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I | I | i‘ !■ • i

ARLIN, Mary Irene, 1939, „ ESQUISSE DE L'HISTORIE DE L'HARMONIE, CONSIDEREE COMME ART ET COMMS SCIENCE SYST^MATIQUE OF FRANCOIS-JOSEPH FETIS: AN ANNOTATED TRANSLATION. 5 Indiana University, Ph.D., 1972 Music

j University Microfilms, A XERGX Company, Ann Arbor, Michigan

© Copyright by MART IRENE ARLIN 1972

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ESQPISSE IE L'HISTORIE BE L'HARMONIE. CONSII8SRBE COMME ART ET COMME SCIENCE SYSTEMATIQOB OF FRAN£OIS-JOSEFH FETISi AN ANNOTATED TRANSLATION

BY MARY IRENE ARLIN

Dissertation submitted to the faculty of the Graduate School in partial fulfillment of the requirements for the degree Doctor of Philosophy, Indiana University February, 1972

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Accepted by the faculty of the Graduate School, Indiana University, In partial fulfillment of the requirements for the degree Doctor of Fhilosophy*

rector of Thesis Ttus Chairman

Doctoral Committeei

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PLEASE NOTE: S om e pages m a y have indistinct print. F i l me d as received. U n i v e r s i t y M i c r o f i l m s , A X e r o x E d u ca t io n C o m pa n y

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ACKNOWLEDGMENT The author wishes to express her sincere appreciation to all who have helped complete this work, to those who have encouraged as well as those who have assisted in a tangible way.

I am particularly grateful

to my committee and especially to Dr. Vernon L. Kllewer, director of the dissertation, for his perceptive guidance, critical suggestions, and personal interest through the years.

To Mary E. Cardillo, Proctor

High School, Utica, New York, my deepest thanks for her generous assistance in a detailed checking of the translation, and to Dr. Marianne Kalinke, University of Rhode Island, for reading the German translations and offering helpful corrections.

I am also indebted to

Mrs. Florence Pfanner and the Interlibrary Loan Department of Ithaca College for helping me procure research material) to Dr. Albert van der Linden, Bibliothecaire, for making the resources of Biblioth&que du Conservatoire Royal de Musique, Brussels, available to me.

Last, but

certainly not least, to my mother, who read and corrected the first draft of the translation, and without whose patient assistance, assurances, and encouragement this work would not have been possible.

iii

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tables o f contests

Page INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . » Biographical Sketch Theoretical Concepts w > c * w i ^ Esqulsse de l*historie de 1'harmonie

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ESQUISSE DE L ’HISTORIE DE L‘'HARMONIE CONSIDER® COMME ART ET COMME SCIENCE SYSTEMATIQUE

21

Foreword 22 Chapter I. The Creation of a Harmonic System « • 25 Chapter II, The Results of the Creation of a Harmonic System , * 103 Chapter III, The Nineteenth Century* The Development of the Art. A Complete and Definitive Formation of the Theory of Harmony . . . . . . . 181 APPENDIX . . . . . . Appendix A: Appendix Bi

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218

A Note in Reply to That of Fetls • How is Pure Gold Changed into Base Lead? >

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219 223 227

BIBLIOGEtAPHY

231

INDEX

239

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iv

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INTRODUCTION In the preface to The Theory of Harmony (2nd ed«s Dekalb, Illinois:

B 0 Coar, 1955), P» vii, Matthew Shirlaw states that "the

Esqulsse de 1*historic de 1*harmonie of Fetis Is a real history of harmonic theory, and of harmonic systems*"

But the Esqulsse de

l'hlstorle de 1 *harmonie is more than a "real" history of harmonic theoryi it is the flrBt history of harmonic theory*

Yet, little is

known about this work because it was published in Paris in 1840 in a limited edition*

there were only 50 copies printed for private

circulation, and few copies are extant *^ Fetis was eminently qualified to undertake such a work*

As "one

of the most lucid musicologists of his time" and "one of the first to consider the past with an artistic interest," the historical aspects of the art*

he knew and understood

As a composer and teacher of compo­

sition, he understood the technical problems of organizing the material of music for the purposes of instruction.

As a theorist, he had

examined and studied the writings of theorists throughout the ages. a result of his research, he expressed the fundamental idea*

As

"art does

not progress, it transforms itself,"^

^Excerpts from the book were printed in the Gazette musicals de Paris. 1840, nos* 9, 20, 24, 35r 40, 52, 63. 68, 72, 73, 75-77? and in Traits dtharmonle, Bk~ IV, pp* 201-54. ^Robert Wangermee, "Fran§ois-Joseph Fetis," Encyclopedic de la musique (Paris* Fasquelle, 1959), II, 52* -^Franjois-Joseph Fetis quoted by Robert Hangermee in Encyclopedia de la musique* II, 52*

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2 In view of Fetis' influence as a theorist, this annotated trans­ lation is hased upon the need to provide access to his ideas on the history of harmony hy making the Esqulsse de l'historle de 1'harmonie available for the English reading public, so that F£tis* contribution to the history of harmony can be interpreted and evaluated in the light of contemporary scholarship.

Biographical Sketch Frangois-Joseph Fetis, b o m on 25 March 1784 in Mons, Belgium, was the son of a musician, Antoine, who was the organist at Saint-Wandru, concertmaster of the theatre orchestra, and conductor of the village concerts,

Franjois-Joseph, destined to follow his father's profession

(F£tis was a musician more by tradition than vocation), began his musical studies at an early age, receiving instruction in piano, violin, and organ from his father.

In the Blographie universelle Fetis wrote,

"The first instrument which was put into his hands was the violinf at age seven he wrote some duos for this instrument, and began to study the piano,

Yet even Fetis was forced to admit that he was not a child

prodigy| he did not demonstrate talent for any particular instrument. Rather, "what engrossed me then was the desire, to be more exact, the need to compose."

2

•^■"Franjols-Joseph Fetis," Blographie universelle des musiciens, 2nd ed., III, 227f hereafter cited as BUM. ^Franjois-Josepb Fetis quoted in Robert Wangermee, Franyois-Joseph Fetis. Musicologue et Compositeur (Brusselst Acadgmle royale de Belgique, l951)t P*

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While young Fetis had had no formal training in harmony or compo­ sition prior to writing some piano concertos and sonatas, a Staaai for two choirs and orchestras, and some string music, he had studied and memorized about 30 symphonies of Haydn, about 20 symphonies of ELeyol, as well as overtures and piano concertos of K.P.E. Bach, Kozaluch, and Mozart.^- Fetis was particularly struck with the works of Haydn and Hozart, because in them he found " . . . the secrets of a new and lively 2 harmony, of which he had no idea at all previously." He imitated these men, and his own compositions were popularly received. In October 1800, yielding to the entreaties of friends, the family sent Fetis to the recently opened Paris Conservatoire where he continued his piano study with Boieldieu and Louis Pradherj he studied harmony with Jean-Baptiste Rey, a devotee of Rameau, and received the first prize in harmony in 1803.

About this same time he developed a keen interest in

the history and theory of music| he read and compared Catel's Traite d*harmonie to that of Rameau.

He studied German and Italian so he could

compare Catel and Rameau to Kimberger and Sabbatini.

In 1806, at a

publisher's request, he began a revision of the plainsong of the Roman Catholic Church and the preparation of a text which was more in keeping with the traditions of medieval manuscripts.

The fruition of his labors

and patient research had to wait, because of interruptions, nearly

30 years. In 1806, Fetis married Adelaide Robert, a young woman of consider­ able fortune, and in 1807 Fetis won the second prize (later called the ^Wangermee, p. 12.

2 "Franpois-Joseph Fetis," BUM. Ill, 227.

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Brlx de Rome) in composition*

Fetis believed that easy circumstances

would permit him to devote his time to composition, but in 1311 pecuniary difficulties, precipitated by the loss of his wife's fortune, forced Fetis to leave Baris*

He settled in Ardennes, where he occupied himself

by studying philosophy which ", . . seemed indispensable for the exposition of the principles of the theory of music, and for the analysis of the facts of the history of this art,""*In December 1813, Fetis accepted the position of organist at Saint-Pierre in Douai and professor of singing and harmony at the munici­ pal school.

Still hoping to earn his way as a composer, he returned to

Baris in 1818 where his stage-works, mostly comic operas, met with more or less success*

In 1821, he accepted an appointment as the professor

of composition at the Baris Conservatoire, and in 1824, at Cherubini's request, he wrote and published his first important work#

Tralte de

contrepolnt et de la fugue, Fetis* concern for the lack of a journal dedicated exclusively to music in France motivated him to fill the void, and from February 1827 to November 1835 Fetis wrote and published, virtually single-handedly, the weekly journal I* Revue musicals.

He continued to contribute daily

articles to 3> Temps and Ia Nationalj on several occasions he wrote three different articles, each dealing with a different aspect of a new opera, and each article appeared in either the Revue muslcale, Le Temps, l"Frangois-Joseph Fetis," BUM, III, 229. ^This was not the first music periodical in FranceI La Revue muslcale was preceded by Journal de Framery (1770), Correspondence des amateurs (1802), and Tablettes de Polymnie (1810), none of which was successful and all of which were very short-lived,

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or

National the morning after the performance.

1

The publication of

Revue musicals and daily newspaper articles helped to make Fetis renowned, and in the succeeding years he had considerable Influence on the Parisian musical life, not as a composer or teacher

but as a critic.

He was

feared and detested) he was frequently the target of considerable criti­ cism by composers, notably Berlioz who considered him to be a ", . « Muster eines in seine Theorien verstrickten Fadagogen, der unfahig sei, sich von der neuen Kunst ergreifen zu lassen."

2

In 1833 Fetis accepted the appointment as Director of the ConserA

vatoire in Brussels and aialtre de ehapelle to Leopold If until his death on 26 March 1871, Fetis held dictatorial authority over the musical life of Brussels as a conductor, teacher, and writer.^

Few men have wielded

greater influence as a critic, historian, and theorist. The fame of Fetis rests on his prolific writings on the history, theory, and literature of musicj these writings span a lifetime of research and study.

In the Preface to the Tralte d'harmonie (p. ix),

Fetis states that he read and studied mere than 800 works dealing with harmony before attempting to write his own.

When he died he was writing

Historic generals de la musique depuis les temps anclens a nos temps (Baris1 Didot, 1869-76)j only five volumes of the projected eight had been completed.

His personal library, acquired by the government at

lnFranjois-Jcseph Fetis," BOM. Ill, 233* 2Bobert Wangeraee, "Frangois-Joseph Fetis," HGG. IV, 130. (". « , an example of a pedagogue who, hopelessly involved in his £own3 theories, could not come to grips with the new art.") 3lbld.. 131.

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the time of his death and housed in the Bibliotheaue royale de Bruxelles, contained more than 5*000 works."*"

Theoretical Concepts Undoubtedly Fetis1 most important theoretical works are the Esqulsse de l*historle de l'harmonle (1840), the first history of harmonic theory, and the Traite d*harmonie (1844), the twelfth edition of which appeared in Paris in 1906,

Through both of these works there is a thread

of continuity and commonality*

tonality.

In the Preface to the Traite

d*harmonie, Fetis contends that the efforts of all theorists who have searched for the fundamental principle of harmony in acoustics, mathe­ matics, aggregations of intervals, or classifications of chords have been p

futile (p. vil),

because tonality is the primary organizing agent of all

melodic and harmonic successions.

This is the first use of the term

"tonality” in music1 the better share of the Traite d'harmonle is devoted to explaining how it organizes music. Scales and tonality* Fetis contends that the primary factor in the determination of tonality is the scale*

the order of the succession

of tones in major and minor, the distances which separate the tones, and the resultant melodic and harmonic affinities (p. 2).

But, as Hindemith

has said*

■*For a complete listing of the holdings see Catalogue de la bibliotheque de F, J, Fetis, acquise par 1'State Beige (Paris* FirminDidot "et" Cle.7l5777.-------- ---- ------------p

All subsequent parenthetical page references refer to the Traite dIharmonle,

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7 Scales are undoubtedly an excellent and perhaps even an indispensable add to theoretical as well as practical music (just as the telephone book Is to the use of the telephone) hut they are not In themselves the material out of which melodies— and harmonies— are made*^ Consequently, FetisT quest for tonal coherence in ascending major and minor scales was in vain, because scales are a posterior fact, theoretical abstractions of melodic and harmonic material*

In and of

themselves, scales are totally useless for determining tonality.

The

first note of the scale, the "tonic,1* is the fundamental note not because of its position, but by virtue of the musical context from which it has been culled.

Hence, one wastes time analyzing and dissecting

scales in search of tonality*

Tonality is

, , , that quality of the musical perception which finds its origin in the organization of a tonal complex about a central point of emphasis. The term "tonic" achieves a terminological importance since it refers to that tone about which all remaining tones of the complex are grouped,^ Tonality existed, thus, prior to the emergence of major and miner scales and is not the result of the emergence of major and minor scales. Scales are not the arbiter of tonality, but the converse,

Furthe? ore,

modality and tonality are not mutually exclusive as Fetis propounds! modality is a particular set of scalar relationships to a fundamental tone— tonic.

1Paul Hindemith, "Methods of Music Theory," Mg, XXX (1944), 24-25, ^William E, Thomson, "A Clarification of the Tonality Concept" (unpublished Ph,D, dissertation, Indiana University, 1952), p, 205*

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8 Twtervals and tonality. In Book I of the Traite d*harmonie. Fetis breaks with the hwhr I taxonomy of intervals In his classification* axe four kinds of consonances:

There

(l) the perfect consonances* the perfect

fifth and perfect octaves* which create the feeling of repose| (2) the imperfect consonances, the thirds and sixths, which do not give a feeling of repose; (3) the "mixed" consonance, the perfect fourth, which lacks finality, is not variable in the major and minor modes# and does not demand a resolution as do dissonances; its usage is, however, very limited; and (4) the "appellative" consonances, the augmented fourth and diminished fifth, are the intervals which "characterise modem tonality,"-* The two remaining intervals, the seconds and sevenths, are classified as dissonances because they " . . . are not pleasing by themselves and only satisfy musical sense by their connection with consonances" (pp. 8-9)* Recognizing the fragility of his definition of dissonance, Fetis clarifies it by stipulating that the two dissonant notes must "touch each other" either at the interval of a second or a seventh*

The disso­

nance created between the fourth and fifth degree of the scale is called "natural” because it is the point of contact between the two tetrachords on the scale (p. 18).

This seventh and its inversion does not have to be

preceded by consonance, as do all of the other dissonant intervals. Having completed his classification of intervals, Fetis proceeds to explain the "laws of tonality," laws which sure rooted in the meta­ physical principle of points of repose in a scale*

Fetis never defines

^Fetis' inclusion of the tritone as a consonance appears to be predicated on the premise that its role in tonality nullifies its dissonant characteristics.

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9 what renders the quality of repose to any scale degree! he merely lists the degrees which possess repose*

The tonic scale degree is the only one

which has the character of absolute reposey the fourth and fifth degrees, points of momentary repose* may have the harmony of a fifth or octave. Neither the triads nor the interval of a third on the fourth and fifth degree may succeed each other because, ", . . having no point of contact between them, £they] present the aspect of tonal absurdity to the musical sense" (p. 17).

This juxtaposition of thirds on the fourth and fifth

scale steps is called fauBses relations. If tonality truly resides in the scale, why exclude the juxtaposition of two of the diatonic intervals contained therein? Fetis is not referring to his limited concept of dissonance and to the two notes which, because they separate two tetrachords, create "natural" dissonance, since he admits the progression 11^ - V in his harmonization of the scale with the justification that " . . . the note of this sixth is at the same time the fifth of the domi­ nant, and that establishes the contact" (p» 17).

If Fetis* objection is

to avoid the parallel fifths which could result from the succession of IV to V, his objection is unfounded1 parallel fifths need not occur in chords which are related by the root relationship of a second*

In the

Esqulsse de l*hlstorie de 1*harmonie, he quotes a rule of de Muris— "we ought to avoid two major thirds by conjunct motion"— a rule which de Muris never stated, but which Fetis misconstrued because he misread de Muris*

There is absolutely no tonal justification for the prohibition

offered by Fetis* The second, third, and seventh degrees, devoid of tonal repose, may not have root position triads built on them because the interval of

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10 the perfect fifth is the interval of repose.

While Fetis is contending,

on the one hand, that the law of tonality resides in the order of the succession of pitches in the scale, on the other hand he is contending that the interval of the perfect fifth is the interval of repose* Fetis never reveals hew the interval of the perfect fifth gains ascendancy as the primary structural unit. ing the perfect fifth

The reasoning which Fetis uses for exclud­

on the third degree of the scalecould also he

used to repudiate hisinclusion of the tritone as a consonance,

albeit

"appellative” : The reasons for this exclusion £of a perfectfifth on the third degree] ares (l) that the fifth of this notewould be formed with the seventh degree, of which the natural attraction towards the tonic cannot be satisfying to the conditions of repose* (2) that this same seventh note would establish a false relation­ ship of tonality with the fourth degree, towards which the third itself has an attractive tendency, being separated only by a semitone.1 What universal law dictates a false relationship with an adjacent note which is not sounding?

Is Fetis implying that the triad on the

fourth degree should follow, enabling each note to resolve a semitone to its "natural attraction"?

I doubt it, because such a resolution would

create the anathema of "good" counterpoint— parallel fifths. The sixth degree of the scale caused Fetis some consternation* while he initially excludes it as a point of repose, he concedes that occasionally it may be considered a point of repose and accompanied by a perfect fifth, because the sixth degree is also the tonic of the relative minor, and in the minor mode the sixth scale degree is also the fourth

^Traite d*harmonie, par, 51» P« 20,

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11 degree of the minor model1

This convoluted reasoning could be applied

to an; of the remaining scale degrees.

For example, the third scale

degree in major is the dominant of the relative minor, a point of reposej the second scale degree in major is the fourth scale degree in the rela­ tive minor, also a point of repose. As for the exclusion of the second degree as a point of repose, Fetis says, If sometimes one accompanies it Kith the fifthy . . . one removes its tonal character and effects a vague change of tonality which opens the way f-r several endings in different keys| for example

V/i,iJij i First of all, the interval of a perfect fifth on the second degree of the scale no more "effects a vague change of tonality" than any other interval or chord unless one wants to modulate via a pivot chord— establish a relationship between two keys through a chord or interval which is mutually identifiable in both.

For example, the interval d-a

could be replaced with "tonal" intervals and Btill cadence as Fetis indicates:

iTraite d*harmonie, par. 48, p. 19. 2Ibid.r par, 51, P» 20.

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12

For Fetis, the only intervals which can give the second degree of the scale *tonal character" are the sixth and third or sixth and fourth. Fetis appears to he stating, in his own inimitable way, that the notes of tonal repose are those notes which can be the roots of cadence chords, chords which normally occur in root position, hence, the perfect fifth. This 1b probably why, in the face of two choices,

(l) accepting the

sixth degree as an occasional point of repose, or (2) denying the existence of the deceptive cadence, Fetis chose the former.

In spite

of his insistence to the contrary, Fetis* theory of tonality is not melodically conceived but harmonically, because a careful scrutiny of melodic cadence pitches will reveal that any note of the scale may be a cadence note, although some notes are more common than others. This lack of concern for an understanding of the "tonal character" of the second degree of the scale becomes one of the main theses in Esquisse de l >hlstorie de 1*harmoniei Fetis censures every theorist who constructs a triad or seventh chord on this degree, and he considers it to be one of the basic flaws of every harmonic theory until his own.*' Chords and tonality. In Book II Fetis* notion of points of repose recurs again in his discussion of chords.

There is only one consonant

*For Fetis, the seventh chord on the second degree of the scale originates from "modification" by substitution and prolongation of the dominant seventh.

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13 chord, that which contains a third and perfect fifth and occurs on the tonic, fourth, dominant, and sixth scale degrees.

This consonant chord

is called a "perfect chord" because it renders "the feeling of repose and of conclusion” (p. 23).

Root position triads can occur on scale

degrees other than the tonic, fourth, dominant, and sixth only in a non­ modulating sequence because ”. . . the mind, absorbed with the contem­ plation of a progressive series, momentarily loses the feeling of tonality and regains it only at the final cadence, where normal order is restored" (p. 26).

Otherwise, the second, third, and seventh scale degrees may

have only first or second inversion chords, depending upon the harmonic circumstances, The "natural" dissonant chord, the chord "invented" by Monteverdi, is the dominant seventh chords it alona, according to Fetis, determines tonality because it contains the "appellative” consonance.

Unquestionably,

the dominant seventh chord used in conjunction with the tonic chord clearly defines tonic and establishes a hierarchy of chords based on this relationship to the fundamental chord.

Fetis’ attribution of the invention

of the dominant seventh chord to Monteverdi can not be substantiated and was refuted by Francois-Auguste Gevaert (1828-1908) in the article "Reponse a M. Fetis, sur l’origine de la tonalite modems" intended for publication in the Revue et Gazette muslcale on 13 December 1868, but printed privately.

The text of this rebuttal is appended on pages 219-26

All the other chords, "artificial chords," are the result of pro­ longation (suspension), substitution— which only occurs in the dominant

^See Shirlaw’s comments on this same issue on pp, 3^5“3^7 of The Theory of Harmony,

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14 seventh or its inversions— alteration, or a combination of these#

No

chord, aside from the perfect chord and the dominant seventh chords is an independent structure representing only itself, and every theorist who has failed to perceive this is in contradiction "with the true principles of the art and the science" (p, 6l).

Every non-dominant seventh chord is

the result of suspension* every non-dominant seventh is simply a triad with an attendant non-chord (embellishing) tone, and every theorist, or composer for that matter, who has placed seventh chords on every degree of the scale as structural entities has "forgotten the law of tonality" (p. 66)r The four phases of tonality. The focal point of Book III is modulation and its concomitant effect on tonality* the latter is divided into four stagesi

unitonlque, transltonlque« plurltonique, and

omnitonique, Unitonlque, the first stage of tonality is, Fetis claims, the necessary result of plainchant tonality which consists mostly of con­ sonant triads* modulation from one scale to another was impossible because the church modes did not have a tritone between the fourth and seventh degrees of the scale to define tonic, and "this lack of tonal determina­ tion Is precisely the cause for the absense of modulation" (p. 155) • This period of tonality existed until the end of the sixteenth century because the composers, ", , , dominated by the nature of the harmonic elements at their disposal, have been unable to escape from the rigorous depotism of this tonal unity" (p, 151). Contrary to what Fetis asserts, the composers did "escape from the rigorous depotism" of tonal unity.

By the sixteenth century, madrigal

composers, following the tenets of Vincentino, were developing pictorial

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15 and expressive writing through extensive use of chromaticism and muslca ficta.

This trend, which hegan with Willaert (c, 1490-1562), reached its

heighth with Marenzio (l553_1599) and Gesualdo (c. I56O-I6I3).

Modulation

was effected, not through the tritone, hut through common chords and chromatic inflection! modulation and chromaticism were inextricably tied together and a "floating tonality," a tonality which shifts from one tonal region to the next, resulted.

Consonant triads, assuming Fetis

means root position triads, are the preferred structure, although, because of the chromaticism, they are often chromatically related, i.e., E-flat major to C major or C major to 0-sharp minor.

The music of the sixteenth

century was definitely not unitonlque, but, on the other hand, it was not "atonal" as; Lowinsky propounds,!

Fetis had allowed himself to be

deceived by a very limited definition of tonality. The second phase of tonality, ordre transitonlque, commenced with the "invention" of the dominant seventh chord by Monteverdi, because the relationship of the fourth degree with the leading tone defined tonic and created period phrase structure.

Fetis is attributing more to Monteverdi

than can be verified! the dominant seventh chord did not emerge as a stylistic mannerism until the late seventeenth century.

The seventh of

the dominant was usually a passing tone, as it was in the sixteenth century, and modulation was effected through a common chord and not by "attacking the dominant harmony" of the new key.

Furthermore, periodic

phrase structure was a manifestation of lats eighteenth-century music and not early seventeenth-century music.

The mere appearance of a

“Edward E. Lowinsky, Tonality and Atonallty in Sixteenth-Centurv Music (Berkeley and Los Angeles; University of California Press, 1961), PP. 38-50.

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16 dominant seventh chord does not create a phrase) at a cadence, chordal succession must be coupled with, for example, longer note values, metric placement, and changes in harmonic rhythm, Fetis explores for the first time the concept of common tone modulation, a modulation which he identifies as an "intuitive attraction" because ", « , musical sense compensates for this implied harmony at the moment of the tonal c h a n g e . T h i s phase of tonality, the third, was called ordre pluritonique. and Fetis contends that Mozart was the first to recognize it as a viable means of expression.

Modulation in ordre

pluritonique is achieved through enharmonic relationships in which one note of a chord is considered the point of contact between different scales.

Herein lies the value of the "attractive" dissonances contained

in the diminished seventh chord, the German and the Italian augmented sixth chords, because the enharmonic resolutions of each chord affords several different possible key relationships. The fourth and final period of tonality, ordre omnltonlque. results from the alteration of the intervals of natural chords and modification by substitution of notes.

The fundamental problem with

extensive alteration and modification by substitution is that the har­ monic aggregation reaches a point of saturation when it is impossible to Identify the original chord.

Fetis gives the following as an example

of the dominant four-two chord of G, with substitution of the minor mode Ce-flat) and alteration of the sixth:

^Traite d*harmonie, par, 270, pp. 180-181,

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Regardless of its "origin*” the above defies empirical verifi­ cation as the dominant seventh chord in G.

This is the basic flaw of

Fetis' theory of alteration and modification by substitution.

Of this

last period of tonality Fetis says* The tendency towards multiplicity or even the univer­ sality of the keys in a piece of music is tbe last term of the development of combinations of harmony| beyond that there is nothing else for these combinations.1 The ordre omnltonique has no other goal than the destruction of tonal unity in music, and in this it foreshadows the twentieth century.

Esqulsse de l'hlstorie de 1'harmonie Although Fetis viewed harmony as a musical science which could be codified and systematized, his presentation of the history of harmony is not objective;

it is permeated with his personal views.

Believing in

the infallibility of his doctrine of tonality, he injects his personal opinion at every opportunity; he praises the theorists whose ideas are compatible with his, and he castigates those whose ideas run counter to

1Traite d*harmonie, par. 282, p. 195*

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18 his.

