n a l than the minor sixth proper to both minor and Phrygian, and which is the minor mode's characteristic half-step tendency tone. Thfe fla t second degree is dire ctio n a l on me other hand, and leads to the tonic its e lf. This last characteristic suggests, however, another source for the N e apoli­ tan re la tio n , and that is through the to n ic iz a tio n process. The fla t second degree in Phrygian is also the next fla t in descending order of the minor its e lf. The presence of the Neapolitan re la tio n thus d ire c tly refers to the scale of the subdominant and to its relative major as w e ll, the submediant o f the underlying scale. This is precisely the

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99

explanation adduced by A lle n Forte fo r the N eapolitan sixth chord, that it arises out of the subdominant. 6

T h isw ou kit end to explain the instance of the N eapolitan com­

plex so frequently as the N eapolitan sixth, since the root o f the harmony from which it derives remains in the bass (E x. Example

8 8

8 8

):

. * s

C m in .: IV Schenker quotes such an example in another connection and calls the chord in ques­ tion the Phrygian II in his analysis^

(E x. 8 9 ):

Example 89. Bach : Organ Prelude, C m inor.

ILLS t