Virtually no theorist escapes his venom.

The EBqulsse de 1'historic

de 1*harmonie is the work of an egotist. As the title indicates, it is an historical outline rather than an exhaustive studyj the sole intent was to provide a succinct record of the facts, errors, and truths of harmony so that future theorists would not have to ascertain them and could avoid perpetuating the "errors," It is a polemic couched in a sincere attempt to evaluate and compare the major harmonic systems until 1840) herein lies its value. (19 April 184-1) to Cousssmaker, Fetis said*

In a letter

"For its object this

[Esquisse] is a new work, and its material is one of the most important of this history of music."

Fetis' history of harmony finds its summation

in his own harmonic theories because* while "Rameau, Sorge, Sehroter, and Gatel found the first elements, I have completed it [the theory of harmony] by putting it on a solid base"*~ this "solid base" is tonality. The annotated translation. Although one of the goals was to produce an English translation which was smooth, readable, and accurate, Fetis* manner of expression is often complex and involved! the temptation to simplify or paraphrase was ever present, particularly in some of the long intricate sentences.

But it seemed more appropriate to retain the

flavor of the original with a fairly literal translation. Some changes in the text have been made, however.

Fetis* spelling

of names of composers, treatises, and musical compositions have been changed from French into the original language or the more usual Latin equivalent.

For example, from Jean Gabrieli to Giovanni Gabrieli; from

Gafori to Gafuriusj from Lucldaire de la musique plalne to Lucidarium ■*-Ses p. 217.

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19 in arte muslcae planae. All musical examples cited by Fetis have been verified and any discrepancies have been indicated in the footnotes; where it was deemed appropriate, another version has been appended*

Likewise, all quotations

have been checked for accuracy, and where phrases or words have been omitted, either intentionally or accidentally, an ellipsis has been added or the omitted word(s) enclosed in brackets*

In addition, if the source,

author, or pages of the quotation were omitted, these have been included in brackets also. and annotator.

All brackets and ellipsis are those of the translator

Since FStls does not indicate when he has italicized

words in the quotations for emphasis, a parenthetical note— (Fetis* italics)— has been added at the close of the quotation by the translator and annotator; The footnotes added by the translator and annotator attempt to clarify the flaws, correct erroneous information, clarification was deemed essential.

and elucidate where

To aid the reader in differentiating

between Fetis* footnotes and those of the translator and annotate^ all footnote numbers of the latter are followed with an asterisk. One last problem had to be coped with in maintaining the integrity of Fetis* work— terminology.

Technical terms (dechant or basso continuo)

have been left in the original language.

The names of the scale degrees

have not been changed into their present equivalents because Fetis states explicitly in the Traitd d*harmonle that the degrees of the scale are -^In Philosophies of Music History (1939I rpt,, New York: Dover Publications, Inc*, 1962)7 Warren Allen claims it is a ", , , notorious fact that when Fetis could not find a date, he invented or guessed at one" (p. 67, n. 11).

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20 deiiigii-- led by their melodic and harmonic tendencies! hence, "third degree" rather than "mediant."

Filially, because Fetis propounds in

the Tralte d'harmonie (p, 6) that only major and minor intervals become augmented or diminished through chromatic alteration, the terms "just" (perfect) fifth, "major" (augmented) fourth, and "minor" (diminished) fifth have been retained.

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ESQUISSE BE L'HISTORIE BE L'HARMONIE CONSIBEHEE COHME ART ET COMHE SCIENCE SYSTEKATIQUE

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22 FOREWORD Voluminous histories of music have appeared in Italy, England, Germany and even in France, where the literature of this art was less cultivated.

These histories, rich in learning as regards times and

facts whose vagueness has not yet "been dissipated because the documents which could raise our doubts have not come to us, have been silent on what in modern art is the most deserving of our interest:

I mean

harmony. Harmony, considered as art, offers the spectacle of transformation in its history, so much more remarkable since the means and end change at times through the appearance of certain facts which, at first sight, do not seem to have such great importance.

Thus, through a simple aggrega­

tion of sounds (one is astonished not to see them introduced into the art before the end of the sixteenth century) one suddenly sees music lose its calm and religious character, acquire the expressive accent which used to be lacking, and even then, having become appropriate to the portrayal of passions, give birth to opera, which would not have been able to exist without this accent. But harmony is not only an art| It is also a science which, by the diversity of its elements and the tenuousness of the bonds which bind them together, presents, perhaps, more difficulties than any other for the formation of a definite theory;

What Is more worthy of attention

than the history of the constant and almost always barren efforts of a vast number of learned men, philosophers, geometricians and great musicians toward the formation of this theory?

Harmony, as a science,

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23 touches everything! from it originates these systems which are so contra­ dictory that some have sought for its principle in the harmony of the spheres, in acoustical phenomena, in abstract numbers, in the measure of the division of the monochord, in the physical makeup of man, and, finally, in Isolated and empirical facts whose influence has been the greatest obstacle to the science's progress for a long time.

Nevertheless, in

the midst of multiple errors there appear, from time to time, a few scattered truths which, although misunderstood, remain the foundation stones of the complete edifice. I believe that the time has come when history ought to set down all the facts, errors, truths! to analyze them, search for their origin, and ascertain the actual state of the art and the science in order to illuminate the path which remains to be traversed* to spare future theo­ rists the useless efforts of redoing what has already been done, and to avoid the pitfalls already pointed out by the failures.

The volume which

I am offering today to a few friends of the science and the art is not this history in all its developments! just as the title indicates, this is only the outline of it.

It has occurred to me that such a great work

could not be released to the public before its attention had been attracted by the indication of essential facts, and I am determined to publish the pieces which constitute this outline in a special paper.

I

am only bringing them together in this volume, which is not designated for circulation, so that my work can be submitted to the criticism of earnest men whose interest it merits by its purpose. The Author Brussels, 12 January 184-1

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2k

Published in fifty copies which are not for sale.

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25 CHAPTER I THE CREATION OF A HAHMONIC SYSTEM1* From the time simultaneous combinations of sound began to emerge from the barbaric system of diaphony, i.e., the long successions of fifths or fourths and octaves, the first elements of harmony present only very crude attempts! it is difficult to distinguish the melodic line from the accompanying part when these imperfect successions recur very frequently*, when the unison constantly happens to betray the harmonic objectives of the compositions, and in which the crossing; of the voices is so frequent. These initial attempts were only in two voices.

We see some of the

examples in Franco's Ars cantus mensurabilis [Couss., Script., I, 117-1353 which, in spite of the conflicting opinion of some writers, belongs to the end of the eleventh century, as I have shown in my Blographie unlverselle des muslcicns fIII. 31^-320J.2* .

1#The title of this chapter is that of the translator, since Fetis did not specify a title. ^*A1though Reese (Music in the Middle Ages [New York: W. W. Norton and Co., Inc., 19^3» P= 289! hereafter cited as MMA) dates Franco as fl. ca, 1250- after 1280, and his treatise as "shortly after 1280," Besseler ("Franco von Koln," MGG. IV, 692) contends that both of these assessments are too late, since the technique which Adam de la Halle demonstrates in his motets (composed between 1262 and 1269) corresponds exactly to the Parisian models of Franco. Besseler asserts that since Adam was not original technically, he copied Francot therefore, Franco's treatise pre­ sumably was written ca, 1260. Strunk (Source Readings in Music History [New York: N. W. Norton and Co., Inc., 19503 p. 13*Uaccepts the latter date, Fetis' dating is wholly untenable and its acceptance " . . . would quite upset the theory of a gradual growth of the mensural system towards standardization." (Reese, loc0 clt., n, 51.) For a complete discussion and critique of Fetis* article, see Coussemaker, L'Art harmonlque aux XII? et XIII? slides (Paris: Durand, 1865)» pp. 22-32,

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26 In this period of the origin of the axt, no harmony other than simple intervals of two tones is recognized, although Franco clearly states that there already was three-voice counterpoint at this time.

But

since he does not furnish an example, and since he does not talk about the uniting of two intervals, there is reason to believe that the third voice was alternatively In unison or in the octave and fifth of one of the other two.

In Chapter XI where he discusses discant or harmony, he

divides the intervals into consonances and dissonances, or rather, concordances and discordances. The concords are divided into three kinds, namely,

(l) the perfect concords which are the unison and the octavej

(2) the imperfect concords or the major and minor thirdsf and (3 ) the mean concords or the fourth and fifth.'*'* Franco also classifies the discords into perfect and imperfect. There are, he says, four perfect discords, namely,

the semitone

(semitonus), the tritone or major fourth (tritonus), the major seventh (ditonus cum diapente), and the minor seventh (semiditonus cum diapente), the imperfect dissonances are the major sixth (tonus cum diapente), and the minor sixth (semidltonus cum diapente). It would take too long to examine the principle which led Franco, or rather his contemporaries* to a similar classification! but it is evident that these principles were arbitrary and false, because the tone, as well as the semitone, is a dissonance, and the sixths bring nothing -*-*The fourth and fifth are intermediary concords— they fall half­ way between the perfect and the imperfect concords in aural perception. According to Franco " . . . they produce a concord better than the imperfect, yet not better than perfect." (Couss,, Script., I, 129.)

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27 "but a sense of concord to the ear and do not imply a necessity for reso­ lution,

With respect to the perfect fourth* ranked here with the conso­

nances, we will see in another place that it has given rise to some lively controversies, Franco gives little information on the use of intervals in the harmony or discant of his timej he only says that this harmony could begin with a unison, octave, fifth, fourth, major or minor third.

Here

is one of the examples which he gives of the harmony of his time, beginning with a fourth,

I have corrected the mistakes of the poor copy

of Abbot Gerbert, according to the British Museum’s manuscript (Scrlptores ecclesiastic! de muslca sacra potlsslmum, II, 13)»

5

£

•3~» 3W

0

5

m

■O

A manuscript from the Royal Library of Paris shows us the didactics of the art of composition and harmony in a most remarkable state of advancement about the middle of the thirteenth century.

Here are some

rules to which I am adding the examples that are lacking in the manu­ script * 1♦ The volume number is undoubtedly a typographical error. should read Vol. Ill,

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It

28 Whosoever wishes to discant £to accompany the chant] ought to know first what the octave and the fifth are* the fifth is the fifth note and the octave is the eighth* He ought to examine whether the chant ascends or descends* If it ascends Tthe chant]* he must take the octavef if it descends* he must take the fifth. If the chant ascends one note* as c-d, one ought to take the upper octave and descend a third as one sees here j-*-

If the chant ascends two notes* as c-e* one ought to take the discant at the octave* and descend one pitch* as in this example:2

If the chant ascends three notes* as c-f, he must take the octave and retain the same note.3

iQulsquis veult desonanter il voit premiers sauoir quest quest quins et doublest quins ejus est 11 quinte note et doubles est la witisme* et doibt regarder se li chant monte ou auale: se il monte nous deuons prendre la double note) sell auale nous deuons prendre la quinte note. Se li chans monte d*une note si comme ut, re* on doibt prendre le deschant du double deseure et descendre deux notes* si comme apert.

2

Se li chans monte 2 notes si comme ut* mi* nous devons prendre li deschant en witisme note et decendre une note. ■^Se 11 chans monte III notes, ut* fa, nous devons prendre la witisme note* et nous tenir au point.

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29

Although these rules and the ones which follow3* for the accompa­ niment of the various movements of the chant appear to have heen made for the improvised harmony which they used to call specifically dechant or or chant sur la llvre, and which the Italians have since then named eontrapunto da mente, they denote a sensible progress which is noticed to the same degree in the harmonic portions of music written in even later time.

But, as is seen, it is not a question of concord with this

harmony, but solely of two intervals, i.e., the fifth and the octave. We observe also that regard for intervallic movement constitutes all of the theory of the art, even in this remote time— a strange anomaly in a time when many imperfect movements were being admitted into the practice. In the same manuscript where there are some rules so consistent with those of a more perfect art, we find some examples where the sequence of fifths and fourths are numerous even in two parts, but where favorable intervallic movement denotes progression towards a more satisfying harmony.

In the three-voice examples,1 the fifths and octaves are used

as actual harmony. In the three-voice chanson by Adam de la Halle at the end of the thirteenth century, and which I have published in the first volume of Revue muslcale (1827) (as well as several other pieces from the

3*An English translation of all the rules for dechant can be found in Hugo Rieraann, History of Music Theory, Books I and II, trans, Raymond Haggh (Lincolni University of Nebraska Press, 1962), pp. 8I-83.

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30 collection where I extracted it),•*•* a few examples of complete harmony of third and fifth, and even of third and sixth, permit; us to see percep­ tible progress in the sense of harmony. It is curious to compare this example with the didactic, written at the same time, which gives us the important works of Marehettus of Padua,

This author, whose fault it is to be obscure by verboseness

rather than conciseness, unduly

expanded certain parts of the science in

which Interest Is moderate in his Lucidarlma in arte muslcae planae,2* and does not expound enough on the objective of our actual researchj nevertheless, we notice there (Tract, V & VI) that the teaching of Franco, with respect to concords and discords, was still in force two centuries after him.^* Thus Marehettus says (in the second chapter of the sixth treatise) that the fourth is not only a consonance, but a divine consonance because it contains in It the sacred quaternary, and because Its parts are, with respect tc music, those which are in other respects the four seasons of the year, the four evils of the world, the four elements, the four gospels, etc.

He must admit that these are

-^This chanson, "Tant que je vivrai," was published in the article "Decouverte de plusieurs Manuscrits interessans pour 1'historic de la musique,” and was extracted from the Vallifere collection, manuscript cote 2736 of the Bibliothfeque Rationale, 2*See Gerbert, Script,. Ill (1784j rpt. Milan, 1931)# 65-188. 3*The premature dating of Franco's treatise led Fetis to a fallacious conclusion. With the more plausible date of ca, 1260 for Franco's treatise, his teachings were still in effect fifty years later. Although the actual date of the Lucldarium has not been conclusively ascertained, the date (1275) given by Gerbert is too earlyf the treatise undoubtedly was written In the early fourteenth century, and probably not later than 1318. (See Heinrich Huschen, "Marehettus von Padua," MGG, VIII, 1627,)

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31 strange reasons for making a divine consonance out of the worst interval of harmony.

The major and minor sixth are a dissonance in Marehettusr

theory, as in that of Franco, of whom Marehettus is, in a way, a commen­ tator,

In the seventh chapter of the fifth treatise, he raises the

question whether the resolution of the dissonance of a major sixth is better to the octave than to the fifth, and answers in the affirmative. The examples which he gives of the two types of resolution furnish us some chromatic successions, so much the more strange and remarkable for the time when Marehettus was writing, because his works have plainsong, where similar successions would be meaningless, for its goal.

Here are

a few of his examples*^*

f ij f 8

$6

8

5

#6

:- f - =

f

8

8

frj*-#6

5

r „ r ; f - ^ d = ^ = « = ^ = f -

These successions, and several others where the boldness of Marehettus* imagination appears in a strange way, remained without significance in his time and only had application nearly 300 years later, because they did not meet any need in the tonality of plainchant. The only one of these things which are found in practice at the close of 1

JkL

^

In each of these examples Fetis erroneously transcribed the treble voice an octave higher them the manuscript indicatesf thus each of the harmonic intervals should be simple rather than compound.

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32 the thirteenth century is the major sixth accompanied by the third and resolved to the octave*

Adam de la Halle furnishes this example of one

preceded by a seventh, entering as the note changes, with a fourth which was not prepared because it was considered a consonance.3-*

r

-o-

m

A treatise of thirteenth-century manuscript music which, after having been passed on from the library of the Abbot of Tersan into that of Perne and today is in mine, provides me ;;ith an example of the major sixth with the major third which resolves down.

Here is this passages

-*-*Both Cousseaaker (Oeuvres completes du trouvere Adam de la Halle £Paris, 1872J, XV Bondeau, pp. 230-231) and Wilkins (The I duas

quintas cum octava et tertia* et duas octavas cum quinta et tei*tia per ascensum vel descensum tenoris."^* This explains the numerous inaccuracies of this genre which we observe in the four-part Mass of Guillaume de Machaut which is believed to have been sung at the corona­ tion of Charles V, King of France* in 1364* being rectified*

The error was not long in

During the first years of the fifteenth century

Guillaume Dufay* Binchois and Dunstable professed to write with more refinement.

The first, particularly, who was one of the singers of the

pontifical chapel of 1380,^* appears to have introduced some noteworthy ameliorations into harmony and the proportional system.

Although he may

not be the inventor of whit*' notation, as some modern writers have believed, it is undisputed that he improved it and propagated its usage. It is not without reason that tht. writers of the fifteenth century have pointed out Dufay as the greatest musician of his time.

A comparison of

his compositions with those of musicians who immediately proceeded alone can give an exact idea of his merit.

There we find the first well-done

imitations and even some two-voice canons which we consider as the first ■^"Likewise we can place [[two fifths with a third in succession* and two octaves of similar fashion and Jf'duss quintas cum una tertia in rota* et duas octavas simili modo, et*)J two fifths with an octave and a third* and two octaves with a fifth and a third, through the ascent or descent of the tenor," (Gerbert* Script., Ill, 307«) 2* Dufay was not born until c« 1400. His name appears in the lists of singers in the Papal Choir on two different occasions! (l) from December 1428 until August 14331 (2) from June 1435 until June 1437« (Reese* Music in the Renaissance |_rev« ed.f New York! W, W. Norton and Co.* Inc., 1959J* pp. 49-50| hereafter cited as MR.)

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43 T* 2* attempts of artificial counterpoint. The fullness of its harmony and the natural stride and melodiousness of the parts are very remarkable. We can judge this by the opening of the "Eyrie" from his four-part Hass entitled Se la face ay pale.

1* While the canon is not employed extensively as a structural element in sacred music until after Dufay, canons and imitations are found in Italian treeentc music. Landings "Del dinmi tu" I (153), athree-part madrigal, has a canon at the fifth between the two lower voices in the first section and a three-part canon at the fourth in the ritornello. In the three-voice caccia "Chosi pensoso" 2 (154)# the two upper voices are in canon at the unison, (Schrade, Polyphonic Music of the Fourteenth Century, IV, 216-220.) For a concise history of the development of the canon in western music see Imogens Horsley, Fugue: History and Practice (New York: The Free Press, 1966), pp. 6-37. 2*Petis uses the descriptive term conditlonnelsi according to Wangermee, Fetis, in an effort to make a finer distinction between the different kinds of counterpoint in existence, used contrepolnt conditionnel in lieu of contrepolnt artlflcleux for canons. (Hebert Wangermee, Franpols-Joseph Fetis. Musicologue et compositeur [^Brussels: Academie royale de Belgique, 195l]» P» 138, n» 3»)

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44 To have reached this point, the harmony of plainchant tonality truly merits the name of art, because in the restrictions of this tonality, we would not know how to write consonant harmony any better. A single dissonance left by the leap of a third attracts attention in the third measure j this is what we call la note changes, because f takes the place of e which would have been the consonant note.

This harmonic

device was in use until the end of the seventeenth century. The purity of Dufay's harmony, of which this example is obvious proof and which we find in some pieces of the same composer in a manu­ script which belonged to Mr. de Pixerecourt

must put us on guard

against the transcription given by Mr. von Kiesewetter (Gescfalchte der europSischabendlSndischen oder unsrer heutlgen Muslk. PI. XIl) of the "Kyrle" from 1*Homme armg Mass of this composer.

The most flagrant

mistakes abound in this piece, and these errors appertain no more to Dufay than to the period when he lived.

There we see an employment of

dissonances which not only was not in use a long time before Dufay, but which has never even existed, and the false relationships there are multiplied.

Space is lacking in this outline to make the necessary

corrections to this piece of music extracted, undoubtedly, from an old faulty copy.

But I am unable to refrain from drawing attention to the

1* Only one work in this manuscript is ascribed to Dufayi "Du tout m'estoit abandonnee"! "Signeur Leon" has only been attributed to Dufay, See the unpublished Ph.D. dissertation (Indiana University, 1959) "Art Edition of the Pixerecourt Manuscript! Paris, Bibliothfcque Nationals, Fonds Fre 15123," Vol. 1 by Edward Joseph Pease, and Dragon Flamenac, "An Unknown Composition by Dufay?", M.Q., XL (1954), 190-200. For Kiesewetter's transcription of the "Kyrie," see Appendix B, p. 228, Ex, 4.

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k5 strange error of the transcriber who has written this piece in the first tone transposed from plainsong, whereas the chanson from l'Homne arml is in the fifth tone, just as the composition of this chant and the Masses of Joaquin DePres, Pierre de la Rue and Palestrina composed on the same theme prove f thus all the minor harmony of the transcription should be major,

I only make this observation in order to show what false ideas

of the history of the art some transcriptions of its monuments, transcrip­ tions made from faulty copies by musicians who do not possess all the necessary knowledge for similar workr can give. A regular harmony and one conforming to the tonality of plainchant was composed in the period of Dufay and merits the name art j nevertheless, a few inaccuracies still appear every now and then in the works of this musician and of his contemporaries or immediate successors: Binchois, Domart and Barblnguant, of which Tinctoris kept a few fragments#

Me

have seen the aggregations of consonances evolve so as to present complete chords of the fifth and third, and of

thethird and sixthj we have seen

the chord successions become regular,

theunison

of the unnatural dissonant harmony in

thesystemof prolongations.

more rare, and the origin

Finally, we have seen the direct succession of fifths and octaves dis­ appear, or at least be concealed by means of passing notes.

However, a

reproach still could be addressed to the composers of the perfections then introduced into the art:

they did not know how to make the voices

sing in narrow limits without confusing them at each instance by the crossings which spoiled the clarity of the harmonic outline.

In this

respect the compositions which belong to the second half of the fifteenth century present a remarkable amelioration.

On this point Ocfceghem,

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46 Busnois, Obrecht, and Tinctoris furnish us some examples of harmony as satisfying by the natural placement of the parts, as by the fullness of the chords and the regularity of their succession,,

Here are Some of them*

n

*7\

5 =

s

$

i flj &

* /?*

xzm

m

This harmonic fragment, extracted from Chapter XVIII of TinctorisT Tralte on the various kinds of points in the notation of music fScriptum

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47 magistrl Johannls Tinctoris super punotis muslcallbus~l, offers several remarkable pecullarltlesr First is the fullness of the harmony* although limited to three voices, combined with the correctness of the successions and the natural movements of the voices*

Secondly* notice

the desire for complete harmony must have been active at the time when this excerpt was written (14-76)* since the tenor is divided into two parts in the fifth measure (probably because it was supposedto be sung by a chorus) in order to have the perfect chord of the third* fifth and octave*

lastly the 4-3 suspension is used with a great deal of elegance,1* The same composer gives us some of the oldest examples of which I

am aware of the combined suspensions of the fourth and ninth in Book I, Chapter V of his Liber de arte contrapuncti (see Ex. l),^* and of the

1*The chromatic alterations in this excerpt are not found in Coussemaker (Script*, IV, 75), and should be viewed as editorial--those Fetis believed were applied by the performers at that time according to the rules of muslca flcta or muslca falsa. In modern scholarly editions of early music,' ail editorial chromatics sire placed above the note in order to dis­ tinguish them from the chromatic alterations given in the original* For some of the rules of and additional information on musica flcta see Reese, MMA, ppa 380-382* Carl Parrish, The Notation of Medieval Music (New York* W, W. Norton and Co., Inc., 19577T"pPr 197-200* Reese, MR, pp. 4448* the article "Musica ficta" in the Harvard Dictionary of Music (2nd ed.; Cambridge: Belknap Press of Harvard University, 19^9), PP. 5^9-551.

2*FetisT predilection for leading tones and penchant for over­ editing transcriptions of early music in accordance with his concept of tonality is revealed again here* there should be no leading tone in the penultimate chord, A careful comparison of Fetis' transcription with Coussemaker (Script.* IV, 85) reveals two glaring discrepancies* (l) the meter signature is clearly indicated as tempus perfectum in Coussemaker and not as tempus imperfectum. (2) The rhythm in the bass line should be parallel to the two upper voices in the first measure* consequently, there is no double suspension. The 4 figure in measure one (added by Fetis) should read For an English translation of this trec^ise see Jean Tinctoris, The Art of Counterpoint, trans, and ed. Albert Seay (n.p.f American Institute of Musicology* 1961).

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U8 fourth and seventh in the same book. Chapter X (see Ex. 2),

This last

example £Couss*> Script., IV, 9?J has the disadvantage of stating several consecutive and exposed quarters at the moment the dissonances resolve jr to avoid this disagreeable effect, contemporary musicians do not resolve two dissonances simultaneously.

1.

At the period of the history of harmony at which I have arrived, the system of consonant aggregations and of artificial dissonances by prolongation is complete, with the exception of the fourth and the sixth which did not appear at all as consonant harmony in the works of the fifteenth-century musicians.

At that time all the harmony was contained

in the chords of the third and fifth, third and sixth, third and octavef and in four voices, third, fifth, and octave. by delaying the thirdl

This harmony is modified

by suspending the fourth into the chord of the

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H9 third and fifth to produce the fourth and fifth (Ex* l)j by delaying the octave in the chord of the third and octave or the thirdr fifth and octave to produce the ninth and third, or ninth, third and fifth (see Ex* 2}t by delaying the sixth in the chord of the third and sixth to produce the third and seventh (see Ex* 3)I by delaying the lowest note of this sane chord to produce the second and fifth (see Ex. 4)j by delaying both the octave and third in the consonant chord composed of those intervals to produce the ninth and fourth, or ninth, fourth, and fifth (see Ex* 5) I finally, by delaying the third and the sixth in the consonant harmony composed of these intervals to produce the fourth and seventh (Ex, 6),

Z.

1

3

— 0--

9 = 4

JL*— —

3

At . J.- o— H K.

9 — *— y.

a----- -9o--- 9 a 9 6

c ■=s-l

Ft

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=

F =

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.

— H---8

7 6 — a---- -j’s..™,.."—

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f c ~ = =

f

=t=j=? — •---- — O--- o A . 5 9 8 JL 4- 3 T — = H &= .s -A —

V

4 ^

6. — H-: --

r= n 4 j --itp| — ©— - ' a

.a..:.«_ 8 7 6 6 43 4 3 “*r— ar-“ — E---- -

\

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

—

50 Such are all the consonant and dissonant harmonies employed hy the composers in the second half of the fifteenth centuryj until the end of the sixteenth century no others were known, because the tonality of plainchant, the basis for all music until this time, was not able to give rise to anything d i f f e r e n t T h i s tonality, which is unltonlque. i.e., does not modulate, in effect does not contain the elements of any harmony other than the consonant chords and some retardations of their natural intervals to produce the artificial dissonances of the second, fourth, seventh and ninth.

Until the

death of Palestrina, music was contained within these bounds. It is easy to comprehend that, limited to such a small number of harmonic combinations, the musicians were destined to search for some elements of interest for their works in a sequence of musical ideas richer in variety.

This was what led them to the discovery

of imitations, canons, and contrapuntal fugue.

I have said that the

earliest traces of these quests can be noticed towards the end of the fourteenth century.

In the first half of the following certury these

imitations become more frequent, and we find the earliest attempts of two-voice canon.^*

A name was needed to distinguish this new genre of

composition of simple counterpoint} it was given that of res facta

^*Tn his Tralte complete de la theorie et de la pratique de l*harmonie ["(Brussels, 1844), Bk. II, par. 132, p. 5 9 Fetis speculates that the need for variety within unity ("the logical relationship of sound”) led to the modification of diatonic chords* ^*See footnote 1 on page 43*

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51 (completed \rork)*^ Jean Ockegfeem seems to have possessed, about 1460, a quality which was superior to all the other musicians of his timer

Xt

is astonishing to see the degree of perfection to which he brought the treatment of three-voice canons, or, as was said then, of the composition of three voices in one.

Here is the beginning of a piece of this genre,

taken from Glarean (Xbdecachordon, p. 454):

■^•Tinctoris defines the differences which existed between the composition res facta, simple counterpoint, and general counterpoint or chanc b u t le livre in Book II, chapter xx of his Liber de arte contrapunctl. Guerson also speaks of it in detail in his treatise entitled Iftillsslme musicales regtale cunctls summopere necessarie plan! cantus. slmplicis contrapanctl, rerum factaram, , . . Lastly, one finds something concerning this composition, but solely with respect to melodic figures employed there, in the Rudiments de musique pratique, reduits en deux briefs Traltez by Maximilian Guillaud (Parisi Nicholas Du Chemin, 155^ PFor a more recent discussion of res facta see Ernest T. Ferand, "What Is Res Facta," JAMS, X (1957), 141-150 J " ^The original version was a third higher with three flats as a key signature! I prefer this transposition because there is no point in using the rather unfamiliar clefs, to musicians, with f on the third line and c on the second,

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52

1^ 1*-—

— fr~f"—

p p - r r

fl— 3 --,---- _ ---

J .

P* n

n

a .----- r

^

f

h

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ft?— --u r

i

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r

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r

J

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r

i

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1

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r

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Si.-:

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f

e

C

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f

=

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l

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The artist who, as far hack as the origin of artificial fconditionnellesl compositions, demonstrated this perfection of structure, truly deserved the admiration which he inspired not only in his students* hut in all the musicians of Europe.

I still find myself compelled to do,

at this point, a critique of the errors which are the pitfalls of music

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53 historians* grave errors which result in a completely false idea of the art in past times.

The piece of which I have just stated the beginning

was written by the composer as a single-line enigmatic canon* with these instructions to serve as the key to the solution of the problems

“Fuga

trium vocum in Epidiatessaron post perfectum tempus** namely, a threevoice fugue (canon) at the fourth above, after a complete measure.

Despite

such an explicit instruction, what does Burney do in his A General History of Music (Vol. II, p.

)?1*

He reverses the order of the voices in such

a way that the canon is at the fifth below, and that wherever there ought to be fifths, there are fourths.

y

jri,

i

© 4 - 3 ------- -— B H k — ------- —

k

w

i

=

Here is his transcription:

=

=

=

---- ....... ~f—

ft-

---- - p — *— "1... — h .. H

b

u

P' ■ .. u r

d

i y f f ~T f I "p J i " p - p - r- ~ f--!~ --= -c-. 1 ™ ■ f f ■ _a___| __ ^4 W 'T 1 -1

■*-*Vol. 1, p. 729 in the Dover reprint (1957) edited by Frank Mercer.

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-

5^ This ridiculous transcription has been, nevertheless, reproduced hy Forkel in Vol. II of his Allegemelne Gesohlchte der Musik [^Leipzig, 18013 (Pr 530)^* and hy Mr, Kiesewetter in his memoire laureating Dutch musicians (Plates, Ex. 0). Once entered upon the path of imitation, canon and pursuit of any genre, the musicians regarded these adjuncts of the art as the principal object, applied all their faculties and, with some very few exceptions where more bold and original trends appeared, all the music, sacred or secular, continued in this manner until the end of the sixteenth century, namely, for nearly one hundred and eighty years* In this long period of time, melody made little progress, because the themes of popular songs served as the foundation for madrigals and for all mundane music, and the singing of hymns and antiphons, or even that of common melodies, was also developed in the scientific combinations of church music,

A small number of compositions, designated by the

name sine nomine, appeared in immense quantity in the works which remain of fifteenth and sixteenth century musicians*

But the themes of these

2* compositions have such little characterization that it is easy to

•*- Fetis apparently did not read Forkel*s explanation for the inclusion of Burney*s transcription, Forkel points out (pp, 529-530) that Burney avoided the fault of Wilphlingseder and Hawkins by correctly tran­ scribing the canon in tempus perfeetum. But Forkel continues1 "On the other hand, Burney side-stepped another of Ockeghemrs instructions, according to which the three voices should follow each other at the fourth above (Epidiatessaron). and wrote it at the fifth below, but however in the Epldlapente instead of the Epidiatessaron under the canon. If there­ fore this canon is to be solved complete according to the instruction of the composer, the upper voice should be changed into the lower, and the lower into the upper." (p, 533») 2*The use of a secular cantus firmus rather than a Gregorian melody was one of the characteristics of Burgundian polyphony. A sine nomine Hass denoted a composed, as opposed to borrowed, melody serving as the cantus firmus.

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55 4magiTin that their composers attached little value to this aspect of the art. With respect to harmony, the tonality of jolainchamt, which was that of all music, sacred or secular, presented nothing at all of the source of other varieties than those which I have indicated previously* Whence it happens that from Ockeghem to the death of Palestrina, namely, in the space of 140 years, the chordal combinations remained nearly the same.

The well-known composer Palestrina attracted attention to himself

in this important aspect of the art only by the perfection which he brought to the voice motion, and by the admirable sense of tonality which shines in his harmonic successions. The theory of harmony ceased to anticipate the practice of the time when Dufay lived.

However; in the second half of the fifteenth

century two men of keen intellect and profound erudition came along to render some eminent services to the science, and established the situation as it was at that time through their works.

The first of

these erudite musicians was Johannes Tinctor, or rather Tinctoris, a Belgian priest who was choir master for Ferdinand d*Aragon, King of Sicily and Naples (prior to 14-75)» and who dedicated his books to this Princes

All Of the musical science of this era is contained in his works.

T# The Art of Counterpoint. that is to say, the art of writing, is particu­ larly well set forth by Tinctoris in the rules concerning the succession of intervals, the sole fault of which is to be very redundant.

But one

^•*Llber de arte eontrapuncti (Coussemaker, Script., IV, 76-153)r one of 12 treatises by Tinctoris, is dated October l4-77» and is divided into a prologue and three books. An English translation of the prologue can be found in Strunk, Source Readings, pp. 197~199» For a complete English translation, see p. 4-7, n» 2*»

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56 must not search for the reasons for the chords taken individually, nor for anything which resembles a system of classification of these chords. The views of musicians were not at all directed towards such considera­ tions, and they did not understand, moreover, the necessity for systematic classification, Gafurius, whose works are a few years later than those of Tinctoris, is perhaps inferior to the latter with respect to the lucidity of insightsf nevertheless^ he has enjoyed greater renown because his works have been printed and several editions have been made of them, while those of Tinctoris have remained in manuscript and copies of them are very rare, Gafurius has given the rules for the art of writing harmony also in his treatise Practlca muslcae £Milan, 1496]f but we find in this work only, as in those of Tinctoris, ideas of intervals and not of chords of three or four tones,** In vain would you search in the writings of the sixteenth century to find science more advanced in these respects} Zarlino himself, this great musician whose works ought to be considered the code of the art at this time, has nothing which could give us the idea of a synoptic science of chords.

Nevertheless, it is in these same works that we find elemen­

tary ideas of double counterpoint, that is, harmony built according to the notion of intervalllc inversion, a notion which was a stroke of enlightenment for Rameau 160 years hence, and which lead him to the discovery of one of the bases of the science of harmony,

Zarlino dealt

^*The Practlca Muslcae of Pranchinus Gafurius, trans, and ed. with musical transcriptions by Irwin Young (Madisoni University of Wisconsin Press, 1969%

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57 with these counterpoints in Book III, chapter 56 of his Lb Istltutlonl harmoniche, the first edition of which appeared in 1558.1*

It is

remarkable that the inversion at the octave, the results of which are all consistent with tonality, did not stand cat in the minds of musicians, and that double counterpoint at the tenth and at the twelfth, much less natural, are those of which Zarlino speaks.

It is also noteworthy that

the examples of artificial fcondltlonnelles"] compositions provided by the celebrated writer are the only ones which ws know of in this era, and of which not a single one is found in all of those of the masters of the Roman School which have reached us prior to the end of the sixteenth century. I said previously that by the mid-fifteenth century all of the harmonic resources which the tonality of plainchant could produce had been fixed, and that this was a sort of foreboding of the impossibility of introducing any new varieties, thus throwing the musicians into the quest for imitations, canons and contrapuntal fugues, in which they manifested a singular skill until the end of the sixteenth century. The direction taken by all the artists during this long interval of more than 150 years attracted them so much that not only were they not at all Interested in the need to vary the forms of the chords, but also that melody itself was considered as nothing more than a part so subsidiary to music that they did not even condescend to invert any; and that the most popular cantilenas were taken by 20 different composers for the themes of their works.

What is more, they attached so little importance to the

^*Book III is available in an English translation in Gioseffo Zarlino, The Art of Counterpoint, Part III of Le Istitutlonl harmoniche, 1558, trans, Guy A, Marco and Claude V. Palischa (New Haven1 Yale University Press, 1968).

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58 significance of the words that they no longer respected either the mean­ ing or the prosody*

The extravagance of the musicians even went so far

that in churchr while one part of the choir chanted "Credo" or "Miserere*" the other musicians said the words of the secular chanson which supplied the theme of the Mass or Psalm, e.g., "Baisez-moi, mon coeur" or else "Robin, tu mJas toute mouillee," To be sure, so complete a destruction of all reason, of all pro­ priety, such a degradation of the art is one of the most remarkable facts of its history.

Taste, simple and true sentiment do not have strong

enough expression to denounce such aberrations.

However, it is to these

atrocious errors that this same art is beholden for the immense mechanical progress which it made then with regard to the purity of the harmonic progressions, the elegance of the movement of the parts, and the magical art of making five or six different voices sing in an easy, natural manner in the most confined space.

This art is unknown in our day, but

was carried to the highest degree of perfection in the sixteenth century, particularly in the Roman School,

When Palestrina came in the second half

of tbls century, he was to refine the style without impairing the skill­ fulness of the art of writing, and to give to church music the noblest character (the most worthy of his objective)! while appearing to be richer in details and more inventive in contrapuntal resources than any of his contemporaries or predecessors, Palestrina imprinted the style of ultimate perfection on the music which emanated from the tonality of plainchant.

No less superior in the madrigal style, he closed the field.

Thus it became necessary to throw the art into new courses, in a word, to transform it| several talented men were excited by this necessity and

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59 made some more or less successful attempts*

But In the midst of them

Monteverdi was conspicuous, especially through some innovations which changed tonality and made it what it is today.

Here I ask for the

undivided attention of the reader, because it concerns the most remarkable era in the history of music and the most fruitful in consequences,

At

first simply a violist, later the music director for the Duke of Mantua, and finally maestro dl capella at San Marco in Venice, Monteverdi (bom in Cremona about 1565) became famous through a number of inventions which gave a new direction to music*

This new direction was in Book III

of his five-voice madrigalsr published in 1598*

where his genius showed

its forwardness by attacking the double and triple dissonances of suspen­ sions in a manner unknown to his predecessors* particularly in the madrigal "Stracciami pur 11 core*" where we notice this passaget**

rfl---------

1

0 - c— ■' - "i

"

... r

j j =4=4= U = H 1UBf''■ '“ f-4 -l

s H =

T - r - f

\ji

- .

4-in a —A------

____

...

------ ■ ar

,

A

r

f r

i

4

J

— - H9------ '•O'-'

&"fl= fl4r

■ i- k . J . 1

- -0-----

“*1598 is the date of the third printing, The third book of five voice madrigals was published initially in 1592 by R, Amadino in Venice* ("Claudio Monteverdi," MGG, IX, 518.) 2*

Claudio Monteverdi, Tutte le Opere, ed, G, Francesco Malipiero (14 vols.f Viennai Universal, 1926-42), III, 26-32,

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60

Nothing similar appears in any of the music which existed prior to the publication of this excerpt*

Some double dissonances had been

noticed here and there in the works of Palestrina, but these instances are rare.

Moreover, these dissonances by suspension are always regularly

prepared and resolved in the works of the illustrious master of the Roman School? whereas the passage which we have just seen Includes not only some triple dissonances of the fourth resolved to the third, of the seventh descending to the sixth, and the ninth resolving to the octave, but also the notes delayed by the three upper voices are simultaneously attacked by the tenor in the third, fourth, and fifth measures? this irregularity produced some Intolerable dissonances of the second in the fifth and last measures.

On the one hand Monteverdi

enriched harmony with new combinations? on the other hand he damaged it by rejecting the evidence of the ear which revolts against such harshness. As regards the seventh accompanied by the fifth and the major third, which we see in the penultimate measure of the cited example, although it was prepared as a suspension, it is not an innovation of lesser importance if we consider it as the origin of modem tonality, because between the leading tone, which forms the major third, and the seventh, which is the fourth degree of the scale, there is an appella­ tion of cadence which precisely forms the character of our tonality? whereas these processes of cadence are never necessary in the consonant harmony which resulted from the tonality of plainchant nor in the

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61

1#

harmony, dissonant by suspension, which derives from it,,

If we

attentively examine all of the music which preceded the remarkable fact pointed out here, it will be seen that what gives it the strange character for our ears is the connection of the phrases to each other without cadence, which never completes the meaning if it is not the end or at some momentary pause.

Thus our ear is incessantly frustrated and

searches in vain for a conclusion— the forming of a phrase before the end of the piece.

On the contrary, as soon as he had made use of the minor

seventh with the major third and fifth, the need for the resolution of these two gravitational notes inevitably led to the cadence, defined the resolution which concluded it, and gave a fixed form to the phrases.^* If we are willing to understand how a seventh chord originating from the delay of a sixth chord differs from the seventh chord composed of a major third, just fifth, and minor seventh, such as Monteverdi employed in the cited example, it should be noted that the resolution of the dissonance being made, he has a sixth chord which does not belong to tonic, because every dominant is, in modem tonality, a note of repose or an intermediary of cadence which is not able to support a chord of the sixth; this is only admissible in plainchant tonality (see Ex, 1 £p, 62]),

Moreover, when a seventh chord with a major third is

•^etis, fervently believing that "modem" tonality resides in the tritone, contends that ", , » having no leading tone in the established music on this ancient unltonlque tonality, since this note only acquires its character by its harmonic relationships with the fourth degree and the dominant in the natural dissonant chords, the cadence properly speak­ ing, that is the rhythmic termination of the phrase, did not exist," (Fetis, Traite d'harmonie, Bk, III, par. 245, P« 152.) 2*Petis asserts that with the advent of the dominant seventh "cadences become frequent and regular," and periodic phrase structure results, (Traite d'harmonie, Bk. Ill, pars. 256-257, pp. 165-166,)

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62

accompanied by the fi£thr it is evident that it does not arise from delaying the chord of the sixth because if the suspension were resolved, we would have a harmony of the fifth and sixth (see Ex. 2).

Thus it is

clear that, following modern tonality, when the seventh resolves down, the major third ought to resolve up which, in the case where the bass is stationary, creates the harmony of six-four (see Ex, 3)«

It is thus that

Monteverdi employed these chords in the cited example#

It is correct to say that Marenzio had made use of the harmony of the seventh with a major third and fifth in the madrigal M0 voi che sospirate" in Book VI for five voices, published in 1591 or seven years before Monteverdi had his third book for five [[voices] published.

But

before long the latter, guided by his instinct, comprehended that the preparation of the dissonance was not necessary in the dominant harmonies accompanied by the major third, and in his fifth book of madrigals he attacked, without preparation, not only the seventh, but even the ninth of the dominant} composers have imitated him ever since.

The relationship

established in these harmonies between the fourth and seventh degree of

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the key Is the principal constituent of modem tonality. in vain in all the music prior to Monteverdi and Marenziox

Me would search it does not

existf it could not exist without destroying the tonality of plainchant. The attraction of these two notes, the necessity of the seventh degree to ascend while the fourth descends, is the peculiar character of the leading tone which received its name because of its tendency.

Thus all

modem tonality is built on this succession which was unknown to all the musicians until the end of the sixteenth century?

As

soon as this succession was admitted into the art,

the eight

church modes from the domain of harmony, and there

were only

two modes

of tonality? major and minor for each note? one or

the otherof

these types was built same way.

it banished

on each note, with each type constructed in the

In both modes the relationship of the fourth degree and the

leading tone is the same? as a result of the appellation of these two notes, the end was always the tonic and harmonic dominant, i.e., the note which is heard in the greatest number of chords is always the fifth degree of the scale. Another phenomenon was the necessary consequence of the dissonant dominant harmony,

I mean modulation, that is to say, the transition from

one key to another by the sole act of attacking the dominant harmony of the latter without preparation, because this harmony immediately creates

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64 the new key hy the double appellation of the fourth degree and the lead­ ing tone.

It Is this faculty of liaison between keys which I have called

ordre transltonique in the course "Philosophic de la musique* which I taught in 1832 in Paris, the church modes.

No counterpart existed in early music based on

Thus it occurred that the efforts of Nicola Vicentino^

and of several other masters of the sixteenth century to create modulatory music were fruitless, because,never having the need for the resolution of consonance and dissonance by suspension in early tonality, any change of key was optional.

Actual modulation was never made apparent.

Hr, von

Winterfeld, who has only a vague notion of all this, like the other music historians, went astray in his book on Giovanni Gabrieli when he proposed to prove the existence of a chromatic genre among the early masters* They had not known this genre because its existence in their system of tonality was impossible, flho would believe that there was not a word of everything we now understand concerning this era which was so important in general history and particularly in music for the change of tonality with all its conse­ quences?

These voluminous compilations abound in nonessential details,

but such a fact is not found in any of the compilators nor enough of practical knowledge, attention or philosophy to shed any light on this obscurity.

All that Burney and Martini saw, copied by Forkel and routine-

minded people, is that Monteverdi had added some new chords to those which were used before himj as for the results of these new harmonies, no one questioned them, ^■L*antica Musica ridotta alia modema nrattica T (Borne 1 Barre,

JTW,----------------

1555)],

2*See pp. 14-15,

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65 Notice, however, that these results were not limited to those whom I have known previously, because as soon as there was a means for modula­ tion and of cadentisl action, the phrases were shaped rhythmically— they had some closes; in their sequence they had as a goal tc move successively from one key into another, then to return to the original key in order to vary the feeling which previously lay on another base.

Now the direction

of all music in a system of alternating modulation led, in the first half of the seventeenth century, to the abandonment of canonic counterpoint and of imitation, which existed only as an object for study, and produced the substitution for it of the real and tonal fugue, the technique of which consists of a regular system of modulation by means of a principal phrase called the "subject,"

The true fugue, then, dates only from this

epoch, although the name was known a long time before;

1*

but the name

misled the erudites in music who, without a solid understanding of the art, confused all of these things.

Insensibly the dissonant harmony of

the dominant wasestablished in the fugue, without preparation, as in all the other music,

The free introduction of this harmony in similar

scientific pieces was slow because the change of tonic was not perceived, and because the actual principles of this tonal change were still a ■*-*For Zarlino rs followers the word "fugue" had a very precise meaning. In The Art of Counterpoint Zarlino points out (pp. 126-13*0 that there are two types of fugues* strict (legate), or what is now recognized as canon, and free (sciolte), in which "the imitating voice duplicates the other in fugue or consequence only up to a point; beyond that point it is free to proceed independently" (p. 127). In both types of fugue the conse­ quence, which had to be exact, occurred at the unison, fourth, fifth or octave.Imitations, on the other hand, while also classified as strict or free, could occur at any interval* the generic classification was retained, but the specific size could vary, (ibid., pp. 135-1^1 ■)

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66

mystery to the musicians.

The rigorous conditions of early counterpoint,

the necessary consequence of plainchant tonality, restrained the masters and created indecision amongst the students.

Even today when students of

some composition schools leave the study of free counterpoint to take up that of the fugue, their professors have a great deal of difficulty explaining the reasons for the free employment of natural dissonances in fugues, sin^e they had prohibited them in counterpoint.

It results in a

sort of trial and error experiment the first few times.

However, I have

explained all of this in the second part of my Traite du conire-point et de la fugue, published 16 years agoj a great deal of time is necessary to make apparent to everyone the facts in an art-science where the consider­ ations are as manifold as in the music, and when the music is capable of numerous transformations. In creating the fugue, the new tonality created double counter­ point at the octave, because the necessity of alternatively passing the subject and the countersubject, which served as an accompaniment, to the lower and treble voices obliges the musicians to turn their attention to the consideration of intervallic inversion at the octave| thus the rules of double counterpoint arose.

We understand from this that these

rules are the prolegomena of the fugue} we can evaluate the shortcoming of criticism and of the order of ideas which lead Berardi, •*£d , Angelo Berardi], Document! armonlci (Bologna:

1

Trevo,

2

1 Fux,-'

Giacomo Monti,

1687). ^ZaccariaTrevqj, II Musico Testore (Venice:

Antonio Bortoli, 1706).

Johann Joseph Fux], Gradus ad Paxnassum, slve nanuductlo ad composltlonem muslcae regularem, methodo nova (Vienna: van Ghelen, 1725).

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Marpurg,^- and Albrechtsberger,^ to d9al with some double counterpoint

only after the fugue, which would not exist without double counterpoint. I have restored the rational order of these studies in my Tralta du contre-polnt et de la fuguei several works, since published on the same subject, have followed this order.

I still ought to cite, with regard

to double counterpoint, a newly proven fact which substantiates that the history of the art in general, and of harmony in particular, is ill-known, even by the most educated men, and that much more is made of some curi­ osities of little consequence than of those which have real value. In the first months of 1830, Hr. de Vos Willemenz, a member of the Instltut des Pays-Bas and secretary of the fourth class of this erudite society, wrote to me, enclosing the volume which contained the report of Mr. Kiesewetter and of myself on Netherlandish musicians*

"You will read

with interest that Mr. Kiesewetter has demonstrated that the Netherlanders are the inventors of double counterpoint."

As a matter of fact, the

second part of Mr. Kiesewetter's report has the title

L*Invention du

contre point artificial, nomme par les modemes CONTKE POINT DOUBLE, peut-elle stre attribute aux Neerlandals?^

Now in examining this question,

Mr. Kiesewetter assumes that double counterpoint is only a division of the ^Friedrich Wilhelm Marpurg], Traite de la fugue (Berlin* B. A. Haude und J. C. Spener, 1753“175t /* [.Fetis has given the facts of publi­ cation for the German edition* Abhandlung von der Fugue nach den Grundsatzen und Exempeln der besten deutschen und auslandischen Keister* the French edition did not appear until 1756*3 ^Johann Georg Albrechtsberger]. Grundliche Anweisung zur Komposltlon (Leipzig* Breitkopf, 1790). %a n n den Niederlandern die Erflndung des kflnstllchen. oder von den Neuren also genannten doppelten Gontrapunktes zugeschrieben werden?

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68

part of the art which he designated hy the general expression contre -point artificial, and the invention of which he attributed to the musicians of the Netherlands

Schoolj from this we are expected to

conclude that he considered these fifteenth century musicians the inventors of this type of counterpoint*

Seeing his error on the subject*

I wrote to Eandler, my corresponding member in Vienna and friend of Mr* Kiesewetter, and showed him that double counterpoint did not arise prior to the second half of the sixteenth century, and that its use in the fugue dated from the seventeenth.

Handler communicated my letter to his

friend, who wrote to me from the Baden resort near Vienna on 27 July 1830. I am indebted to you for a very ingenious observation, one worthy of a professor of counterpoint as well as a scholar in the historical fact of the musical art} an observation which was truly never so clearly presented to me. It is that the old masters were not yet aware of this type of counterpoint which the academy of our day called contre point double *3. and of which the first traces are only found towards the end of the fifteenth century, I acknowledge that immediately I went through about 100 scores of my collection, and I am convinced of the accuracy of your assertion, because everything that I have been able to find of the same kind was indeed only simple imitation. If anywhere there was a short passage which seemed to announce the outline of inversion, the latter did not follow, and immediately I noticed that it was only suggested accidentally and not intentionally in the forms. In rendering justice to the sincerity of Mr, Kiesewetter, who brought about recognition of an important error with so much candor, we can not help regretting that the study in which he indulged after having read my letter had not preceded his work} nonetheless, this error remains Why "of our day"? Didn't Zacconi deal with De* Contrappantl doppii in the second part of his Prattlea dl musica* published in 1^22? Isn't it also the same for 1. Penna in Book II of his Prlml alborl musical!* pub­ lished in 1656? for Bononcini in Part II of his Muslco prattlco (1673)? for Berardi in his Document! armonlcl (I687), and for 20 other seventeenthcentury writers?

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69 published in a hook whose Influence is much greater because it has been awarded a prize by an academy.

And it is precisely this Influence which

compels me to stress the fact that the natural harmony of the dominant created modem tonality? that the latter led to the destruction of imita­ tive counterpoint in the tonality of plainchant, and to the creation of the fuguej finally, that the inseparable conditions of the latter gave birth to double counterpoint at the octave.

All of this is connected,

but, once more, the philosophy of music history has been completely unknown until today. Because the expressive accent, composed of the appellation of the leading tone resolution in conjunction with either the fourth degree of the two modes or with the sixth degree of the minor mode, was contained in the natural dissonant harmony, music assumed a dramatic character with the birth of this harmony? all the new forms which transformed the art— true opera, the cantata, the air with instrumental accompanimentarrived at almost the same time. But everything in an art which is transformed is not conquered. The bold spirits who discovered the new harmony and all its consequences could no I— be satisfied to sake do with the rigorous rules of the art of writing to which the composers of the older schools were indebted for the admirable purity of style which renders their works imperishable, and which made them models of disheartening perfection.

Inaccuracies of every

kind began to be profuse in the works of the musicians of Venice and Naples? Home alone resisted and retained the excellent traditions which more than two centuries have not destroyed completely.

Proud of its

success, the new school did not delay invading the church? the dramatic

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70 style was Introduced Into sacred music and took the place of the ponderous, solemn and devout tone of the works of Palestrina*

Then the absurdity of

this secular expression began to be applied to holy things) in lieu of being prayers and professions of faith, the "Gloria," "Credo"Sequences" and "Psalms" became dramas*

The symbol of the suffering of the Savior was

transformed into a representation of carnal agony, and one went to church to experience emotions rather than to pray with meditation*

That was not all*

Instrumental harmony had become necessary

since vocal harmony ceased to constitute every single kind of music.

Thus,

instruments assumed the same importance in the church as in the theatre and became Indispensable especially for the accompaniment of motets for solo voice or for two voices) motets were published in tremendous quantity in the first half of the seventeenth century*

The concert# style followed the

simple (osservato) style of the old sacred music) from that moment the genres were merged and, as Abbot Bainl said, church music was destroyed*

Since

then some beautiful works in the system of expression applied to sacred music have been composed) some models of perfection of this genre are the Psalms £l724[] of Marcello, the Miserere of Jomelli, some of the compositions of Alessandro Scarlatti, [[Leonardo (1694-1744)3 Leo, Pergola si and, in more recent +imes, the Masses of Cherubini.

But the genre itself is an abase­

ment of the primary object of the art in the service) the degradation was so heartfelt that even Rossini, this leading propagate

of the

brilliant art, frequently told me that the only profound and lasting impression of a sacred character in church music which he had experienced was made by the works of Palestrina, which he heard in Rome 25 years ago* For the worldly people, and even for the musicians who hear only this

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71 music in Earis concerts* with mediocre execution* there probably will be more than some exaggeration in these words of the composer of Guillaume Tell | but for whomever has made a serious study of the immortal works* objects of his admiration* there is a conviction that true church music ceased to exist with the appearance of theatre music* and that it lost -n the second half of the six­ teenth century with these wordsi

da cantare e suonare, that the instru­

ments performed the same parts as the voice.

As to the figures and symbols

placed above the bass, these are not noticed at all prior to the year 1600. The idea must be credited to either Emilio de Cavalieri or to Guidotti, Concerning Viadana*s invention, the error of Mr, Baini results from knowing neither the passage where this musician himself speaks, nor the testimony of Cruger, From all of the preceding it follows that (l) the initial idea of a continuous bass accompaniment arose with the first attempts of solo voice songs supported by an instrument, about 1580} (2) this bass, becoming more animated and varied in its forms, was adapted to organ for vocal accompaniment by Viadana, and received the name basso continuo from him about 1596t

(3) about the sane time, the usage of figures and symbols

above the bass for raising or lowering £notesJ was introduced by either Emilio del Cavalieri, his editor Guidotti, or perhaps by some other unknown •j jt

Arnold states that "The term was first brought into general notice owing to the large and immediate circulation of Viadana*s Concerti, 1602, though the term Basso continuato was used by Guidotti in 1600," (Arnold, I, p, 6, n, 27)

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78 musician.

Soon afterwards many sacred works with figured basso continuo

were published, and from that moment the genre madrigal-liker i»e*, unaccompanied music for several voices and in imitative style, ceased to be in vogue in order to make way for what the artists themselves called the nuove ausiche. It would be deceiving to believe that the formation of a harmonic system was the immediate result of the double invention of basso continuo and figures intended to denote its accompanimenti quickly.

things did not move so

The slowness of scientific progress in that respect is likewise

one of the remarkable facts of the history of music*

Some directions

which appeared in the first half of the seventeenth century concerning the art of figured bass accompaniment only reinforced the observations of detail, which showed no consideration at all for the generation of a chord.

At last a fact of some importance appeared 25 years after the

publication of Viadanars workj in this fact the first elements of ths science of harmony are found.

It merits all of our attention*

Agazzari

from Siena, a contemporary of Viadana, had worked out Guidotti's instruc­ tions on the use of the figures, and had extended its applications in his Del sonare sopra 11 basso con tutti 11 stromentl e dell* uso loro nel conserto, placed at the front of the fifth book of his motets which appeared in 1607? he did not show the nature of the chords which ought to have belonged to such and such degree of the scale.

Galeezzo Sabbatini,

maestro di cappella to the Duke of Mirandola, went farther In his small work entitled

Regola facile e breve per sonare il Basso continuo nell*

Organo, fMana cordo.l £ altro Simile Stromento * * . (Venice, 1628, 30 PP*)r because he gave the first rule of the octave which is known, namely, the

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79

harmonic formula indicating the proper chord for each note of the scale.

I*

It is of this formula that Sahbatini intends to speak when he says that he is the inventor of its method; but the ambiguity of his words leads one to believe that this musician was the first who had a treatise on accompani­ ment.

This is an unmistakeable error. It does not seem that Sabbatini's method had much success in its

newness, because Lorenzo Fenna, who had a treatise on basso continuo and on the manner of accompaniment r said not one word about the rule of the octave in Book IIr Chapter 23 of his Li prlml alborl musical! (Bologne , 1656),

Most of his rules are arbitrary; he does not go into the consider­

ation of the formation of chords at all.

Things remained in this state in

Italy until the end of the seventeenth centmry

with respect to harmonic

theory; but the practice of accompaniment made considerable progress particularly in the schools of [^Bernardo] de Pasquini in Rome, and of Alessandro Scarlatti in Naples.

For their students these great musicians

wrote a great many figured basses, to which they gave the name partimentl1 instead of striking some chords, following the French and German usage, these masters demanded that the accompanist have all the parts of the accompaniment sing in an elegant manner.

In this connection, the Italians

maintained an Incontestable superiority in the art of accompaniment for a long time. The book of Francesco Gasparini, maestro di cappella in Venice, made progress in the methods of exposition, although it ought not to be regarded as a systematic treatise of this science.

This book, LtArmonlco

In the rule of the octave each note in the diatonic scale was given a preferred figure; this figure would be used only when that particu­ lar scale degree appeared in an unfigured bass.

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80

Pratleo al Olmbalo, appeared for the first time in 1701

In a very

few pages it gives the proper principles for the accompaniment of various bass movements, progressions, and especially for major and minor scales* It is in this last part where Sabbatini's rule of the octave is found again.

This rule, different in many respects from the usage set up in the

French and German schools, offers this pecularityi

the ascending fourth

degree of the minor scale is accompanied by a perfect chord, while the same note in the major mode is accompanied by the six-five chord, Gasparinl*s book, frequently reprinted, remained the vade mecum of all the Italian schools during the entire century, and has been profitably replaced by Fenaroli's treatise, of which I will speak later, on harmonic practice. The systematic works of Tartini, which came to light about 50 years after L'Armonlco Pratico, exercised no influence on the practice of Italian schools.

I will discuss these work3 later.

I have cited Cruger as the author of a method on the accompaniment of a basso continuo, published as a supplement to the second edition of his book Synopsis muslca (Berlin, 163*0*

This method is important because

■^According to Martin Ruhnke ("Gasparinl," in MGG, IV, 1414-17), the first edition of Gasparinl*s treatise was published in 1708 in Venice by Bortolij the sixth and last edition was published in 1802, nearly a century after the initial printing. In Biographle unlverselle. Ill, 414-15, Fetis gives 1683 for the first edition and 1708 for the second edition. If the date 1683 is accepted, Gasparinl's treatise was published when he was only 15 years old, ^*A1though Gasparinl states (p, 68) that the fourth degree in minor, whether approached by step or by leap, must always be accompanied by a minor third, he neither states nor consistently illustrates this scalar degree with a "perfect chord," i.e., a minor chord in root position! occasionally he uses six-three or six-five. See Francesco Gasparinl, The Practical Har­ monist at the Harpsichord, trans. by Frank S. Stillings and ed, by David L. Burrows (New Haven* Yale University Press, 1968\ pp. 29, 69, 74, 75*

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81 for the first time the isolated construction of consonant chords, desig­ nated by the author as triade harmonlque, is foundj the name was retained by his successors*

It does not seem that Cruger understood the formation

1* of natural dissonant chords --at least he says nothing at all about them-although they were known as early as his era.

The other German writers of

the same period who discussed basso continuo at the end of the seventeenth century and the beginning of the eighteenth century, particularly Prinz and Werckmeister, added a few changes to the chordal successions in the examples furnished by Cruger, but did not try, any more than their prede­ cessors, to form a synoptic classification of these harmonic groups. The theoretic science of harmony made progress in the hands of Friedrich Erhardt Niedt , a German musician who was at first a notary in Jena, and then settled in Copenhagen where he died in 1717,

A poor

writer, he often tires the reader with his diffuse style, but it can not be denied that he inspired a salutary impulse to the theory of harmony in the first two parts of his book on this science.

The first, Guide musical,^

contained a treatise on basso continuo. The formulas for the harmonic cadences which are still in use today, the way to realize them, and the theory of passing tones which can be substituted for the struck chords are found there.3* The natural dissonant chords of the seventh and ninth are 1# 'L A "natural dissonant chord" is one in which the seventh and ninth are unprepared, The dominant seventh and dominant ninth chords fall into this classification, ^Musikalische Handleltung, oder der Grflndllcher Unterricher. Yermlttelst welchen eln liobhaber Edlen Music in kurtzer Zelt slch so welt perfectionlren kan , , , (Hamburg1 [b. Schillers], 1710), ^Chapter VIII.

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82

presented there In their true character, namely, as able to be attacked without preparation* but the ninth chapter, with reference to these chords, gives a false ascending resolution of the seventh in these examples. This fault recurs in several places.

It is remarkable that Mattheson, to

whom we owe a second edition of the second part of Niedt*s book, has said nothing at all about this irregularity.

The last chapter of the first

part also attracts attention by the formulas for harmonic modulations,

1*

which no author had indicated previously. In the second part of his Guide musical. Niedt presents many varia­ tions of simple bass movements, with some rules for the ornamentation of the harmony in the upper voices.

Mattheson issued a second enlarged and

improved edition of this second part (Hamburg, 1721), to which he added the dispositions of 60 of the best German organs.

The third part of the Guide

musicals relates to counterpoint. The development given by Niedt to the science of harmony and accompaniment doos not go so far as to search for the normal generation of chords* the idea of such a generation had not occurred to any harmonist before the genius of Rameau conceived it.

Thus, we ought not to expect to

find more progress in this connection in a few books published in Germany after Niedtrs work* but the method of exposition and the natural classifi­ cation of the principal variations of harmony received some considerable improvements in the basso continuo treatises that the choir masters Heinichen and Mattheson revealed soon afterwards.

The first of these

authors published the first essay on his method in Hamburg in 1711 in a

■*-*Chapter XI contains the harmonic progressions for modulation* there are 12 chapters in the first part.

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83 book entitled Instruction fondamentale et nouvellement inventee qulr par une methods certalne et profitable, peut condulre un amateur de muslque a la eonnaissance complete de la basse continue,-1- This work is divided into two parts, each one of which contains five chapters.

In the first

the author arranges all of the true chords into two classes, consonant and dissonant.

The perfect chords— major, minor, and the chord of the

sixth— are the only ones admitted into the first classification by Heinichenj the six-four chord does not appear in the exercises which he gives for the use of consonant harmony, because he does not accept an unprepared fourth.

In the second classification all dissonant harmonies

are listed according to the prolongation of intervals.

In this classifi­

cation of chords, Heinichen confused those of the dominant harmony which exist by themselves, independently of every circumstance of prolongation. In this respect, he is less advanced than his predecessor Niedt.

The third

chapter deals with passing notes and the way to distinguish them from the actual notes, by means of the duration of each genre of tacten,

Heinichen

manifests great sagacity in this matter, and presents some delicate con­ siderations which have been very neglected by modern harmonists.

In the

following chapter we find the application of all the rules which concern chords, figures and passing notes in all the keys.

The last chapter has

some exercises for accompaniment. In the second part of the book we find some rules for the accom­ paniment of an unfigured basso continuo. 2* a very good chapter on the *1

H

Neu erfundene und Grundliche Anwelsung, wle eln Muslc-llebender auff gewlsse vortheilhafftlge Arth kSnne zu vollkommener Eh"lera»n|g des General-Basses L(Hamburg! 17 Schillers, 1?10)J, 284 pp, 2*Heinichen expands upon Sabbatlni's rule of the octave, focusing on the role of chords in a musical context.

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84

accompaniment of a recitative, another chapter on modulation, and some general exercises for accompaniment. Seventeen years after the publication of this book,the author produced a new edition, which seemed more like an entirely new work on the framework of the first.

This new work of Heinichen, an immense

encyclopedia of the art of accompaniment at the beginning of the eighteenth century, which forms a volume of more than a thousand pages, appeared under the title

La bases continue dans sa composition, ou instruction nouvelle

et fondamental. , .

I have just said that this new book is realized

from the same plan as the first, although the material is developed with a great deal more depth;

nevertheless, the ideas of Heinichen had been

modified with respect to the dominant seventh, because he gives some examples of it as a natural dissonant chord (pp. 145-46). Mattheson, whose name dominates music literature in Germany during all of the first half of the eighteenth century, benefited from the work of Niedt and Heinichen for the composition of his booki

Die exemplarische

Organlsten-Probe im Artikel vom General-Basse (Exemplaire de I1examen de 1'organists en ce qui concerns la basse continue), published in Hamburg in 1719.

Just as this title Indicates, the principal part of this work is

composed of figured basses or Partlmenti, accompanied by analyses of the

• W General-Bass in der Composition, oderi Neue und grundllche Anwelsung, wle eln Muslk-Llebender mlt besondern Yorthell, durch die Brlnclpla der Composition . . . (Dresden. 1728). The errors are so numer­ ous in this volume that he had to make 10 pages of errata in small print. ^*George J.Btelow does a his dissertation! "Johann David position ! A Critical Study with (unpublished Hi.D. dissertation,

comparative study of the two editions in Heinichen, Der Generalbass in der Com­ Annotated Translation of Selected Chapters" New York University, 1961), Chapter 2,

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85

various harmonic circumstances which are encountered there*

A theor­

etical introduction precedes these exercises; some rather vague principles of harmony, mixed with calculations of the numerical proportions of inter­ vals, a n found there*

Although approaching Heinichen in the details of

analysis, Mattheson’s hook is absolutely different from that of the former as to plan, because no classification of chords can be seen there*

As in

most of Mattheson*s books, in this one we encounter a multitude of things foreign to the subject, or which have no direct bearing on it.

The second

edition of this book, Grosse General-Bass-Schule, is no more methodical than the first, although more developed. In his Petite ecole de la basse continue (ELeine General-BassSohule),

an absolutely different work from the one which preceded,

Mattheson created a truly methodical treatise of harmony, preceded by the principles of music.

2*

He goes farther than Heinichen in the division of

chords into different classes,3* and examines with much care the harmonic 'Ircumstances of the preparation of resolution of dissonant chords; it can be said, nevertheless, that this book, published 13 years after Rameau’s Tralte de l’harmonle, is not on the level to which this great man had just ^Hamburg:

Kiszner, 1735*

^*The book is divided into four "classes"; (l) lowest, (2) ascending, (3 ) higher, and (4) upper; each class is divided into seven lessons (Aufgaben), and on occasion the lesson is subdivided into smaller units called Abthellungen. 3*MatthesonTs division of chords into classes is based solely upon the frequency of occurrence of each chord. Mattheson orders chords into three groups: (l) common or most harmonious, (2) less common, and (3) un­ common. Hhile the first class professes to deal solely with consonances, the diminished triad, augmented triad, dominant six-five, supertonic sixfive, and augmented sixth chords are included in this classification; the latter two groups of chords are dissonant.

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86

raised the science.

In all his vork Mattheson observes profound silence

on the theory of the generation of chords conceived by the French theorist. Moreover, it is evident by the analysis which he gave of the Tralte de l'harmonle in his Grltlca muslca (II, 7-H)

that he understood neither

this work nor the theory of the inversion of chords discovered by Bameau. Several treatises on accompaniment and on basso continuo were published in France before Bameau had entertained his theory of chords and of fundamental bassf the main ones are those of Saint"Lambertr Boyvinr Couperin and Dandrieu.'1"* All wrote in the same spirit, i.e., without criticism and with the unique end of accompanlmental practice.

In order

to have an idea of the utility which could be drawn out of these works, it suffices to glance at the division of harmony which Boyvin placed in his Tralte fabrege] de 1*Accompagnement (Paris*

Ballard, 1700).

There are,

he said, three kinds of chords, namely, perfect, imperfect, and dissonant. The perfect is composed of a third, fifth, and octavej the imperfect contains the fourth and the sixths1 the dissonant includes the second, seventh, etc. Although ordinary usage demands that dissonance be pre­ ceded by consonanceg one may dispense with this rule now and then and create some which are not precededi this is recognized by good usage and good taste.^* (Fetis* italicsT)

Saint-Lambert, Houveau Traite de 1' accompagnement du clavecin, de l'orgue et de quelques autres instruments (Paris* Ballard, 1707)* Jacques Boyvin, Tralte abrege l~Accompagnement pour l*0rgue et pour le Clavessln (Paris* Ballard, l?00ji Frangois Couperin, Begles pour 11accompagnement (MS, c. 1698). Jean-Frangois Dandrieu, Princlpes de 1*accompagnement du clavecin (Paris* The author, 1719)* ^*Jaeques Boyvin, in Archives des Maitres de l*0rgue des xvig xvllg xviiif Siecles, VI, 75*

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87

Thus

this poor Boyvin knew no other way to

distinguish natural

dissonances

than hy "good usage and good taste"! Such was the state of

the science

when Bameau gave it a real existence*I will expose presently

the creations of

this man of genius

and the results which they produce in

all of Europe* While the science was stationary* the art was enriched hy several types of new effects*

Thus the composers had introduced into the chords

some momentary alterations of natural notes, and hy this means had estab­ lished some ascending and descending tendencies which multiplied the expressive accents* new aspect*

The chords affected hy these alterations assumed a

Moreover, as well as substituting the sixth degree for the

dominant in the dominant seventh chord and its derivatives*

1#

they gave

rise to some new dissonant chords which were attacked without preparation, like those from which they originated.

The substitution in the seventh

chord had produced the chords commonly called the "major dominant ninth" in the major mode, and the "minor ninth" in the minor aode.^* The same substitution in the first derivative of the seventh chord had created the leading-tone seventh chord in the major mode, and the diminished

1* A "derivative" is an inversion* It is "derived" by transposing, for example, the dominant (root) note to an upper octavej the resultant structure, a dominant six-five chord, is the second combination of a dissonant chord* (Fetis, Traite d'harmonle, pp. 40-44.) ^*To obtain a complete triad on the resolution of a dominant seventh in four voices, F£tis propounds that the root of the dominant seventh be doubled and the fifth omitted* The substitution of the sixth degree for the doubled note creates the dominant ninth* "The substitution of the sixth degree for the dominant is always a melodic accent placed, for this reason, in the upper voice* * * (Fetis, Traite d'harmonle* Bk* II, par. 120, pp. 47-48.)

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88

seventh in the minor,1*

Lastly, each one of the seventh chord derivatives

had undergone a double transformation consistent with the mode in which they were to he employed,

The result of the minor mode substitution and

the ascending alteration of the natural notes in the perfect chord and its derivatives had been to create the transition, i.e., the optional modulation which can be resolved into several keys, and leaves the ear irresolute until the completion of the resolution. It is from this new order of facts that the emotion of surprise, so frequently called forth, arose, and in that very way is weakened in modem musicI One must not believe that those who found these novelties were passionately fond of themj they employed them only very guardedly, and their appearance in compositions was so unusual until the era when Mozart took possession of them

that they were considered, as it were,

only as exceptional licences. Such was the situation of the art and of the science when Rameau undertook to join together the elements of harmony in order to give them a systematic foundation.

B o m in Dijon in 1683 of a father who had more

inclination for music than knowledge of its theory and practice, he had 1# In each inversion of a dominant Beventh in which substitution will occur, the root of the chord must be in the soprano. Fetis explains it thusly i "It [[the sixth scale degree[] is a melodic accent in the inversions as in the fundamental chord, and consequently it is always placed in the upper voice, and is found at the distance of a seventh from the leading tone. . . . " (Fetis, par, 121, p, 48.) ^*Having no discernible root and with each note a potential prime, the diminished seventh chord can easily adapt into any key* the listener is dependent wholly upon context for its harmonic role, The German and Italian augmented sixth chords share, to a lesser extent, some of this same ambiguity* only within a context can its identity as an augmented sixth or a major-minor seventh be ascertained. This is the factor to which Fetis is referring here, and upon which he elaborates in Ek» III, oh, iii of his Tralte drharmonie.

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89 no other master for composition} a few clavecin lessons from an organist, a family friend, constituted all of his elementary education.

Later he

visited northern Italy, but it seems that the sojourn which he made to Milan was of such little profit to his education that he was more than 30 years old when he learned the rule of the octave from an obscure musician named Lacroix in Montpellier, on harmony date.

2*

It is from this moment that his meditations

Going to Paris in 1717 but not being able to locate there

suitably, he was obliged to retire to a province, and for four years he ful­ filled the duties of organist at the Cathedral of Clermont at

A u v e r g n e , 3*

These four years were spent perusing the books of Mersenne, Kirchner, Zarllno, some treatises on accompaniment, and drafting his Traite de l'harmonle, which he finished at the age of 29 and published in 1722 in Paris.

1 There is no existing evidence to substantiate Fetis' assertion that Bameau had only a few clavecin lessons from a family friend. A musician and organist himself, Jean-Philippe's father instructed each of his 11 children in music. Of this instruction Hugues Maret (ELogue historlaue de MY Bameau £Dijon* Causse, 1766], p. 43) states, "He taught them music even before they had learned to read} the rewards, suitable to what they desired, were given to those who knew their lessons well, and a lack cf attention or slothfulness was severely punished." 2*When Bameau actually learned the rule of the octave can not be conclusively ascertained} nothing was written of his early years during his lifetime. Sr. Michaela Maria Keane, S.N.J.N., suggests that during his return home from his short sojourn in Italy ". . .he may have met M. de la Croix who was to teach him at the age of 20, the then popular rule of the octave." (The Theoretical Writings of Jean-Phllippe Rameau (^Washington* The Catholic University of America Press, I9&J, p. 10.) 3*Cuthbert Girdlestone (J ean-Phllippe Bameau* His Life and Work fLondon* Cassell and Co., Ltd., 1957J* p. 5) insists that when Bameau arrived in Clermont and how long he stayed is unknown, while Sr. Michaela Maria asserts (p, 14) that he was there for a period of six years, probably 1716-1722.

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90

This hook created little sensation when it appeared! the novelty of the material and the obscure and verbose style of the author made it intelligible to only a small number of readersj it is even permissible to say that people did not exactly understand the importance of it.

This was

the creation of a new science which merited the attention of educated musicians.

But where can the latter be found in a time when education

for artists was lacking, and when all their learning was restricted to the knowledge of the contemporary practices of their art?

If they had

been able to understand that Rameau's work did no less than lay the founda­ tions of a philosophical science of harmony, the idea to them would have seemed so ludicrous that it certainly would have been the object of their taunts.

There was a great distance between this frame of mind and that

which would have been necessary to welcome the Tralte de l'harmonle with the interest which it was worthy of inspiring.

At last I have arrived at

the analysis of the theory, so new then, set forth in this book, Zarlino, Mersenne, and Descartes Introduced Rameau to the cognizance of applying numbers to the science of sounds| his ardent soul was enthused by this science which revealed the possibility of giving a positive founda­ tion to music theory,^-* From that time the regular divisions of the mono­ chord appeared to him to be the point of departure of a harmonic system, and all his attention was turned towards the development of the logical consequences of the facts revealed by these divisions.

In the very

1#In the preface to the Trait! Rameau writes, "Music is a science which should have definite rules. These rules should be drawn from an evident principle, and this principle cannot be known to us without the help of mathematics," (Rameau, Tralte de l'harmonle redulte a ses prlncipes naturels, dlvise en quatre llvres (Parisi Ballard, 1722), preface, n.p,)

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91

■beginning of his hook he established that the Identity of the results of the science of numbers, whether the application be to the divisions of a single string or whether it have as its objective the lengths of strings corresponding to these divisions, the dimensions of the tubes of wind instruments, or the rate of vibrations, makes evident the utility and infallibility of this science in its connections with music.

His

efforts have, then, for a goal to settle that the sound of one string in its entire length, represented by 1, is identical to the ear with the divisions of the same string corresponding to the numbers 2, 4, 8 which produce the octaves of this entire string.

This identity of octaves,

to which he returns later in his other works, notably in one pamphlet of which it is the particular object,seemed to him rightly the foundation for the system of the fundamental bass which he wanted to establish, and of which he will speak later. On the other hand, a statement from Descartes1 Compendium musicae became the criterion for the generation of chords for himj it is stated this way* I can still divide the line A B [[the monochordj into 4, 5 or 6 parts, but not further, because the ear is not able to distinguish any further the differences of pitch without considerable effort.2

•^Extralt d ’une reponse de M, Bameau a M, Euler, sur l'ldentite des octaves, d'ou result des verites d'autant plus curleuses qu'elles n*ont pas encore ete soupoonnees (Parisi Durand, 1743), 41 pp. ^Rursus possum dividers lineam A B in 4or partes vel in 5e vel in 6, nec ulterius fit divisiot quia, scilicet, aurium imbellitas sine labors majores sonorum differentias non posset distinguere, (Compendium musicae [[Trajecti ad Bhenum, I650J, pp. 12-13)«

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92

Now the numbers 1, 2, 3, 4. 5, 6 give the perfect chord, with the intervals doubled and arranged on the monochord in the following way*

c

c

g

c

e

1*

g

c

1__________ 2___________ 2__________ 4_______ 1______6________ 8 octave

double octave

triple octave

After having developed at great length the demonstration of this principle of the existence of the perfect chord in the laws of numbers applied to the divisions of a resonant string, Rameau envisions a theory from which all the other chords are generated by a supposition or superposition of a certain number of major or minor triads,

or derived by

the inversion of these original aggregations of thirds, ^ One difficulty arises, however, in the outcome of the division of the monochord taken as the basis for consonant harmony* the perfect major chord,

it gives only

Rameau understood that it was very grave to

reverse his system if he stuck to this general resultf he dodged it by only drawing upon the proportions of the major third (4*5) and of the minor third (5*6), given by the notes c-e and e-g, in order to form all of its combinations of thirds.

This combination became his point of

departure, He says t

^*With the exception of the number eight which Rameau has added, this arithmetical series of numbers represents the senaxlo of Zarlino, Rameau has omitted the number seven because he is trying to confine his theory to the consonance of the major chord as it is contained in the senario. ^Jean-Philippe Rameau, Traite de l*harmonlet p, 33• 3Ibld,, pp. 34ff.

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93

As a matter of fact, in ordsr to form the perfect chord, one third must he added to the other, and in order to form all of the dissonant chords, three or four thirds must he added one on to another| the difference between these dissonant chords only proves the different position of these thirds. That is the reason w© can Attribute to them all the force of the harmony in reducing it to its first degrees, One may prove it hy adding a proportional fourth to each perfect chord, from which two seventh chords will arisef and hy adding a proportional fifth to one of these seventh chords, from which a ninth chord will arise. The ninth chord will contain the four preceding chords in its con­ struction,1 From this theory Bameau established that there are two perfect chords, one major, the other minor, and that each of these chords generates hy Inversion a chord of the sixth and of the sixth and fourth.2

To the

perfect major chord which he transposes, no one knows why, he adds a minor third and forms the dominant seventh chord (g-h-d-f) which, hy inversion, gives the chords of fausse quinte (minor fifth and sixth), petite sixte (leading-tone sixth), and the triton,^ By the addition^ of a minor third to the perfect minor chord (a-c-e), he forms the minor seventh chord (a-£-e-g) which has for deriva­ tives, hy inversion, the chords of grande sixte (fifth and sixth), petite sixte (third and fourth), and the seconded

1Ibid,, p. 33, 2Ibid,, pp. 34ff. 3Ibid,, p. 37, ^This paragraph as well as some of the following were extracted from my Revue muslcale, XIV, 114. -Rameau, Traite de l'harmonle. p. 39,

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94 The addition of a major third to the perfect major chord provides the "major seventh" chord (c-e-g-b) for Rameau,3A minor third added below the perfect minor chord (g-b-flat-d) produces that of the seventh with a minor fifth, which we designate by the name leadlng-tone seventh (e-g-b-flat-d), and its derivatives by inversion.2 The aggregation of two minor thirds gives Rameau the fausse quinta chord (minor third and minor fifth or perfect diminished chord), and that of two major thirds, the qulnte superflue chord (augmented fifth chord), as well as their derivatives by inversion. Rameau finds the origin of the diminished seventh chord (c-sharpe-g-b-flat) in the addition of a minor third below a fausse qulnte chord (e-g-b-flat). He calls accords par supposition those which were created, accord­ ing to him, by exceeding the compass of an octave by the addition of one or more thirds under any seventh chord.

It is thus that he explains the

origin of "ninth” and "eleventh" chords, which we now consider as the retardation of the octave by the ninth, and the third by a fourth--^* In the ignorance which he had of the technique of prolongation, he demon­ strates a rare sagacity to find a reasonable explanation of the difference of this quarte dissonants, an object of so much embarrassment for former harmonists,^ •^Ibid.. pe 40, 2Ibld,. p, 41, ^*By "we," Fetis is referring to his theory of the origin of ninth and eleventh chords. ^Rameau, Traitg de l'harmonle, p, 73,

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95 If we were to put ourselves into the situation In which Rameau found himself, i.3.f in complete absence of any harmonic system at the time when he was writing, we could not help but admire the powerful mind which created all that he invented, and of which I have just stated a summary, although the system is essentially false*

Fascinated by certain

properties of combinations which the intervals employed in the construction of chords possessed, this genius of a man seized hold of them to form the foundation of his theory.

When he published his Tralte de l'harmonie. he

had still not focused his attention on the phenomenon of the production of harmonics in the resonance of a sonorous body,1* which subsequently made him modify his ideas, and which successively gave rise to the publi­ cation of his Nouveau system de muslque theorique and his other works.^ This was still only the principle of superposition and supposition of thirds which guided him in his system when his first book of harmony appeared.

Now, in order to make the application of this principle to

all chords, he found himself obliged to abandon the whole idea of tonality, because he did not always find the thirds disposed as he wanted them in his system, for each dissonant chord, on the notes where these chords are placed according to the tonal principle. minor seventh with a minor

For example, the chord of the

third,the eternal stumbling-block of all the

If

A "sonorous body" mental tone is accompanied

is theacoustical phenomenon in which the funda­ by itsovertones in a harmonic ratio,

^Nouveau syst&me de musique theorique ou l’on decouvre le principe de toutes les rfegles necessalres k la pratique, pour servlr drintroduction au TraitS d’harmonle (Paris: Ballard, 1726;j generation Harmonique. oil Tralte de musique theorique et pratique (Paris! |Prault j, 1737)f Demon­ stration du principe de 1*harmonique, servant de base a toute l*art musicals theorique et pratique (Paris: Durand, 1750)I Nouvelles reflexions |_de M, fy»»eauj sur la Demonstration du principe de l'harmonie (Paris: Durand, 1752).

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96 false harmonic systems, this chord, I say, which we commonly call the • '’seventh chord on the second degree" because It is formed on the second note of a major scale (d-f-a-c), could not arise from the minor perfect chord of this second note, because he knew very well that in the system of modern tonality this chord does not occur on the note from which it is set into motion.

He was thus obliged to take, for the origin of this

"chord of the minor seventh with a minor third," the perfect minor chord of the sixth degree (a-c~e), so that his seventh chord (a-crl-s) seems to be connected to this last note. By operating in this manner for most of the dissonant chords, Hameau was obliged to consider these chords as isolated occurrences, and to separate all of the rules of succession and tonal resolutions established in previous treatises of accompaniment and composition. Because these rules, consistent with the natural laws of tonality, assign certain positions to the chords, they were incompatible with the doctrine of the generation of chords by the superposition or supposition of thirds. Such, then, was the radical vice of the harmonic system conceived by Rameauj it consisted of destroying the rules of sequency based on aural impression, although it might be qualified as arbitrary, in order to substitute a certain order of generation, fascinating for its regular aspect, but the effect of which was to leave all the harmonic groups isolated and without connection. Too good a musician not to understand that after having rejected the rules of the succession and resolution of chords incompatible with his system he had to supply some new rules which would not be contrary, Rameau conceived his theory of "fundamental bass," the system of which

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97

he set forth in the second article of the eighteenth chapter of his Tralte de l'harmonie. This article was captioned

"The way to compose

a fundamental "bass underneath every kind of music. The fundamental bass in Rameau's system is only a means of verify­ ing the harmonic regularity and not an actual bass; this is why he pointed 2 out that one ought not to stop writing successions of consecutive octaves or fifths,3* The principle rules of this bass are

(l) to form harmony

with the other parts, there can only be perfect chords on the tonic, fourth degree, dominant, and sixth degree; and seventh chords on the dominant and second degree.

During certain successions of the fundamental bass, it was

sometimes necessary to restore cadential action to this bass with a sixfive chord on the fourth degree moving to tonic.

This difficulty led

Rameau to consider this chord on the fourth degree as a perfect chord, to which one will sometimes add the sixth; he gave it the name "added sixth,"

But considering the perfect identity of this chord with that of

the second degree which he designated as the grande sixbe, he also gave the former the name chord of duplication f d*accord de double emplol], and supposed that until it makes its resolution on the perfect chord of the dominant, it is the grande sixte chord and is derived from the minor seventh chord (Example a); that until it is followed by cadential action towards tonic, it is the added sixth and fundamental chord (Example b). ^Rameau, Tralte de 1'harmonle, p, 13^. 2Ibid,. no. 7, p. 135, 3*According to Rameau, the fundamental bass of a chord is the tone corresponding to the string length which, when divided harmonically, will produce all the tones of the chord. Consequently successive fifths and octaves are permissible in the fundamental bass, because they are not occurring between composed voices.

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98

a

b.

fe— * ---- • 1

=4-—

^ — «----- — -a v 1

z±-— : -fl-------

_A _______

Now, It Is evident that this so-called fundamental chord, which one would not know how to compose of superlmposed or supposed thirds, destroys the economy of Rameau's system from top to bottom| but such is the effect of prejudice, that the inventor of the system of fundamental bass deceived himself on this capital shortcoming, and that his partisans did not even perceive it. The other rules for the verification of harmony by the fundamental bass were

(3)^"* In every perfect chord on the tonic or dominant at

least one of the notes which composes the chord should be found in the preceding chord,

(4) The dissonance of a dominant seventh chord also

ought to have been heard in the preceding chord,

(5) In the six-five

chord or the "added sixth," the bass, its third, or its fifth must have been prepared in the preceding chord, but the dissonance formed by the sixth against the fifth is free in its movement,

(6) Every time the

dominant is heard as the fundamental bass it ought to descend a fifth or rise a fourth*

(7) When the fourth degree is in the fundamental bass,it

ought to rise a fifth or descend a fourth, l'*For some unknown reason, (2) is omitted by Fetis,

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99 These rules, given by Rameau for the formation of & hass different from the actual bass of the music and for the verification of good employ­ ment of the chords, could only be established by him in an arbitrary wayi it would have been impossible to state a rational theory founded on the very nature of harmony.

Moreover, they have several essential defects

which all the inventor's sagacity was not able to rectify.

One was the

inadequacy of these rules for a number of circumstances, an inadequacy that has become much more apparent since a great quantity of harmonic combinations unknown in Rameau's time have been introduced into music. But it is not only for their inadequacy that the rules of fundamental bass fall short, it is also in their contradistinction to tonal gravities and to the judgement of the ear in most of the successions.

According to

this doctrine, several of these successions have been rejected, in spite of musical instinct and the laws of tonality.

Thus according to the

fourth rule, the dissonance of a dominant seventh chord or its derivatives must have been prepared by the preceding chord, whereas what distinguishes these natural dissonant chords from dissonances by prolongation is precisely that they could be attacked without preparation.

Following the fifth rule,

the bass, the third, or the fifth ought to have been prepared, while the dissonant sixth is free in its attack.

Now in this chord it is not at all

the sixth which is the dissonance, but the fifth, and the pecularity of this dissonant fifth is precisely of being able to be employed only with preparation, while the bass and the third are free.

Moreover, the specified

succession of this six-five chord to the perfect tonic chord is not good, and although it has been employed in these times by Beethoven and some other musicians of the German school, it is, nevertheless, a harmonic

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100

absurdity since the dissonance has no resolution. The doctrine of the fundamental bass was, in the origin of Rameau's idea, only an accessory* or, if you will, a complement of his harmonic system.

In the sequel when this great musician became enthusiastic for

the phenomenon of the production of the perfect chord in the resonance of a sonorous body of a large dimension,1* he feels somewhat embarrassed about the subject of the perfect minor chord when the experiments had not found the product at all in this phenomenon,

A little self-satisfaction for his

theory of fundamental bass has made him contrive the double emplol of the six-five chord.

A condescension analogous to his new ideas made him

discover^ some quivering or other of aliquot parts^* of the sonorous body which produced to a weaker degree this minor perfect chord which he needed.^* •*-*Rameau appears to have studied the acoustical theories of Mersenne and Sauveur before wilting the sequel. ^Demonstration de •principe de l'harmonle. [jpp. 6kff/]» 3*An "aliquot part" is that part which will measure the whole without a remainder— an exact divisor. Thus k is an aliquot of either 12 or 16, whereas 5 would be an "aliquant part" of 12 or 16, ^*Rameau's explanation is rather confusing. He says, "The minor third then will be of necessity generated from the difference cf the effect between it and the major. "The ear also indicates clearly the operations of the principal generator C in this circumstancef it chooses there, itself, a fundamental sound, which becomes subordinated to it. . . . "In forming the minor third of this new fundamental sound, that one judges must be the sound A, the principle C still gives its major third for a fifth. . . . This new fundamental sound that one can regard in this case as generator of its mode is not a generator except by subordination! it is forced to follow, in all points, the law of the first generator, which cedes to it its place only in this creation, in order to occupy there that which is the most important," (Demi>nstratlon, pp. 70-71.)

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101

As a matter of fact, he would have been able to find many other resonances and quiverings in a sonorous body of fixed shape and dimensions, but I will demonstrate elsewhere that these have no coincidence with the real system of harmony.

However that may be, we noticed from that time on

that Bameau insisted less than he had formerly on his doctrine of the generation of chords by the combination of thirds, while his system of fundamental bass pleased him more each day.

Therefore it was this part

of what he called "his discoveries" which were most successful,

Many

people who did not understand his theory of the generation of chords set forth in the Tralte de l'harmonle were enthused over the fundamental bass, by means of which they believed they learned "composition," thanks to some short formulas. One observation which has escaped all the critics who spoke about the fundamental bass system is that, even in admitting his rules as infallible and conforming to what we call the laws of tonality and our musical consciousness, they would not have been able to take the place of the older practical rules, because the application of the latter gives some immediate results, whereas the fundamental bass was only a means for verifying the faults which had been made, Notwithstanding the radical shortcomings of the various parts of Rameau’s system, it is none the less true that this system could only be the work of a superior man, and that it will always be noteworthy in the According to Krehbiel (James W. Krehbiel, "Harmonic Principles of Jean-Philippe Rameau and his Contemporaries” £unpublished Ph.D. disserta­ tion, Indiana University, 1964], pp. 51-52) the fundamental C is using one of its overtones to produce an interval which is foreign to the original fundamental. Yet Rameau does not explain how A becomes a generator, even a "subordinate" generator. Is the implication an arithmetical derivation of the generator A from E?

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102

history of the art as a creation of a genius.

There isr moreover, in

this system an idea which alone would immortalize its author, might he otherwise not have any titles to famei

I wish to speak of the considera­

tion for the inversion of chords which belongs to him, and which is fruitful in favorable results.

Without it, no system of harmony is

possible! it is a general idea which can be applied to all good theory, and which can be considered the basic foundation of the science.

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103

CHAPTER II

THE RESULTS OF THE CREATION OF A HARMONIC SYSTEM Each era has its tendencies in every matter) we observe that the need to search for a theoretical base for harmonic practice was a pre­ occupation of the eighteenth-century musicians and scholars, and that this general tendency became more pronounced immediately following the publication of Rameau*s Nouveau system de musique theorique in 1726.

The

first who rushed into this research was the illustrious geometrician Euler. If it is true, as Forkel says and as I have repeated on his authority, that a first edition of his Essai d*une nouvelle theorie de la musique^ was published in 1729 in St. Petersburg, he was only 15 years old when this book appeared)

I only know the edition which he put out in 1739*

Even

then he was considered one of Europe's greatest mathematicians, and his admirable intellectual proficiency appeared in several academic memoires and in his Mecanique analytique. Using as a point of departure the principle expressed by Leibnitz, that music is an arithmetic secret of sound relations that man makes without realization, he concluded from this that the most simple ratios are those which ought to please more, because they are more easily understood) he formulated some tables of degrees of

■'•Leonhard Euler, Tentamen novae theoriae ausicae ex csrtissimis harmoniae principlls dilucidae exposltae (St. Petersburg, 1739)• Charles Smith refutes the existence of this alleged edition with the following statement* "Euler had almost completed the Tentamen in 1731. about eight years before publication. This information plus a succinct statement of purpose is contained in a letter (pa) May, 1731 to Daniel Bernoulli, . . . " ("Leonhard Euler, Tentamen npyae theoriae musicae, trans. and commentary"by Charles Samuel Smith [^Unpublished Ph.D. disserta­ tion, Indiana University, i960], p. 8.)

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11* because he wanted to have the greatest

degree of agreeableness possible, he avoided the complication of the kij ratio, which is that of the fourth g-c,

According to this, when we

judge the excessive complication of all these combined ratios and of the separation, the perfect chord, composed of all its intervals and their doublings, ought to be in the first degree of agreeableness! Such is the inevitable result of the agreeableness of harmony founded on the simplicity of numerical ratios. Fascinating at first sight because it seems to have something philosophical, the initial idea of this theory does not support the development of its consequences.

But before

an opinion about the worth of Euler's system can be formulated, it is imperative to consider it in its other applications,

I am going to proceed

to this examination, and I will continue to restrict myself to pointing out that this system is the most complete negation Imaginable of the reality which Rameau, as we saw (Gazette muslcale, no, 40), finally summed up in the production of the harmonic intervals of the perfect chord through the resonance of large sonorous bodies.

Following Rameau, the objective fact

in harmony is identical with the subjective conscience, while according to Euler's doctrine, the aggregations of sounds are only some fortuitous constructions to which we only consciously gave numerical ratios; and the feeling of the harmonic ratio of sounds grows weaker in the same The inclusion of k raises the exponent to 2k, and hence to the sixth degree of agreeableness, as opposed to the fourth. The inclusion of 5 would raise the exponent to 60 and move it into the seventh degree of agreeableness.

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113 proportion that these ratios become complicated and lose their simplicity. It is this prevailing idea, the premise of all his system, which has not permitted him to deal Hith the construction of truly complete chords, according to musical sense, in the chapter of his book devoted precisely to this aim*

His scale of degrees of agreeableness is already

so high for simple chords of two tones admitted by the ear with pleasure, that for him the construction of chords of three and four distinct notes had seemed a source of intolerable confusion for hearing, as well as complexlng for the mind,

I admit that I am not able to explain why this

circumstance did not enlighten him on the lack of his principle if, as Mr. Fuss (his son-in-law and biographer) said, music was one of Euler's principal pastimes| it is true that he added

that when cultivating it,

Euler brought all of his geometrical mind to it.

It is very likely this

mind, which conceived nothing if it was not in the form of computation, and which absorbed the pure feeling of harmony with Euler, put his intelligence at variance with his ear. In spite of the insurmountable difficulties which he encountered for the construction of his chords by subjecting them to his scale of agree­ ableness, he dares to treat the laws of the succession of two chords in the fifth chapter of his book.

The incisiveness, the precision of his

mind had made him see that the succession of harmonies ought to be one of the causes of the pleasure which it procures, as well as the composition of the chordsf and in that he had seen farther than Rameau, whose views had only spanned the combinations of isolated chord.

The order which we follow demands that we investigate now what will be the nature of two tones or two chords which succeed

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114 one another, so that the condition of pleasure he fulfilled. Because, to obtain an agreeable succession of sounds or chords for the ear, it Is not sufficient that each of them please individuallyj moreover, it is necessary that they have a certain relationship for one another, that can best be defined only by calling it an affinity. (FStis* italics.) Nothing is more true nor more thought-provoking than this passage | to see it singly, we would believe that the writer had preceded or sur­ passed his century in the theory of music, and that he had penetrated its secret} but before long disenchantment arrives.

And at first, instead of

beginning to examine the succession in the two sounds of which he spoke and to build a scale of these sounds, i.e., the scale or rather the scales, he enters into the subject with chords} if he speaks of the isolated sounds of which they are composed, it is only to consider the particular numbers to establish the ratio of the succession. (chap. v, par. 2)

After having said

that in order to know with what facility a succession

can be evaluated, it is necessary to express the simple sounds which make a part of it by the numbers which represent them, and to form the smallest multiple} next, to find this multiple in his table of degrees of agree­ ableness, and that degree which will correspond will make known how much ability is necessary to perceive the proposed succession.

He arrives at

this necessary, but monstrous for a musician's intelligence, conclusion in his systemi

"Hoc igitur capite ordo requirit, ut investigemus, cuius modi esse debeant duo soni vel duae consonantiae, quae se invicem sequentes atque successive sonantes suaves sint perceptu* Non enim ad suavitatem successionis sufficit, ut utraqu9 consonantia seorsim sit grata; sed praeterea quandam affectionem mutuam hebere debent, quo etiam ipsa successio aures permulceat, sensuique auditus placeat»" (Euler, ch. v, Cpar. l], p. 77.)

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115 For this, the two chords which compose the succession ought to he considered as if they sound togetheri the exponent of the composite chord which arises from this hypothesis will indicate the degree of agreeahleness to which the succession itself is raised,-*- (Fetis* italics,) Leaving aside the impossibilities that a similar conclusion necessarily includes with the musical ratio, Euler*s embarrassment becomes evident in realizing the inevitable consequences of his system, because he contradicts himself twice on the same page.

As a matter of

fact, it is easy to understand that if the calculation of ratios becomes complicated with respect to a single chord, it is more so in the combina­ tion of the ratios of two chords.

This consideration led the great

geometrician to express himself thus* Just as the simplest chord of three tones is more difficult to perceive than the simplest blsona, so the difficulty of per­ ceiving any chord, be it the simplest of its kind, will be increased with the number of tones which compose it. Nonetheless, the agreeableness of multi-tone chords will not only be equal to, but will surpass that of a simple tone or that of a chord which would be created from only two tones, (Fetis* italics,) A little farther on these words are found*

"However it can not

be denied that the simpler the exponent of a succession of two tones,

-^"Ambae igitur consonantiae successionis tanquam simul sonantes considerari debebunt, huiusque consonantiae compositae exponens declarabit, quam suavis et perceptu facilis sit ipsa [consonantiarunQ successio," (ibid,, ch, v, par, 3r Cp * 77l«) ^"Quemadmodum enim simplicissima consonantia trisona magis est composita, quam simplicissima blsona) ita ex quo pluribus sonis constet consonantia, magis etiam erit composita, etiamsi sit simplicissima in suo genere. Hoc tamen non obstante suavitas non solum eadem, sed etiam maior percipitur ex consonantiis duobus tantem sonis constantibus," (ibid., ch. v, par. 5, [pp. 77-78].)

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116

the easier it is to comprehend the order which governs it,

Now, if the

fundamental principle of all Euler’s system was that the agreement which results from harmony is a corollary of the simplicity of ratios, this last passage is in flagrant contradiction with the preceding, I shall not examine the laws of the succession of chords according to this theory any further} what we have seen will suffice to make clear to what results it is hound to lead,

I still have to expose the forma­

tion, from this same theory, of what Euler calls "genres of music."

The

eighth chapter of Tentamen novae theoriae muslcae, where he has dealt with this subject, is one of the most curious of the work. Euler calls "genre of music" a system of composition where one would make use of only certain pre-determined chords.

Thus the first

genre, according to him,contains no harmony at all other than the octave| but owing to its very great simplicity, it is not employed.

The second

genre of music is that which contains only the tones 1, 3r and their multiples, i.e., the fifth, octave, and their doublings,

"By representing

the lowest tone with 3» the form of the harmony will be 3*4i6 (£-£-&) t where the lower interval is the fourth, and the upper interval is the fifth,"

(Ch. viii, par. 14.)

He adds that this genre is still very

simple and that it has never been employed} but in this he is deceived, because the diaphony of the tenth and eleventh century was nothing else. The third genre is that in which the sound 5» i.e., the major third, is introduced into the harmony, but without preserving the sound 3r which ■^"Interim tamen negari non potest, quo simplicior fuerit successionis duarum consonantiarum exponens, eo facilius etiam ipsam successionem et ordinem, qui in ea inest percipi." (ibid,, ch. v, par, 7, Qp, 78].)

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117 is the fifth, so that the chords there are composed of only the third and the octave* Apropos of this novel harmonic element, Euler makes this remarki Until our day we have admitted into music only the chords whose exponents are composed of the prime factors 2, 3 and 5f la fact, in the formation of chords, musicians have not gone beyond the number 5«^* A century before Euler, Descartes had said in his Compendium muslcaei Bursus possum dividers lineam A 3 in quatuor partes vel in quinque vel in sex, nec ulterius fit divisioi quia, scilicet, aurium imbeclllitas sine labors majores sonorum differentias non posset dlstinquere.2

This principle which is still held by many geometric theoreticians had been refuted by Euler himself about 30 years after the publication of his Tentamen novae theoriae muslcae in his Memoirs entitled

"Conjecture

sur la raison de quelques dissonances generalement regues dans la musique," inserted into the anthology of the Academy of Berlin (1764)*

This

Memoire purported to discover the principles of the rational construction of the dominant seventh chord (g-b-d-f) and of the six-five (f-a-c-d) » After having remarked that the character of the chord &-b-d-f is contained in the ratio of b, expressed by the number 45, with f represented by the number 64, he points out that the latter number undergoes a modification

1*

"In Musica ad hunc usque diem aliae consonantiae non sunt receptae, nisi quarum exponentes constant numeris primis dolis 2, 3 et 5, adeo ut musici ultra quinarlum in formandis consonantiis non processerint," (ibid,, ch» viii, par. 15.) ^Descartes, pp, 12-13.

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118 through the attractive affinity of the intervalj and he adds (par. 13) that the ear substitutes 63 and 64, so that all the numbers of the chord are divisible by 9,

such a way that (par. 14), for the audition of the

tones g-b-d-f, expressed by the numbers 36 *45 *54*64, we think we hear 36*45*54*63 which, reduced to simplest terms, gives 4:5»6s7«^* Euler terminates his MemoIre with this outstanding paragraph! It is commonly contended that only the proportions composed of the three prime factors 2, 3 and 5 are employed in musicf the great Leibnitz has already remarked that in music we still have not learned to count beyond 5» a remark which is unquestionably just for string instruments following harmonic principles. But if my conjecture is founded, we can say that today in composition we count up to 7r and that the ear is already accustomed to iti it is a new genre of music which is being employed and which was unknown to the ancients. In this genre the chord 4*5*6*7 is the most complete harmony, since it includes 2, 3, 5 and 71 hut it is also more complex than the perfect chord in the common genre, which contains only 2, 3 and 5« * » «2* (Fetis* italics.) Euler returned to the same subject in his Memoirs. "Du veritable caract&re de la musique modems" (Academy of Berlin, 1764)r and confirmed his conjecture by some extensive theoretical

^ Euler, believing that the ear is not capable of perceiving complicated proportions (e.g., 36*45*54*64 or 2° x 3^ x 5)» finds his justification for alteration in equal-tempered tuning. "In equal temperament where all of the 12 intervals of an octave are equal, there are no exact consonances except the octavest the fifth there is expressed by the irrational proportion of 1* , which Is somewhat different from that of 3*2. Nevertheless, . . . the ear is not jarred by this irrational proportion, and hearing the interval G*G does not fail to perceive a fifth, or the proportion 3*2." ("Conjecture sur la raison de quelques dissonances generalement recues dans la musique," in Memories de l rAcademle Berlin. XX [[17643, 168.) 2*Ibid., p. 173*

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119 1*

developments.

It is necessary to render justice to this great manf if

the fascinating hut false hypothesis on which he hullt his system led him astray, and if his weak view of harmony caused him to construct some intolerable aggregations of sounds in his chapter on genre of music, the philosophy of this art is none the less indebted to him for the discovery of a truth as irrefragable as new in the paragraphs of the Memolre I have just cited.

He was the first who saw that the character of modem music

resides in the dominant seventh chord and that itB determining ratio lies in the number 7l but until today his words have not been understood. ratio constitutes the genre which I have named transltonique,

This

I will show

in my Philosophle de la muslque that the ordre plurltonlque, which is that of actual music, raised its exponent to the prime factor 11, and that the ultimate limits will be attained in the ordre omnltonlque when the exponent will be raised to 15,

I have already indicated this in some of my articles

on this philosophy, notably in the outline of my work (Gazette musicale, 1840, p, 4) in the fourth paragraph of the second book, I will not begin an examination of the bizarre and inadmissible combinations of the various genres of music exposed by Euler in his Tentamen novae theoriae musicae, because all of these things are actually incompatible with the true art.

But I will remark that before having found

the important truth of which I have just spoken (with reference to the numerical expression of the dominant seventh ratios), it was impossible 1*

To recognize seventh chords, particularly the dominant seventh, as more consonant and thus give them a lower ratio, Euler uses the seventh partial as well as the second, third and fifth to derive scales. Beginning on F and using the formula 2n ,3^»5^,7r Euler derives a 24 tone chromatic scalei he calls the 12 new tones tons strangers.

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120

for him to give the exact formation of the diatonic-chromatic genre , the eighteenth of his table, which he correctly considered as characteristic of modern tonality. Here I have to stop the analysis of this system, the first to be produced after that of Bameau, In spite of the illustrious name of its author, it has remained unrecognized, and I think I can affirm that until today no musician, not even one of those who has made the theory of this art the object of his studies, has either known or understood,-*-* although copies are not scarce.

Only d'Alembert expressed a somewhat favorable

opinion of this system in a note placed among his mathematical works.

It

is probably of this note that Mr, Puss (Euler’s biographer), erudite geometrician and president of the Imperial Academy of Science in St. Petersburg, wished to speak when he put it this way, This principle of the inadequacy (the one of agreeableness procured by the harmony) of the simplicity of numerical ratios has been accused correctly; but since no mathematician is able to sub­ ordinate the relative qualities of the sound to the stringency of his calculations, it is difficult to prove the solidity of it. Granted this principle, we would be obliged to acknowledge that it is impossible to make better use of it, nor to reason with more solidity and penetration. Besides, all the objections against this principle are not liable to do harm to the actual work; they could only be inclined to regard it as a perfect edifice in all its parts, but built upon shifting terrain* while admiring the competency of the architect, they would lament not having been able to construct it on a more solid base. This conclusion is precisely that one at which we arrive after an attentive and intelligent study of Euler’s work.

It Is an important

^•*Fuss said, "Euler's Tentamen novae theorlae muslcae had no great success, as it contained too much geometry for musicians, and too much music for geometers." N. Fuss quoted in Robert E, Moritz, Memora­ bilia Mathematica or The Philomath's Quotation Book (New York* KacMlllian Co., 1914), P. 156.

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121

subject of reflections on the employment of mathematics In the theory of music, although the complete disappointment of a greater intellect which had overcome so many other difficult subjects. Once the thought occurred of the possibility of an exact and complete theory of harmony, we see systems of every kind presented as the realisations of this thought) they followed one another rapidly.

After

those of Bameau and Euler came the one which the celebrated violinist Tartini set forth in a book titled Trattato dl muslca secondo la vera sclenza dellrarmonla (Padua*

[[stamperio del Seminario]r 175**, 175 PP»)»

This book is divided into six chapters, the contents of which follow* (l) "Of harmonic phenomena* circlet

their nature and their usage") (2) "Of the

its nature and its use") (3) "Of the musical system*

consonances,

dissonances, their nature and definition") (4) "Of the diatonic scale, of practical musical genre*

its origin, its usage, and its consequences")

(5) "Of the modes or ancient and modem keys") (6) "Of the intervals and of the modulations of modem music." One of the most remarkable phenomena of the human mind's lack of consistency comes in this book, because there we can see a man acquainted with all the secrets of his art search outside of the structure of this art for the principles which serve as its basis,

and wear himself out

^*Helmholtz believed that the fundamental problem in Euler's theory was his failure to express how the mind " . . . contrives to perceive the numerical ratios of two combined tones," Helmholtz endeavored to rectify this deficiency in his investigations of the physiological processes, and concluded that the human mind " . . . perceives only the physical effect of these ratios, namely the continuous or intermittent sensation of the auditory nerves." (Herman Helmholtz, On the Sensations of Tone, trans. Alexander J. Ellis |_2d ed.) New York* Dover Publications, Inc., 195*0* P. 231.) ^*Fetis is referring to Tartini’s physical-mathematical system as evidenced by the harmonic, arithmetical, and geometric systems.

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122

in barren efforts to draw them from a doubtful form, and from calculations whose mechanism he did not know*

Repelled by the obscurity which prevails

in all the work, the critics have reproached Tartini for not having presented his ideas in a clear enough manner and attributed the lack of clarity which they noticed to his style »•*•* With more attention, they would have seen that the obscurity is in the ideas themselves, and that if ingenious views are not lacking in the system which the author endeavored to coordinate, rigorous liaison does not exist between them; finally, that the results which he draws from them have no soundness at all.^* But do not anticipate, and let us only say that Tartini’s system having exercised no influence on the formation of a system of practical harmony, it will suffice to give a succinct analysis of it here. The first chapter of the Trattato dl musica contains an account of the various phenomena of harmonic his time.

resonance as they had been observed by

He accepts those harmonics forming the octave of the fifth and

the double octave of the third of the principal sound of a deep sonorous body, and does not reject the resonance of the octaves and double octaves 1* Recognizing that , there are many parts of the original very complicated and difficult to comprehend*" Tartini’s follower, Benjamin Stillingfleet, attempted to explain these principles "in a more easy way" (p. iii) in Principles and Power of Harmony (Londons J, and H« Hughs, 1771), ^*In his remarks about Fetis’ critique of Tartini, Rubeli states, "If Fetis maintains that not only Tartini’s manner of presentation, but also his ideas are obscure and unclear, he certainly is not wrong. But it is severely exaggerated when he accuses Tartini of not having correctly understood his own methods of calculation, because the computations which are contained in the Trattato are almost without exception correct," (Giuseppe Tartini, Traktat uber die Musik gemass der wahren Wlssenschaft von der Harmonle, ubersetz und erlautert von Alfred Rubeli |_Dusseldorfi Im Verlag der Gesellschaft zur Forderung der systematischen Musikwissenschaft e. V,, 1966], p, 28,)

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123 of this sound, to the weakest degrees) hut he does not immediately draw any conclusion from this fact.

Moving ahead to the examination of the

natural tones produced hy the horn and the trumpet at the harmonic points of the division of their tubes, he devotes himself to some reasonings and to some hazardous conjectures to explain the conflicting results which the production of harmonics give in the resonance of deep sonorous bodies, and the progression generated hy these tuhes.

Moreover, neither does

he draw any fundamental conclusion from this fact, and passes rapidly to the examination of the harmonics produced hy the monochord called the tromba marina. He deceives himself radically in this examination when he affirms that hy lightly touching the point which cuts this string in half, its harmonic can not be produced.-*-* If he worked out the experiment, some extraordinary circumstance must have prevented the production of this harmonic, the existence of which other tests have proven. reasoning which he sets up on this fact thus fall wrongly.

All the

In addition,

he begins to hint here at his system according to which it would not he the fundamental sound which would generate the harmonics, hut the combi­ nation of the latter, from which the fundamental sound would result.

And

for proof he took the organ stop called foumlture, where on each touch ^ Rubeli disagrees with this statement which Fetis attributes to Tartini) Rubeli says, "Such an assertion, however, appears nowhere in the entire first chapter, although it would fit later theoretical state­ ments, Here Fetis has obviously carelessly read a passage which contains, indeed, a significant misobservation for the system. For Tartini main­ tains (p, 12 of the first chapter), that if a string were touched at the point 2/5 (thus not at the point 1/2), then no higher tone would arise, as is the case with contact on the point 1/5, but only an 'ambiguous hum’ (*um certo tal qual ronzamento'). In reality there appears on all points which delineate a divisible distance through l/5 of the total string length, . , . the fifth overtone which corresponds to the basic sound of the string," (ibid., p, 29,)

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124 a series of pipes tuned In a harmonic manner produce the perfect major chord with various doublings of their intervals because of the number of pipes of which they are composed, and which to the ear produce only the sensation of a single tone.- But this demonstration is deceptive, because the harmonics of the fouralture are not absorbed into the sensation of a single tone any more than adding some large pipes such as the bourdon or the eight foot open flutej so that here, as in all, it is not the har­ monics which generate the deep sound, but the latter which contain the harmonics. Tartini likens to the phenomenon of a plain pipe of the organ that which had been pointed out previously by Serre from Geneva, Romieu from Montpellier.

Oik

He said.

1#

and by

"One has discovered a new harmonic

phenomenon which admirably proves the Bame thing, and indeed a great deal more."3

This phenomenon, designated by the name "third sound," is the

product of an experience in which two treble sounds, forming between them a harmonic interval, produce, when they resonate loudly and with perfect justness, a third low-pitched sound which is the fundamental of these

Jean Adam Serre (1704-?) deals with difference tones in intervals only in Essais sur le principes de l'harmonle, ou l*on traite de la Theorie de l'Harmonle en general. . , . p^risi Prault fils, 1753)* Krehblel discusses Serre's findings on pages 116-129, 01k

" Jean Baptiste Romieu (1732-1766) explains difference tones in an article entitled "Nouvelle d6couverte des sons harmoniques graves, dont la resonnanee est trfes sensible dans les accords des instruments A vent" published in the Memories de la Societe des Sciences de Montpellier, 1752, (Fetis, Biographle unlverselle. VII, 304-305.) ^"Si e poi scoperto un nuoco fenomeno armonico, che prova mirabilmente lo stresso, e moito di plu." Giuseppe Tartini, Trattato dl Musica secondo la vera scienza dell'armenla (Padua: Staaperia del Seminario LG. Manfre], 1754), P» 13.

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125 harmonics, although it may be the product in experience.1* The phenomenon which it concerns is the irrefragable proof for Tartini that harmony reduces itself to unity, represented by the fundamental sound of the whole chord, and in his enthusiasm for this discovery he exclaims, "Therefore, the unity considered in all its ratios, is inseparable from the harmonic system? the harmonic system itself enters into unity as into its principle." It is evident, from the preceding, that the point of Tartini's departure for the formation of a harmonic system is exactly the anti­ thesis of that of Rameau?^* one will see by the following that this point of departure leads to some much less complete and much less satisfying results than those of the theory of the illustrious French musician. The second chapter of Tartini's book has for its aim to profit by the properties of the circle in favor of the harmonic theory, and in particular of the circle considered as inscribed in a square.

The idea

of the application of these properties to music was not new, because Ptolemy devoted himself to the examination of this question in chapters

As a general rule, Tartini is mistaken in the placement of the difference tones* they are an octave too high for first-order difference tones, ^"Dunque dal sistemo armonica e inseparabile la unita considerata in qualunque rispetto, ansi il sistema armonico si rlsolve nella unita, come in suo principio," (Tartini, Trattato, p. 13.) 3*Shirlaw denies this, asserting that through mathematical and scientific principles Tartini attempts to demonstrate the correctness of Rameau's theories. (Matthew Shirlaw, The Theory of Harmony ^DaKal'b, Illinois* Dr, Birchard Coar, 1955]r P« 293.)

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126 two and nine of the third hook of his Harmonicsi^~ the same subject has been discussed in great depth by Othon Gibel in his Introductlo muslcae theoritlcae dldacticae (pp. 125ff),

The obscurity is more profound in

this chapter than in all the rest of Tartini*s book, and one can see in the rash propositions, uniquely founded on some arbitrary ratios, that the author himself did not understand, and that he had so little con­ fidence in the results which his pursuits had produced, that he ended by asserting that these speculations are not necessary for the comprehension of his system.

Here are his words:

"Pero mi son dilatato, e

ho divagato dl molto in questo secondo Capitolo per cose non affatto necessarie al sisterna Musicale" (p. 47),

Thus it would be fruitless to

follow him in his divagations on this subjectf yet it is perhaps not fruitless to give an example, taken from this same chapter, of the lack of soundness of the celebrated artist's reasonings! one will find it in the following paragraph. Tartini recognizes that he has demonstrated by algebra and by the most detailed method of arithmetic that the three terms,

1, 2, x,

•^Pages 229, 252 and following of the Wallis edition (Oxford, 1682), or Vol. Ill, pages 129 aad 141 of the mathematical works of the latter. ^Rubeli takes exception to this paragraph: ", , . Fetis* deroga­ tory criticism about the second chapter of the Trattato is unjustified. Although Tartini often makes it difficult for the reader, one must still grant him that he arranges the content clearly. The added mathematical examples at the end of the second chapter— and this intention is expressed clearly in the text many times— were to persuade the mathematicians of the correctness of his new harmonic way of thinking. Undoubtedly this manner of thinking is absurd in itself, but it is a gross misunderstanding if one believes he himself doubted his proofs! It is wholly misleading to believe he himself didn't quite trust these proofs, and that for this reason, he explained, that they £the proofs] were not absolutely necessary for the comprehension of his system," (Hubeli, p. 29.)

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127 being supposed to form a harmonic proportion, x designates an infinite size by relation to the first two terms.

He adds that the circle gives

the proof, because, he says, the radius represents the first term, 1 , the diameter the second term, 2, and the circumference expresses the third, x.

In the enthusiasm which this discovery prompts, he exclaims,

"Oh, what will the results of a similar conspectus be, if it is true!""1" Farther on, however, he is obliged to recognize that it would be absurd to pretend that the circumference of a circle is, in this circumstance, a size equal to x, namely, an infinite size,^

To summarize, the long

garrulousness of this chapter, which fills up 28 pages in quarto size and which d'Alembert himself found to be an impenetrable obscurity, has no other aim than to establish the similarity and relationship of harmonic unity with the unity of the plan of a circle in which all the arcs and their angles converge to its formation, as the harmonics tend to resolve into the fundamental tone.

He does not doubt that there would not be

the demonstration of his proposition, if the problem of the quadrature of the circle had been resolved.

In supposing, for an instance, the

reality of these relations, one does not understand why Tartini had the idea of rushing into this research without possessing the most elementary elements of analysis,3* ^"Oh quali, e quante consequenze da tal vista, s'e veral" (Tartini, P. 27,) ^Moreover, Serre has proven that this is not at all the circum­ ference of the circle, but the hyperbole, considered between its asymp­ totes, which contains exactly the conditions demanded by Tartini,

J Some of Tartini*s arithmetric premises are discussed and analyzed by Alejandro B, Planchart in "Theories of Giuseppe Tartini," JMT, IV (April, i960), 4i-^7» and. in Serre, Observations sur les principes de l'harmonle, oceaslonnees par quelques Ecrlts modemes sur le sujet, , . , (Geneva t H, A. Gosse and J, Gosse, 1763), part 3,

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128

The results of Tartinirs musical unity is precisely contrary to that of the phenomenon of the sonorous body adopted by Rameau as the basis of his system, because the first ^Tartini] departs from the har­ monics to go back to the low tone, while the latter {^Rameau} follows an inverted course.

T'fc

Whence it follows that Tartini*s system lacks some

criterion for the generation of chords, and that it can not match the beautiful theory of inversion discovered by Rameau.

This consideration

alone demonstrates the superiority of the French musician's work, with respect to the practical didactic and the barrenness of the principle of unity, so highly spoken of by the Paduan violinist.

This radical vice

was noticed neither by J. J, Rousseau in the erroneous analysis which he made of Tartini's theory in the article "Systems" in his Dictionnalre de Muslque, nor by d'Alembert in his article "Fondamental" of the Encyclopedle.

2* who wrote a special report on this subject, nor even by Suremain-Myssery and who had grasped the question much better than d’Alembert and Rousseau. The point of junction fails so completely in Tartini between his principle of unity and the facts of the practice of the art, that having come, in the third chapter of his book, to the musical deductions of his speculations, he further finds only arbitrary rules of which the first are insignificant, and of which the others are contrary to the known rules of the art of 1* While Rameau emphasizes the multiplicity of sound with the sonorous body, Tartini " . . . considers multiplicity a function of unity and regards the division of unity into multiplicity and the resolution of multiplicity into unity as parts of a complete cycle," (Planchart, p. 36.) Thus, Tartini and Rameau are not, as Fetis would lead one to believe, in direct opposition to one another.

2*

Antoine Suremain de Missery, Theorie acoustico-muslcale, ou De la doctrine des sons rapportee aux principes de leur combinalson (Pariss Didot, 1793).

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129 writing; since he can not form a complete table of the recognized chords In harmony, and In those which he does cite

he falls Into some gross

errors with respect to their constituent intervals. Tartini has nine practical rules.

With the first he recommends

that the tones of chords be arranged so as to form, as much as possible, a harmonic proportion, i.e., that they be arranged in this orderi

1 , 1/2,

1/3, 1/5 . This can be good for the complete chord in which the octave can be effectively placed near the bass, then the fifth and finally the third in the top parti but this is not always practical in the others, and besides, it would be in manifest opposition to the considerations of succession, considerations on which the whole art of writing rests.

1*

The second rule of Tartini is that in a perfect chord one ought not to redouble the principal tone at the double octave in following this progression:

1 , 1/2 , l/3, l/4, 1/5, etc., i.e., in composing the chord

£•-£-£-0-67 etc. The motive of this bizarre rule, absolutely contrary tj practice and which would make all harmonic music Impossible, is explained only with much effort and in an almost unintelligible manner by Tartini) but a patient study of his obscure phrases discloses that the rule intends to avoid the fourth which is found between the two terms l/3, 1/bt i.e., between g and £, and constitutes the proportion kij which, according to Tartini, is a principle of dissonance.

2*

This false theory has classified

^•*Fetis has misconstrued the intent of this rule which is to preserve the spacing of the harmonic proportion and not the ordering. ^*This is a complete distortion of the rule set forth by Tartini, which he expresses thuslyi "Tones that are in proportion as l:l/2:l/^ ought not to be employed as harmony, although they are contained in the

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130 the fourth among the dissonances, as if the intervals could not he generated hy the inversion of intervals of the same nature, and as if the fourth was net, in this connection, the product of the fifth which is a consonance.

‘ This theory, I say, has no other cause than the

appearance of dissonance where the fourth occurs, when through the result of prolongation, it strikes the fifth at the second} hut in this case, it is not as a fourth that the note which forms the interval is dissonance, hut as a second, the fundamental principle which has heen recognized hy none of the authors of harmonic systems. Having not perceived at all, or rather not heing ahle to accept 1* the theory of inversion since his principle of unity makes him follow a contrary route, and yet understanding the necessity of harmonies derived from the fundamental chords, Tartini could not present a regular system of the generation of chords} hut he compen­ sated with his third rule which permits all the tones which are part of the chord to move in an arbitrary way to the low-pitched or to the

harmonic sextuple 1, 1/2, l/3» l/4» l/5» 1/6 . . . . The tones 1, 1/2, l/4 contain merely the possibility of a harmonic chord, hut they still form no defineable harmony, as for example the chord 1 , 1/3, l/4 fC-£ Cl, or 1/2, 1/5, 1/6 fc-e'-gl, or 1/3, 1/4, 1/5 fg-c'-e'J, and therefore they also produce no satisfactory chord. Moreover, 1, l/2, l/4 is a geometric progression, and we will see later that the geometric progressions are at the hase of the dissonant chords." (Tartini, Traktat, trans. Rubeli, p. I67.) While the geometric progression (A geometric progression is a series of numbers which progress hy multiplying each preceding term hy a fixed number called the "common ratio." The common ratio of 2 , 8, 32, 128 is 4} of 2 , 6, 18, 54 is 3*) 1» l/2» l A is not dissonant, the geometric progression li3*9 fc-g-d] is dissonant, 1* To the contrary} while Tartini recognizes the theory of inversion, he employes the word "position" instead, (Planchart, p. $1.)

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131 high-pitched.

But fruitlessly Tartini took a great deal of trouble to

shroud this rule with the appearance of obscurity, which hs knows how to put everywhere j it is no less true that after having taken a direction opposed to that of inversion for the construction of his system of unity, he was obliged to come back into the veritable order of generation, without which there would be no possibility of harmonic variety. But here is a much more striking contradiction.^* After having forbidden the doubling of the octave (1/^) in his second rule because of the fourth which it forms with the third term of the harmonic progression (1/3), Tartini establishes in the fourth rule or musical laws The intervals of the octave, fifth, fourth, major third and minor third, as essential parts of the system or of harmonic resonance which forms the most complete and most perfect consonance, are all consonances, because they are all of the same nature or of their integral unity, which is the sextuple harmonic progression. 3 Thus, here the fourth, which was dissonance, becomes the consonance it actually is.

But observe that the major sixth, of which he does not

speak, is also contained in the harmonic progression 1, 1/2, 1/3, l/^» 1* The third rule has nothing to do with voice-leading. Tartini says, "But it would contradict the mathematical and physical facts of my system, if one were to add notes which can not be brought to unity with a note of the sextuple through transposition by one or more octaves. . . , Rule threes Hie tones of the harmonic sextuple may be transposed by one or more octaves." (Tartini, Traktat, trans. Rubeli, p. 168.) ^*Fetis' unfounded interpretation of Tartinifs second rule led to this seemingly "striking contradiction," ^"Che gl* intervalli di ottava, quinta, quarta, terza magglore, e terza minore, come parti integrant! del sestuplo armonico sistema, ch'e la perfettissima eonsonanza integrals, sono tutti consonant!, perch! sono della natura del suo tutto, o sia della sua unita integrale, ch'e la sestupla armonica." (Tartini, Trattato dl Musica, p. 65.)

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132 l/5, l/6, since it is found between gy the third term of this progression and er the fifth term,

As for the minor sixth, it could not be found in

the six terms of his progression.

Notice further, that the sixth term

would be useless (since it is only the doubling at the octave of the third term), if it did not aid in the production of the minor third* now, since Tartini understood the necessity of this doubling, we do not see 1* why he stopped at this sixth term. I will say nothing of the two following rules, because

they are of

no practical use* but 1 notice in the seventh a fact borrowed from Bameau for the formation of dissonant chords, because this rule is stated thusi ", . . it is not possible to have a dissonant chord which is not based on a consonant c h o r d , I t is, as one sees, the generation of dissonances by the addition of thirds, an abstraction based upon determinations of tonality.

One can see what I have said of the drawbacks of this system

in the examination of Bameau*s theory. By his eighth rule Tartini wishes ", , , that the dissonance be prepared by a consonant note on the same degree, and that it be resolved by descending a tone or a semitone,"^* It is apparent with this very rule that he did not understand that there are some natural dissonances which result from the tonal relationships* others arise from the artful ^•*Tartini undoubtedly stopped at the sixth term because these are the terms contained in the resonance of sonorous bodies. Like Bameau, Tartini is confining his theory to the consonance of the major triad as it is expressed in the senarlo, 2**, , , che non si dAf ne pub darsi posi?lone alcuna dissonate, se non fondata sopra la posizione consonate," (Tartini, p, 77*) 3*", , . che la dissonanza sia apparecchiata da nota consonate unisona, e sia zisoluta in nota consonanta, che a ragguaglio della dissonanza discenda per tuono, o semituono," (ibid., p» 84.)

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133 device of prolongation, and these are the ones which have to he prepared* This rule and the preceding one suffice to destroy from its foundation a whole harmonic system* I have nothing to say about the ninth rule which is found in the following chapter, and which is of little Interest because it concerns only the discordances or notes which never create harmony.

To touch on

such a subject, I would be obliged to say that there are no disoordances other than the tones which lack exactness, and that every interval, what­ ever it may be, is harmonicj but this subject, which contains a complete order of new considerations about harmony, would not be proper here, and would involve me tee such w Quo will find it dealt with theoretically in my Philosophle de la muslque

and practically in my Traite complet de

l'harmonle* The analysis which I have just given of Tartini's system seems to me to show the radical vices of its conception, its inadequacy as a practical method, and its inferiority with respect to a system of funda­ mental bass.

It is often said that its obscurity gave rise to successj

I believe that the opposite is more exact, because when one does not understand, one assumes profundity*

If he had been more intelligible,

the defects would have been more easily perceived* I am obliged to go back a few years prior to the publication of Tartinirs system for another theory which, having not been noticed when it appeared, was reproduced later in various forms*

Levons, maltre de

muslque of the Bordeau Cathedral, was the first who made it known in a ^•*It is noteworthy that Fetis, with his obsession for the prepara­ tion and resolution of dissonance, did not comment about Tartini's reso­ lution of the augmented twelfth down rather than up* (Ibid*, p* 82.)

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134 ■book entitled composition,

Abrege des regies de l'harmonle pour apprendre la

avec un nouveau project sur un systfeme de muslque sans

temperament, nl cordes mobiles (Bordeaux*

Chapters, 1743, 92 pp»)«

In

the first part of this work Levens proves that he was a good musician and that he wrote more correctly than most of the authors of musical treatises,

The first part concerns the practice of harmony, such as it

was known in France at this time, and according to Rameau's doctrine, which the author does not always understand and sometimes contradicts. The second part of the book offers more of interest with the plan for a new system of which Levens turns out to be the inventor, as he himself says, because first he substitutes the arithmetical progression for the harmonic progression employed until his time for the generation of intervals,lie noticed that the harmonic progression can not generate a complete scale, and thi? fourth note was not necessarily its product because, he said, none of the numbers of this progression could find some other which would be in the proportion of 4*3, which is that of the fourth, with it.

This consideration leads him to propose to turn to the

arithmetical progression extended jointly with the harmonic progression to the tenth term, the latter one rising, the other descending.

From this

progression he divides two strings of which the first gives him an ascend­ ing series of tones whose intervals are those of the natural tones of the horn and of the trumpet, 1#An arithmetical progression is a series of numbers in which each term, after the first, is obtained by adding a fixed number called the "common difference" to each preceding term. In the series 1, 3, 5, 7 the common difference is 2, and in the series l/2, 2, 7/2, 5 the common difference Is 3/2, A harmonic progression is a series of fractions in which the numerator is common and the denominators create an arithmetical progression. For example, 1, l/2, 1/3, l/4, l/5l or 60, 30, 20, 15, 12 (60, 60/2, 60/3, 60/4, 60/5),

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135 Example of the ascending progression 0, c, 1

£,

1/2

1/3

c’, e ’f g', b-flat*,

c” , d* *,

e»'

1/4 1/5

1/8

1/10

1/6 1/7

1/9

Proceeding in the reverse for the second string hy means of the arithmetical progression, he finds a descending series which gives him the fourth degree and the sixth degree lowered hy a half-tone* Example of the descending progression c"\

c” , f',

1

2

3

o',

4

a-flat, 5

f,

d,

c,

6

?

8

B-flat, 9

A-flat 10

In this system Levens found three different £whole] tones, namely, the "major tone" in the proportion 7*81 the "perfect tone," in that of 8*9i finally the "minor tone," as 9*10*

Prom the experiment which he

had done, he said there resulted in this diversity of tones a very acceptable variety*

To complete the chromatic scale, he had only to

divide the major tone into two unequal tones in the proportions of I4il5 and I51I6, the perfect tone intotwo semitones of which the proportions are 16*17 and17*18* finally, the minor tone into two minor semitones, as

18*19 and 19I20*1* The main defect of this system, a defect which crumbles it at its foundation, is, on one hand, that it does not correspond to the formation -*-*To derive each of the new tones it was imperative to find the harmonic mean of each proportion, which is the average obtained by doubling (the new tones will occur an octave higher) the ratio of each proportion. This harmonic mean becomes the "mean proportional" in the expression a*b=b*c, In 2(7*8), the harmonic mean Is 151 therefore the expression aib=bsc is 14 *15=15*16* In 2(8*9), the harmonic mean is 17, and the expression is 16*17=17*18, Consequently, Levens ends up with six differ­ ent semitones, ranging in size from 119 cents (l4*15) to 89 cents (19*20),

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136 of any tonality, and,on the other handr that with regard to their pro­ portions, the intervals do not coincide in the various octaves, and consequently false sensations stimulate the ear.

For example, at the

two extremes of the scale^* one finds the distance from c to d represented by a major tone on one side, and by a perfect tone on the other side.

But these difficulties do not stop Levens, and do not

preclude him from building, with the harmonic progression, a minor seventh chord (c-e-g-b-flat) on tonic, although the note which forms the seventh may not be of the keyj a six-five chord (£”e-g-a) on the same note, although this chord never occurs theref a dominant seventh chord (g-b-d-f), forgetting that this last note does not exist in the first 10 terms of the harmonic progression, and that what he substitutes is not the true fourth degree of the keyj the six-five chord (f-a-£~d), although the first two notes of this chord are also missing in the harmonic progression of the first 10 terms? finally, the dominant seventh chord from the harmonic progression, although the third of this dominant may be formed with a note lower than the true leading tone.

2*

Nevertheless, it is the essential feature of this system that was taken up much later by Balli&re, geometrician and member of the Academy of Science of Houen, and even much later by Jamard, Canon of 1#In this case the two extremes of the scale are the upper portion of the ascending harmonic progression and the lower portion of the descending arithmetical progression, 2*In the harmonic progression the minor second from b-c (15*16 or 112 cents) is larger than the minor second 17*18 (c-sharp-dT which, at 99 cents, is an approximation to an equal-temperament semitone of 100 cents. The solution to this disparity of just intonation was propounded by Andreas tferckmeister who, in his Husikallsche Temperatur (1691), formulated the principle of a twelve-tone scale in which all the half-steps were equal.

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137 Sainte-Genevieve. If we compare Levens' "book with the one which Ballifcre published under the title

Theorle de la muslque (Parisi Didot, 176*0,

we recognise at first glance that the latter was a much less competent musicisn, but a better educated mathematician*

After having established

in nearly the same way as his predecessor the necessity to extend the harmonic progression beyond six in order to complete the scale, he arrives at Bameau's objection against the natural sounds of the horn and of the trumpet*

"The tones l/7, l/ll and l/l3» being not at all harmonics of 1

1* 2 or 3, are always false in these instruments*-" Bameau, like all those who have expressed the same opinion, had not seen that the ratio of the number seven was precisely the one which could give the fourth note of the key its attractive character with the leading tone, as Euler saw so well*

(See the above mentioned.)

Balllere did not see so far; he was

content to make this weak reply [to Rameau's statement]]* If by the word "false" one means that they deviate from the principles which musicians have established, flnef but if one means to say that they deviate from the natural laws, the wnrd "always" fails to apply to the proposition. How can I "believe, in fact, that a sound which nature "always" presents is not that which it ought to present? One Is more justified In believing that the principles of musicians lack some exactitude. 3* In place of "principles of musicians," he ought to have said "principles of geometricians," because musicians have felt precisely by instinct the necessity of the number seven, not for the seventh degree Bameau undoubtedly means that these are out-of-tune to just intonation, because he omits the tone l/9*

2 Bameau, Generation Harmonique, p. 62. 3* Charles Louis Denis Balllere de Lalsement, Theorie de la muslque (Paris* Hidot, 176*0, P81* 7* P» 5*

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138

but for the fourth j Euler saw quite well that for that [^reason] he had to change the generating note from the tonic to the dominant. Balli&re, like Levens, understood the necessity for the descending progression which he calls sous-doubles t he does not create it in the same way.

In order to arrive at the same result while avoiding the

monstrous defects of Levens' inverted progression, he employs a geometric progression which furnishes him for each note proportional lengths of strings, and which, on the whole, is identical to the properties of the sections of the circle analyzed by Gibel in his Introductlo muslcae theoreticae didactlcae. By the artificial introduction of the inter­ mediary sounds to the products of the inverted progressions, Balllere forms the two following ascending and descending scales i c,

d, e,

f , g,

a,

b-flat, b-natural. c.

c

b, b-flat,

a, g,

f,

e,

d,

c.

That settled, Balllere searches for the principle of chords in the combinations of numbers, and after having established that the sounds 1, 3, 5 (which form the perfect chord) are the principle of harmony, he does not see any difficulty, Mthe impression of 1 being given," with this chord being followed by those ofthe same form*

7, 9» H (b-flat-d-f)

or 9, 11, 13 (d-f-a). He does not concern himself for even a minute with the dreadful succession of these tones, absolutely strange to one another, because he is satisfied that there was a geometric progression in the formation of these chords.

1*

And so one may not believe that I

^*The progression which creates these chords is arithmetical, not geometric.

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139 attribute a meaning to his words which they do not have, and that Balliere may consider these chords in isolation and not in succession, here it is as he expresses its The. perfect chord 1, 3r 5 is therefore the principle one of harmony, and every piece of music begins with this expressed or implied chord* The impression of 1 being given, one can follow this chord with other notes of the progression, such as 7 9 11 3 9 15 9 11 13 b-flat - d - f f g - d - b s d - f - a j and this is what one calls the succession of chordsT^ (F^tis'-italics*) After this specimen of BalliSre's harmonic organization and the basis of his system, I do not believe I have to give an analysis when it is easy to perceive the results.^* Becherches sur la Theorle de la musique of Jamard would not occupy me further, if I did not find there a hint of the logic which is lacking in Levens* and Balliere*s books. Jamard, who possesses a veritable philosophical method, retreats before none of the results of the principle of the arithmetical progression, and carrying them to the end, he arrives at the destruction of the system of fundamental bass, which was the object of Levens* adoration. On this subject he says, Mr* Levens only pushed his division as far as e l/lO; no doubt he was frightened of f l/llf this is what I do not under­ stand at all. One continually repeats that the ear is the great judge in music* Now, what could concern me more than to support this proposition, since I do not know a single experiment done on these sounds which does not seem to be amenable to my •^Ibld.* Ijpar, 73,] P. 36* Krehbiel (pp* 179-197) believes that Balliere*s theory cf har­ mony, demonstrably inconsistent in logic, is not only a diatribe against the musical practice of his era, but also against a theory of modulation, since the scale structure as Balliere states it is nonmodulatingt the principal note (C) determines the scale structure of all the tonalities,

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140

principles, and which does not serve to confirm them? This tone f l/ll certainly is not at all offending on the hunting horn when it is not accompanied hy other instruments which play f 3/32* Why then, should it he called more falBe than the latter? If these two f's sound together, the ear is torn, I agree, hut does it follow from that that one of the two is false? No, unquestionably, hecause if these two sounds heard apart elsewhere create a good effect f it simply follows that they are not at all intended to he heard togetherj it is this, I believe, on which everyone agrees* Let us return to Hr, Levens, and admit therefore that so skillful a musician, who was occupied with our ordinary practice, and who regarded the rules of the chords and the system of fundamental hass as the foundation of all harmony, or rather of all music; admit, I say, that he went to a great deal of trouble to reject f 3/32, subdominant of the mode, from his system in order to admit f 3/33 in its place, since hy rejecting f 3/32 he had to renounce absolutely the whole system of fundamental bass.3We think we are dreaming in reading such things, and we can not refrain from deploring the blindness of this fad for a system which, since that of Rameau, possessed the mind of a multitude of scholars and men of letters, or musicians jealous of sharing the glory of the author of fundamental hass.

At first we only intended to explain

what was the art which existed, hut since the philosophical school of this time was preoccupied with the sole thought of searching for the origin of ideas in lieu of first sticking to doing a severe analysis and rigorous classification of Intellectual faculties, we stuck to the search for the origin of the scale in lieu of ascertaining as a fact that it was next necessary to analyze the properties in order to deduce finally the systematic results.

What was happening in this research,

too delicate and too difficult for the scope of those who had given themselves to it?

It is that, encountering some insurmountable diffi­

culties in the point of view where they stood, they ended up hy refusing l£canonJ Jamard, Recherches sur la Theorie de la musique (Paris 1 Jombert, 1769), p p * 34-36, L n *J»

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141

to explainr in order to establish something more or less different, which they did not fail to represent as real music, if only by means of some experiment or calculation they could give an appearance of regularity to the system.

The various viewpoints under which the construction of a

scale from the arithmetical progression is produced in the books of Levens, Balliere and Jamard are some of the most striking examples of the facility which certain writers have to take appearance for reality, and of the degree of absurdity to which a false system can come, pushed to its ultimate consequences,

Undoubtedly the sounds 1/11 and 1/13

enter into the combinations of music today, but with neither the form nor the name which Jamard gives them,

I do not need to explain the

harmonic results of the letter's system} we see at first glance where they would lead*

Moreover, I am net finished with the principle of this

system; we shall see it again treated by a musician as wise a harmonist as a mathematician, and yet who strayed into a false path* In the midst of systems of music and harmony which followed each other and clashed since the publication of Rameau's, a methodical harmony book, freed from every consideration of number and physical phenomena, appeared} this book is the Traite des accords, et de leur succession, selon le systems de la basse fondamentale by Abbe Roussier* divided into three parts, two parts,

1*

It is

I have nothing at all to say about the first

because they only contain a classification and analysis of

chords following Rameau's principles} I will at least remark that

1*

For a succinct and concise discussion of the first two parts of this treatise, see Mitchell, pp, 87-90} for a detailed discussion see Richard Dale Osborne, "The Theoretical Writings of Abbe Pierre-Joseph Roussier" (unpublished Ph.D. dissertation, Ohio State University, 1966),

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142 although he had little skill in the art of writing and his early educa­ tion as a musician had been neglected,. Roussier shows much more sense of method than the Inventor of the systemr and he was the first in France who supported his [^Rameau'sj views on the very important consideration of harmonic succession*

But we are astonished by the third part* if we

consider the time of its publication* because Abbe Roussier proposes Introducing into music a certain number of chords then unknown.

Here

is what he saysi Those who absolutely wish to confine themselves to the circle of their knowledge will probably not like my daring to propose some chords which have never been heard oft others will reject them as too harsh* dissonant. As for the first* . . . by expressly declaring here that this third part is not intended for them* they have nothing more to say to me. As for the others* I ask them to be willing to examine attentively the thing before making a decision,^ (Fetis* italics.) Truly there was astonishment that* guided by analogy and musical sentiment* which was weak then, Roussier foresaw the possibility of making good use of certain harmonies which only Mozart's genius and a small number of his contemporaries and their successors had known how to bring into play.

Thus it is that the "augmented sixth*" or as was said then*

superflue* is controlled by the law of the inversion of a chord of a diminished third and just fifth* and that of the minor third and minor C&nter Birkner claims that it was Rameau's Traite de l'harmonle which provided Roussier with an introduction to the theoretical facts of musici furthermore* the Traite de l'harmonle (discovered at age 25) was the agent which stimulated his future endeavors into the new-found science. ("Pierre-Joseph Roussier*" MGG* XI* 1018-19.) ^Pierre-Joseph Roussier]* Traite des accords* et de leur succession* selon le systeme de la basse-fondamentale (Parisi Duchesne* 1764)* p. 158, n. 59.

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sixth with the major fourth (tritone)f thus it is, moreover, that hy moving from the chord of the "diminished third and just fifth" to that of the dominant seventhr he conceives the possibility of altering its third as in the perfect chord.

If he had been content with the altera­

tions of the natural intervals of the chords, either original or modified by prolongation or substitution, he would have rendered the greatest service to the advancement of the art and of the science| we would be forming the highest opinion of his instinct, his taste, and his experience. But it is not at all so, because the barbarity of his ear made him imagine some intolerable harmonies in which the feeling of any tonality is destroyed, for example, a chord which he calls the "diminished eleventh" (g-sharp-d-sharp-f-a-e), a chord of the "augmented fifth with a fourth and minor seventh" (g-d-sharp-f-a-c), a chord of the "augmented seventh with a minor sixth and minor ninth" (e-d-sharp-f-a-c), etc.^* Moreover, scarcely able to discern the true dissonances, he makes some very poor applications of a good rule which he had found by reasoning, and which he expresses thus:

"All major dissonance ought to rise one

degree} all minor dissonance ought to descend one degree" (p. 42). The first part of this rule would be incontestable if Roussier had applied it to dissonances formed by the leading tone or by ascending alterations} but the double empioi of Rameau led him so far astray that he finds no application of his rule except for the six-five chord on the fourth degree (f-a-c-u), because he takes the sixth for the dissonance, while analysis shows that in two notes which clash in

^■*Each of these "new" chords was created by "supposing" a root below the German seventh chord.

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144

secondsr it Is the lower which is the dissonance.

1*

In spite of the extensive faults which I have just pointed out, the Traite des accords and the complement of this work which Boussier pub­ lished under the title L*Karmonle pratique, ou Bxeaples pour le Traite des accords (Barisi

1775) could have rendered eminent service in Prance

to the theory of harmony by calling attention to the consideration of the succession of chords, which Rameau’s system had forgotten, if Roussier himself had not lost sight of his practical vrorks by a return to a theory of numbers applied to music, of which he gave the first indication in the notes of his Observations sur dlfferens points d’harmonle,^ then in his Lettre a 1*auteur de Journal des beau-arts et des sciences, touchant la division du zodlaque et lfinstitution d*une semalne planetaire, relativement a une progression geomdtrique, d'ou dependent les proportiones muslcales,^ and which is found developed in his Memoire sur la musique

l*According to Roussier, the consonant intervals are those formed by disjunct degrees of the scale. There are, therefore, four consonant intervalsi the third, fourth, fifth and sixth. The dissonant intervals, formed by conjunct degrees of the scale, are the second and the seventh, the seventh being conjunct to the octave. (Traite des accords, p* 6.) While the sixth as such is consonant, within the context of the super­ tonic six-five the sixth is dissonant because it forms a conjunct degree with the fifth. Moreover, if one accepts the concept of interval roots and the value-order of interval roots set down by Hindemith in Series 2 as a valid basis for root determination, Roussier1s analysis is sub­ stantiated because the best Interval in the chord is the perfect fifth) hence, d is "dissonant." As for his rule about tfeo resolution of disso­ nance, Roussier states explicitly (p. 42) that the major dissonance is the sixth of the subdominant, and the minor dissonance is the seventh of the dominant or the leading-tone seventh) all other dissonances are "accidental." ^Parisx

d ’Houry, 1765, pp. 217-25.

3paris, 1770-71.

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145 des anclensy^ and in his notes on the memoir of the Jesuit Aaiot con­ cerning Chinese music,^

A passage from Timaeus of Locris, reported by

Plato, and the dreams of Censorin had turned Roussier's head; thence he drew the idea of a geometric progression of 12 terms, which he calls the "triple progression," because the proportion of the just fifth 3 *1* tripled from fifth to fifth, gives him the following descending pro­ 1 13:9 *27:811243«729^ , etc,, which, pushed as far as the

gression!

twelfth fifth or the thirteenth term, gives the figure 531*441, an expression, according to him, of the comma between c-flat and b, his point of departure being this last notej because in the order of the planets corresponding to the hours of the day to the days of the week, Saturn is first*

He arrived at Roussier's system just as we have seen

in all the others:

in wishing to explain music, he defines it.

Finding

this numbers game in opposition with the results of the indicated harmonic proportions, he denied the reality of the latter and threw in many sup­ positions which do not hold up in the most cursory examination.

For

example, an antique bronze cited by Montfaucon in Antlquite expllqueer where one saw the series of seven main divinities beginning at Saturn and ending at Venus, gives him the scale which he considers fundamental: b, c, d, e, f, g, aj it is from here that he departs to make his •^Memoire sur la musique des anciens, Ou l'on expose le Principe des Proportions authentiques. dites de Pythagore, et de divers systemes de Musique chez les Grecs, les Chinois et les Efeyptiens (Paris: Lacombe, 1770), Notes et observations sur le memoire de Pj_ Amiot concemant la musique des chinois! (Paris, 1780),

-

,

8

f a _ o ----------

.... ~

- - 0 “ —

—

'

V

l>

-

a --------

»

A

>fl

1 ----- ------- 1-------- -

to

lHI

U

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-

146 progression of 12 fifths:

b-e-a-d-g-c-fy etc. Furthermore, by long chains

of reasoning he arrives at the opinion that the tones of the musical scale of the Greeks were interpreted as descending! this opinion mas expressed already by Pepusch** and is reproduced these days by von Drieberg,

as I have said in my Resume philosophique de 1*historic de

la musique,3* Finally he Insists in many places on the necessity of making all music modem, as if the arts could be created by similar processes, and as if the forms of the scales were not determined by laws set much higher than in the arbitrary properties of numbers-r-as if in this same system the numbers had not been reduced to a hypothetical value I Finally, we conceive the reality of facts represented by numbers, when these numbers are the expression of the dimensions of sonorous bodies or of the frequency of vibrations from which the measure of the intervals of sound result, But where can one find the law of this triple progression on which one wishes to place the criteria of the art and of the science, if it is not in the alleged analogies with a planetary system and an ancient calendar? theory?

Have we founded a real science or a useless hermetic

Are we musicians or ought we to form a sort of gnostic sect, a

l*John Christopher Fepusch, "Of the various Genera and Species of Music among the Ancients, with some Observations concerning their Scale,” Rillosophical Transactions of the Royal Society of London, XLIV (Oct.Dec, 1746), 266-274, Fepusch is arguing to consider a descending scale as well as an ascending scale, and that "the first Sound in each was the Rroslambanomenos" (p, 269)» 2*pri©drich von Drieberg, Die praktische Musik der Griechen (Berlin: Trautwein, 1821), ^*Fetis chastises von Drieberg and Pepusch on two accounts: (l) the hypate is the first and lowest note of the scalef (2) Greek scales (to which he ascribes the name of the church modes) consist of an ascending series of sounds, (Fetis, R6suml philosophique de l'histolre de la musique in Biographic unlverselle, I j_Paris: Fournier, 1^35J, cv-cvi, n»y~

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1k7 new breed of Illuminati?

What would make one believe that the Abbe

Roussier leaned towards this last conversion of the seekers of harmonic theories is that, in many a place in his Memoire sur la musique des anciens and in his Lettres on the relationships of the zodiac and the planetary week with the scale, he speaks very highly of the wisdom of the Egyptian priests who revealed the secrets of their musical doctrine only to those whom they initiated into the mysteries of their theophilosophy.

But it is enough for us to be Interested in the aberations

of Intellectual research inserted in the creation of a harmonic system} let us return to theories more positive and more consistent with the object of the art. But first, before turning our attention to the work undertaken outside of France to accelerate the development of harmony under the double relationship of the art and the science, let us say a word about the last two systems which closed the sphere of speculation on this subject at the end of the eighteenth century.

One was published by

1♦ Mercadier de Belestat under the title Nouveau systems de musique theorlque et pratique, the other by the Chevalier de Lirou, Engrossed with the idea of a reconciliation between the requirements of mathematical theory and musical feeling, Mercadier begins by establish­ ing the necessity of the absolute justness of the octaves} that is why, taking two strings in unison, he leaves the first in its entirety so as to make it a constant point of comparison, and cuts the second into two

Jean-Baptiste Mercadier was commonly surnamed "de Belestat" because he was born in the borough of Belestat, (Fetis, Blographle universelle, VI, 90»)

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148 equal parts, whence results the octave in the proportion 1/2•

He also

stresses a great deal the almost absolute analogy of this octave with the principal sound, and thus borrows.the idea of the identity of octaves from Hameau.

For the other intervals he consults the ear, and finds his

£own] sense of accord with experiment and theory in the accuracy of the fifth resulting from the proportion 2/3«

The inversion of this proportion

gives him the interval of the just fourth 3/4.

For the other intervals,

he says, mathematical exactitude is no longer rigorously necessary for the earj thus the major third employed in music will not always be represented exactly by the proportion 4/5, nor that of the minor third by 5/6.

But

this matters little, because the ratios of the accordance of sounds are often determined in practice by considerations independent of that of absolute accuracy.

The farther away we get from the simplicity of the

primary ratios, the less rigorous accuracy is necessary, according to Mercadier*s doctrine. This result attained, the author of this theory forms the perfect chord from the sounds produced by the proportions 1/2, 4/5, and 2/3, and seeks the other consonant chords which can be formed in the same way, with the sounds produced by the less simple proportions.

Next taking the

1* Having demonstrated that there are but two perfect consonant chords (major and minor), Mercadier uses the proportions for the major, claiming, "Experience teaches us actually that it likes the perfect chord a great deal more in this disposition 1 5/4 3/2. than in the other 1 6/5 3/2." (Jean-Baptiste Mercadier, Nouveau systems de musique thfeorique et pratique £Paris: Valade, 1776], pp. 20-21.) Hindemith opposed this irrational procedure of measuring intervals, claiming, "Construction by means of a series of fifths and thirds does not represent a primeval method of erecting a scale. One is simply taking the scale already pre­ sent in practical music and trying to explain the intervals of the series. . . r" (Paul Hindemith, Graft of Musical Composition, trans. Arthur Mendel, Bk. I £4th ed.j New York: Associated Music Publishers, 1945], P* 33»)

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sounds 2/3 from the division of the first string as the generator, he proceeds in the same way and, finding the fifth in 2/3 of the total length of the string tuned to the unison of the sound produced hy this proportion, and the third in 4/5 of this same string, he obtains two new sounds necessary for the formation of the scale.

Now, let us assume

that the sound 1 is ci the sound 1/2 will be its octave, the sound 2/3 will be g, and the sound 4/5 will be e,

Taking next the sound g as

the new generator on a new string, 2/3 of this string will be d and the sound 4/5 will be b,

To tell the truth, these sounds will not have a

mathematical justness, but they will satisfy the ear, and following Mercadier's doctrine, this is the exact result at which he wanted to arrive.

The sounds obtained thus far are, therefore, c, d, e, g, b, c.

Let us go on, and taking for the third generator the sound 2/3 of the second generator, i,e,, d, tuning a string to the unison of this new sound, next by taking the sound 2/3, i.e., a, fifth of this d, nothing more will remain but to find the last tone in order to have the complete scale.

But here a difficulty arises, because the sound 4/5 of the

third generator is not f which we need, but an f-sharp which does not appear in this scale.

We remove this difficulty by taking, with the

compass on the first generating string, a fourth equal to that which is found between the sound 2/3 and that of 1/2} then bringing all these sounds forward in their natural successive order by means of the identity of octaves, we will have all the scale i c, d, e, f, g, a, b, such as the ear Indicates, with a natural temperament which will avoid the comma from

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150 the succession of a dozen fifths,^-*

To he "brief in this expose* I have

omitted all the calculations viti. which Mercadier backed up his system. Thus, after having formed his scale, an object of complete satis­ faction for him, Mercadier has no difficulty at all borrowing from Rameau his generation of chords by the addition of thirds, his inversion of these chords, and even his fundamental bass, after having made a harsh enough critique of this learned musician’s system in his preliminary 2* discourse. There is something more, because on examining it closely, his melodic generation is drawn, like that of Rameau, from the fundamental movement of fifths.

Mercadier had not noticed that he put into contra­

diction the two principles which he wished to make the foundation of his system.

Obliged to turn to the impression of the ear in order to oppose

the laws of calculation, which indeed can not generate any scale, he ought to have perceived that sound structured from numbers became useless, I would say almost ridiculous.

The good tuners of keyboard instruments

do not do it so many ways? they also satisfy the ear, but they do not pretend to make any system.

What is more, on the relation of harmony

considered in its theory as in its practice, Mercadier lagged behind 1* Barbour declares that Mercadier is propounding a Pythagorean-type of temperament in which some of the fifths are tempered by either l/6 or l/l2 comma? the latter is the temperament of the fifth of equal tempera­ ment. In summing up the labyrinth of calculations into which Mercadier leads the reader, Barbour saysi "He directed that the fifths from C to E should be flat by l/6 syntonic comma j^80i8lj, and those from E to G-shaxp by l/l2 comma. Then G-sharp is taken as A-flat, the next three fifths axe to be just, and the fifth F-C then turns out to be about l/l2 comma flat" (p. 168). (James Murray Barbour, Tuning and Temperament! A His­ torical Survey ^2d ea. ? East Lansingi Michigan State College Press, 1953], PP. 167-169.) 2*Mercadier also adopts Rameau's theory of chords by supposition (Pt. VI, ch. H i , pp. 20^-210) and his double emploi (Pt* VI, ch. ix, P. 235).

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151 Roussier, although he may he a few years later.

His system did not have

any success* As well as Mercadier de Belestat, the Chevalier de Lirou, author of Explication du systems de 1*harmonic, pour abreger 1 ’etude de la composition, et accorder la pratique avec la theorie,^* attempts to extract the scale from the harmonic construction of several perfect chords, hut he proceeded differently, as we will see presently* Jamard had proved that the logical results of Levens’ and Balliere *s systems had to lead to the abandonment of fundamental hass; hut this consideration is all theoretical in his hook.

The Chevalier de Lirou was the first French

author who, in a hook on harmony, completely hroke away from this system of fundamental hass and became the guide for all harmonists amongst us, and who resisted the attacks of his adversaries for a long time.

An

exact enough observation of the phenomena of chordal succession had led de Lirou to search for the laws of these successions in the affinities of tonality which are, actually, the unquestionable basis of all music. Unfortunately, the author’s ideas lacked clarity with respect to this criterion of the science and the art*

In lieu of the quest for the

principle of the tonal affinity of sounds by the order of succession, he takes the harmonic resonance of sonorous bodies, supposedly uniform, as his point of departure.

He says c produces ef g generates b, dj moreover,

c can be considered the fifth of f, whence f , a, c.

Thus c being placed

■*-*[\Tean Franjois Espic] de Lirou, Explication du systeme de l'harmonle, . * . (Paris* Londres, 1785)* Since de Lirou contends that ", * * the fifth is the basic inter­ val of harmony, the interval par excellence," (p. 18) and that music emanates from a "common center," tonic, in effect, represents duality* it

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152 as Intermediaryr we find the following tones in the harmonic resonance of f, c, and g*

e, f, g, a, b, c, d, e, which include all the Intervals of

our major scale

and correspond to the two tetrachords of Greek music*

e, f, g, a| and b, c, d, e.

And because he had succeeded in finding the

notes which compose the scale with a mechanical and arbitrary processr he believed he had tonality and persuaded himself that he only needs to change the disposition of these notes by beginning on c instead of e. He does not know that all the difficulty is precisely in the determina­ tion of the first note of the scale,

Having reached this result, de Lirou

arranges the notes in a circle which represents the two progressions, ascending and descending*

£-e-£-b-d-f-ai c-a-f-d-b-g-e. He takes these

as the basis of all chordal constructions, all harmonic successions, modes, n* and modulation. Like most of the French harmonists of his time, de Lirou had only a very imperfect command of the art of composition, so that several successions of chords which he presents as admissible are not, but he is, I believe, the first who specified sufficiently good rules, although incomplete, for the circle of keys and modes in modulation* After de Lirou*s book, which appeared in 1785 and which was scarcely noticed, the eighteenth century was closed in France with respect to the science with the Traite d'harmonie et de modulation, which was published is the generator of a triad as well as the fifth of a triad. This concept is not unique to de Lirou, but is reminiscent of Rameau's triple progres­ sion (1*3*9) where tonic is represented by the number 3r the subdominant by the number 1, and the dominant by the number 9* (Rameau, Nouvelles Reflexions sur le princlpe sonore [[Paris, l76o[Jf p. 196, n.) This concept recurs again in Hauptmann when he talks about ’'having" a dominant and "being" a dominant, (Shirlaw, pp. 352-62,) 1* The arrangement of the pitch material in this disposition permits, according to de Lirou, each note of the scale to be either a tonic, a third or a fifth of a perfect chord, (de Lirou, p, 26,)

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153 in 1797 by Langler librarian of the Conservatoire, An unquestionable reputation as an expert harmonist had been made by this artist through the school of Naples*

In this book one no longer sees either the

numerical theories of intervals nor the acoustical phenomena as the basis of a system of harmonic generation} Langle5s declared pretension is to search for the true foundations of the science in the practice of the art.

From the first words of the preface which he put in the front of

his treatise, we are tempted to believe that he had grasped the true principles of this science, because he protests against the previously published books in which the chords are considered in an isolated way, without regard for the laws of succession which rule them,

But immedi­

ately following, we see him advance this singular proposition!

there is

only one chord, that of the third, the combinations of which produce all 'T* the others. And for the proof of this principle, he offers this series of thirds as the example1 f-a-c-£-g-b-d-f} then he extracts from it the perfect chord on the fourth degree (f-a-c), the perfect minor chord (a-c-e)t the tonic chord (c-e-g), the relative minor chord of the dominant (e-g-b), the dominant chord (g-b-d), the major seventh chords (f-a-c-e. and c-e-g-b), the minor seventh chord with the minor third (a-c-e-g), and the dominant seventh chord (g-b-d-f), Now, in this classification Langle confuses everything in making, through his generation of thirds, the classes of seventh chords, e,g,, of every kind, as if these relations 1-*Langle has been misquoted by Fetis } here is what he saidi "I recognize in harmony only one unique interval, generator of all chords} this interval is the third. By its multiplication it produces a sole chord which contains all of them," (Honore Francois Marie langle, Traite d'harmonie et de modulation ^Barisi Cochet, 1797^* p. 1*)

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15k existed by themselves in music and apart from every consideration of the formation of chords by alteration, prolongation, and substitution.

For

this very reason he finds himself in obvious contradiction with the beginning of his book.

This fault, which, although not analyzed by its

readers, just the same cast much obscurity onto his system, is detrimental to the success of the book.

Moreover, some of the shocking imperfections

in the chord successions which he gives as examples

caused his book to

be rejected in the examination which was made of various harmonic systems in 1800 by the assembly of Conservatoire professors, and from that time his system fell into oblivion. After the examination I have devoted to the efforts made in France in the eighteenth century for the formation of a rational system of the science of harmony, I need to cons?.der the influence which the idea of a similar creation exercised on Germany and Italy. Although I have said the idsia of a theory of the generation of chords had not occurred to any German harmonist prior to the publication of the Traite de 1'harmonle and the Nouveau systems of Rameau, Mattheson, who published the second edition of his Grosse Generalbass-Schule in 1731* had not only not adopted the idea which brought so much honor to Rameau, in spite of the errors into which he let himself be led, but in lieu of a reasoned discussion, Mattheson condescended to write these coarse insults in his Crltica musical

In the works of the organist of Clermont, one generally finds 1000 hundred-weight of toilsome research and feeble observations! 500 vain pretensions at originality! about 3

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155 pounds of original, or borrowed knowledge! 2 ounces of common sense and scarcely a grain of good taste*** The Treullcher Unterricht im General-Bass of David Kellner* published in 1742 and frequently reprinted*

had not pulled the science to

the empirical point of view where Helnichen and Mattheson had left it} the middle of the eighteenth century had therefore arrived before the German harmonists became interested in a theory of the art which they taught by practice*

But one is going to see that from this time on* all

Germany was under the stress of agitation for the creation of a similar theory, Sorge* organist at lobenstein* was the first who* without adopting anything from Rameau's theory* seemed to be won over to his idea of the necessity of a scientific base for the proceedings of the art} he states his principles in a book which is entitled

Vorgemach de muslcalischen

Composition, oderi Ausfuhrliche, ordenliche und vorheutlge Praxin llhlangllche Anwelsung zum General-Bass (Lobenstein, 1745-47)»3 Although Sorge dees not account for this subject* there is reason to believe that the reading of Euler's Tentamen novae theoriae musicae l*Since Fetis gives neither the volume number nor the page on which he found this quotation in the alleged work* he probably was relying on his memory and meant instead to refer to the ELeine General-Bass-Schule where Mattheson castigates Rameau with these very words in a footnote on pages 220-21} the latter appeared later (1735) than either Gritlca muslca (1722-25) or Grosse Generalbass-Schule (1731), Since no edition of Kellner's treatise appeared in 1742* this date of publication is a misprint for either the first edition (Hamburg1 Kissner, 1732)* or the third edition (Hamburg 1 Christian Herold* 17^3;I the eighth and last edition appeared in 1796 ("David Kellner*" MGG* VII* 817)* 3Georg Andreas Sorge* Antichamhre de la composition muslcale* ou Instruction detallee, regullere et suffisante pour la pratique actuelle de la basse continue (Lobenstein* 1745-4-7) ,

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156

had made an Impression on his mind, and that it is from this book that he drew the idea of a system founded on the numerical ratios of sounds, But instead of adhering to the consideration of the degrees of the agree­ ableness of chords because of the simplicity of the numerical ratios, he adopted the opinions of musicians concerning the consonant and dissonant qualities of the chordsr Like them, he divided the chords into consonant and dissonant harmony, and he considered as consonant every chord which is composed of any three sounds from the intervals of the third, fourth, fifth, or sixth of diverse nature*

But because several of these chords

are not the product of pure harmonic progression, he resorted to the arithmetical progression in which he found the approximate expressions of these same chords.

In the ratio 4*5*6 he finds the perfect major

chord, and he remarks (ch. vi, p. 14) that experiences of various kinds prove that this chord exists in the resonance of several sonorous bodies. The natural sounds of the trumpet give him the perfect minor chord, which he calls trlas minus perfects (ch. vii, p. 17) and which he represents by the numbers 10il2xl5 of the arithmetical progression.

The same instru­

ment gives him the chord e-g-b-flat which he calls trlas defIdeas, and which is commonly called a "perfect diminished chord" in the modem school.

In reference to this b-flat, he introduces the number seven into

the calculation without any difficulty, and represents the chord following the proportions 5*6*7 (ch. vili, p. 18).

For the perfect chord with the

augmented fifth, he is obliged to raise the terms of the arithmetical proportion to 48*60*75 (ch. ix, p. 20)j finally the perfect chord with the diminished third, as a-sharp-c-e. leads Sorge up to the numbers

180*225*256.

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157 In the second part of his book* he deals with the chords of the sixth* and of the fourth and sixth* derived from the preceding perfect chords which he calls fundamental (Haunt-Accords)i hut in this distinction of fundamental and derived chords* he does not mention Bameau* to whom it belongs* and does not call to the attention of his readers what is important in the consideration of inversion* The third part of Sorge's hook is devoted to dissonant chords* Just as the sounds of the trumpet have given the perfect diminished chord (c-g-h-flat)* they provide that of the minor seventh* c-e-g-h-flat* or its transposition* g-h-d-f, represented hy the numbers 4151617* Sorge also forms dissonant chords of the same kind hy adding the minor seventh to the perfect minor chord, following the arithmetical proportion 10ii2 *15iI8 | to the perfect diminished chord* following the numbers 45 i54i64i80j to the perfect augmented chord* in the proportion 48i60i75*85i finally to the perfect chord with the diminished third* in the proportion 180t2251256*320* All these chords and their derivatives are ranked hy Sorge among those in which the dissonance is natural* l*e«* is attacked without prepa­ ration} as to the other dissonances* they appear to enter into the category of transitional notes or of prolongation* following the ancient theory formulated hy Johann Cruger in these words1 "Dlssonantlae* concentuum musicum magnopere exomantes* ingrediuntur karmoniam duobus modisi enim celeritate obliterantur* vel syncopationihus

vel

Observe this care­

fully* because here we have arrived at one of the most important facts of

Johann Cruger], Synopsis musica (Berlin* 1630)* ch, 12* p. 127* £"Dissonances* greatly elaborating a musical resolution* attain harmony in two ways 1 they a m obliterated either quickly or by syncopation."]

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158

the history of harmony i it Is the second period of genuine discoveries in this science, and the glory of this discovery belongs to the humble organist of Lobenstein, neglected by all the music historians until today.

For the first time he has established that a dissonant chord

exists by itself, apart from any modification of another harmony, and he states that this chord is absolutely different from other dissonant harmonies,** It is true that he is mistaken in granting the same character to the chord of the minor seventh added, according to him, to the perfect minor chord, although this chord is only formed and used as a product of prolongation and of another kind of modification of which I will speak later.

But if the aspect of regularity in the formation of

chords led Sorge astray, he has nevertheless grasped the fundamental character of the dominant seventh chord and of modem tonality.

In this

he deserves to take a place in the history of harmonic science immediately after Rameau, who first had perceived the foundations of this science, and had established them in the theory of the inversion of chords,

I do not

own the books of this expert musician, and I had not read them when I wrote- in the article "Kimberger" in my Blographie unlverselle des musicians, that his theory of prolongation in the succession of chords was the only real thing done for the advancement of the science of harmony from Rameau's classification of fundamental and derived chords until Catel's work,

A fortunate accident having put these very rare books into

my hands, I saw there, with as much astonishment as pleasure, the fact

1* ^ Shirlaw points out Fetis' factual error, Rameau, not Sorge, was the first to recognize that the dominant seventh chord could be taken without preparation, (Shirlaw, p, 307*)

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159 that I have just pointed out~a fact which ought to become one of great importance subsequently* The works of Rameau, having reached Germany, made a profound impression on Karpurg,

A trip which he had made to Paris in 1746, more­

over, had made him see the enthusiasm that the theory contained in these books excited among musicians*

Back in Berlin, he devoted himself to

the study of this theory, and it was from this, with modifications, that he drafted his Bandbuch bey dem Generalbasse und der Composition* pub­ lished in 1755*

There he reproduced the principle of the generation of

chords by the addition of t h i r d s t h e inevitable consequence of this principle is to isolate all the chords and remove their actual formation from the lavs of tonality and succession*

As regards his particular

ideas, they consist of a multitude of particular cases in which he let many errors slip amongst some factsf in a classification of the dominant seventh, leading-tone seventh, and dominant ninth chords, he considers them as "quasi-consonants," although in reality they are of a character exactly opposite to consonance, since they are more gravitional than any other harmony and demand resolution imperatively,2* l*While Rameau suggests building chords by the superposition of thirds, Karpurg establishes it as the fundamental principle of chord construction* (Erehbiel, p* 166*) 2*In the Traite complet de la theorle et de la pratique de l'harmonie* Fetis includes the following paragraph about Marpurgi "Without entering into greater expositions* it is easy to understand the spirit of this system. Its advantage over that |_system] of Rameau consists of keeping for the har­ monies the place which they ought to occupy on the degrees of the scale, in lieu of searching for the formation on arbitrary notes* Harpurg removes from his theory the considerations of numerical proportions and acoustical phenomena f he replaces them with that of tonality, conserving from the system of his predecessor only the mechanical formation of dissonant chords by the addition of thirdsf this is why he himself qualifies his theory

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160 Fifteen years after the publication of his first work, Sorge issued an abstract freed of all numerical considerations, with the title Compendium harmonicum. He attacked Rameau's theory in its double eaplolr in the chord of the eleventh, which he proved to be only a suspension of the third of the perfect chord) lastly in the construction of the dominant seventh chord, which he maintained is an immediate product of nature a necessary result of the arithmetical progression 4:5«6i7« received a large share of the criticism in

and

Marpurg also

Compendium harmonicua, as

having, said Sorge (in his preface), added some new errors to those of his model. The response was not expected, because less than six months after the publication of Compendium haraonicum, Marpurg published an analysis of this book^ in which he attacked his adversary on the substitution of the arithmetical progression for the geometric progression, and on some inaccuracies of the examples given by Sorge concerning harmonic successions. But he did not touch at all on the fundamental things and really did not make any solid objection to the facts established by the latter, although it seemed he must* have devastated him by his epigraphs voulu, Georges Dandin."

"Vous l'avez

Nevertheless, the influence of well-known names

and the confidence which they inspire is such, that Marpurg evidently "eclectic” in his preface. But,like Rameau, he confuses by this procedure the natural dissonant chords with those which can arise only from the cir­ cumstances of succession, and he makes of it so many facts, that it is impossible to perceive their application ahead of time. Diis mechanical formation of dissonances is absolutely arbitrary, and has no connection with the procedures of the art," (Bk. IV, par. 301, p. 211.) •^Friedrich Wilhelm Marpurg,J Herra Georg Andreas Sorgens Anleltung zum Generalbass und zur Composition mii Aamerknngen (Berlini Gottlieb August Lange, 1760).

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161

conquered In this conflict; passed as the victor, and the editions of his Handbuch hey dem Generalbasse multiplied, while the misunderstood hook of the poor organist from Lobenstein fell into utter discredit and did not sell at all, A contemporary of Sorge and Marpurg, Daube, a musician in the service of the Duke of Wurtemburg, was worried, as they were, about the need of a systematic theory of harmony* but cutting himself off from every consideration of numbers and acoustical phenomena, he conceived the usefulness of this theory only by making it conform to practice. Actually, what he published under the title Qeneralbass in drey Accorden, gegriindet in den Begeln der alt-und neuen Autoren (Harmonle en trois accords, d rapres les regies des auteurs anciens et modernes)

is less a

theory than a classification of chords because of their functions in tonality.

Although this work only appeared in 1756, he had finished it,

nevertheless, two years beforehand, as the preface, dated 28 December 175^ in Stuttgart, provest consequently, Daube wrote it before knowing Marpurg*s Handbuch bey dem Generalbasse, Sorge*s book, published nine years previously, does not seem to have been employed! either he was too un­ acquainted with the science of calculations to read it with profit, or he simply wished, as he indicates in several places, to replace the empirical and obsolete works of Heinichen and Mattheson with a systematic treatise. By the title Generalbass in drey Accorden gegrflndet, Daube proposes three fundamental chords, existing by themselves, as the consequence of tonality and under a law of close connection with their constituent intervals.

These three chords are the perfect chord, the dominant seventh

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162 chord* and the six-five chord on the fourth degree*

It is a long way

from that of the unique perfect chord of Hameau to the construction of other chords hy addition of thirds and the suppression of intervals. However* it is evident that Daube borrowed his chord on the fourth degree from the double emplol of the French harmonist, as he perhaps owes to Sorge, of whom he does not speak, the idea of the original existence of the dominant seventh chord.

Lastly, Hameau also gives him the theory of

the inversion of fundamental chords.

Daube does not explain the motive

which makes him accept the six-five chord as fundamental, rather than that of the seventh on the second degree.

But after what he says in

the second chapter concerning the dissonance of the second which generates the seventh, and not the seventh giving rise to the second, there is reason to believe that it is this motive which makes him consider the chord of the six-five as fundamental, because the interval of a second exists between the fifth and the sixth. The three chords of which he has just spoken appear to the author of the system to constitute all harmony, because he says (ch. iii, p. 20) they and their derivatives suffice to accompany all the degrees of the ascending and descending scale.

And to demonstrate it, he gives this

tonal formula with the harmonies drawn from these very chords* but some of these harmonies are as poor with respect to the feeling of tonality as with that of the succession of intervals.

For example, Daube places

the six-four-three on the ascending sixth degree, followed by the chord 1*

Daube, in giving the figured bass six-four-three as the first inversion and as the third inversion^ (johann Friedrich Accorden gegrflndet {^Leipzig * J. B. Andra,

for the inversions, lists the the fundamental seventh chord Daube, Generalbass in drey 1?56]r p. 17.)

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163 of the minor fifth and sixth on the leading tone? whence it follows that the dissonance of the chord on the sixth degree has no possible resolu­ tion.

This fault, and the six-four chord on the dominant which deprives

this degree of its chord of repose, makes the harmonic formula of the author of this system inadmissible.

Marpurg sharply criticized this

scale and many other things, under the mask of anonymity, in the second volume of his Historisch-kritische Beytrfige zur Aufnahme der Muslk (p* 465). Daube considers all the other chords either as complete prolonga­ tions of the primary chords, or derived from cadential activity, or as alterations of the natural intervals of these chords— a system in which Sorge had preceded him. Let us not be astonished by Daube*s error as regards the six-five chord on the fourth degree, because this harmony, derived from certain modifications of which he spoke later, had been the stumbling-block of all harmonists until today.

By considering it as primary, the whole

concept of a complete rational system is made impossible.

Actually Daube

added nothing to the fundamental base of these systems laid down by Rameau and Sorge; nevertheless, in his book one finds some good modulatory formulas which have enjoyed a certain vogue in Germany, After Daube, a remarkable book appeared which, nevertheless, escaped Germany's attention, or which at least was not appraised at its just worth.

I wish to speak of what Schroter, organist in Nordhausen,

published in 1772 under the title Deutllche Anwelsung zum Generalbass (instruction claire sur la basse continue).^* An educated man, not only

"L* « Christoph Gottlieb Schroter, Deutliche Anwelsung zum Generalbass. in bestandiger Veranderung des uns angebohmen harmonlschen Dreyklanges mlt zulMngllchen Exempeln (Ealberstadti J. H. Gross, 1772).

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16k In music but in letters and in sciences, Schrflter strengthened his ideas on a theory of harmony, object of so many fruitless efforts, in meditation and in the calm of a small village.

He had read everything which had been

published on this science, had analysed the work of his predecessors with care, and had resumed his observations and analysis in a history of har­ mony of which the manuscript, unfortunately, perished in a pillage of Nordhausen by the French army in 1761. Too elderly to begin a similar work, Schrflter limited himself to giving a resume in the excellent preface 1*

of his Deutliche Anwelsung,

Nevertheless, I ought to acknowledge that

this interesting man had a moment of error in taking part in the discussion of Sorge and Marpurg, and in declaring himself in favor of this latter by the observations inserted by Marpurg into his Kritische Briefe, but later he severed himself from the latter in the most important points of his theory. Schrflter establishes in the eighth chapter of his book (p. 38) that there is only that perfect chord which exists by itself, and that all the others are the products either of the inversion of this chord, or of the substitution of the seventh for the octave for the formation of the dominant seventh chord, or of prolongation, for the construction of the seventh on the second degree and the harmony which derives from it, or finally, of the anticipation. Well then, here is a great step in the true theory, in that the harmony of the minor seventh and those which derive from it are considered in their real aspect, i.e., [as] a prolongation which retards the natural

■^Sections of the resume have been translated by Arnold, I, 295301.

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165 intervals of a consonant chord.

In this phenonomen, Schroter considers

only the effect of a suspension! this is why he gives it the name Yerzoger»ng (retardatio). If one had demanded of him what this retardement isr he would have met with a great deal of embarrassment to find a satis­ factory response, because it is evident that if the prolongation ceases, e.g., in the chord d-f-a-cr one will have d-f-a-b for a resolution, which is not a consonant harmony.

Now, there is another circumstance which, in

the chord d-f-a-c, unites with the prolongation of c, but Schroter* s analysis did not dig so deeply— it stopped at the discovery of the fact of retardement. One can not deny that this discovery is of great impor­ tance, because it furnished the first element for a classification of dissonant chords which do not exist originally as the consequences of tonality.

This was the first blow raised at the false theory which had

ranked the seventh chord with a minor third in the same class of harmony as that of the seventh with a major third. Concerning the latter, Schroter took a step backwards in consider­ ing it as the product of the substitution of the seventh for the octave of the perfect chord, because this chord of the dominant seventh, characteristic of modern tonality, exists by itself in this tonality of which it is the generator,

This is what Euler and Sorge saw so well.

In chapters nine through 17, Schroter develops the results of the theory set forth in the eighthf the eighteenth is devoted to alterations, and the nineteenth to the retardements of all natural and altered har­ monies,

In the latter the author gives proof of a great sagacity-

Some

of his views are more advanced than the state of the art of his time, and he foreshadows by instinct some of the harmonic aggregations which Mozart,

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166 Beethoven, Weber, and Rossini subsequently introduced into practice. I have said in my article on Klrnberger (Blographle universelle des muslciens, V, 341)

that we have conceded too much to this theoreti­

cian in improving the theory of harmony,

I myself conceded too much in

this work, because I was acquainted with neither Sorge's books nor that of Schroter, and because the music historians who spoke about him had not understood his merit. book

All the writers who have cited Kiraberger's

Die wahren Grunds&tze zum Gebrauch der Harmonle (Les vrai nrlncipes

conceraant 1 *usage de 1 'harmonle )•*•* say that he reduces the fundamental harmony to the perfect and the seventh chordsf even he, in the prefaces of his various works, and especially in that of his Grunds&tze des Generalbass [jBerlin, I781J, which are the practical development of the preceding work, even he, I say, congratulates himself on having arrived at this simplicity.

But just as he considers the three-tone chord under

its three tonal forms, namely,

the perfect chord with the major third,

the same chord with the minor third, and lastly the chord of a minor third and minor fifth (on the leading tone), so he considers the seventh chord as original, whether it has either the major third, as g-b-d-f, or whether this third is minor, as in a-c-e-g, or,finally, whether the third and the fifth are minor, as in b-d-f-a, or even whether the third and seventh are major, as in e-e-g-b.

These forms, Kiraberger says, differ

only in the quality of the intervals, but the quality of the intervals is precisely what establishes the difference in the natural or artificial existence of the chords with respect to tonality.

If the difficulty were

not that, there would be nothing at all in it* l*Berlin und KBnigsberg:

G. J. Decker und G. L, Hartung, 1773«

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167 As for the seventh chord on the second degreer Klrnberger derives it from the retardation of the sixth in the chord of the sixth on the same degree, derived from the chord of the minor third and minor fifthr His chord thus formed is composed, therefore, of d-f-c, retarding d-f-bj sometimes the fifth is Introduced, he says, to fill up the harmony.

But

by virtue of what law and by what technique is this strange note introduced into the chord?

This is what he has not seen at allr and what he does not

even attempt to explain, restricting himself to point out a fact of experi­ ence.

Indeed, this difficulty is the most extensive of all the rational

theory of harmony, and it has been the stumbling-block of ail the harmonists After Klrnberger, several musicians who enjoyed honorable reputa­ tions in Germany wrote some harmony and basso continuo treatises during the last part of the eighteenth century, but among them only one appears to me to have wanted to create a system which does not appear to have been realized on a single attempt, if we judge by the first writings intended for propagation, compared to his last works in which he set forth his system.

The author of this system was Abbe Vogler who, having

instituted a school of music in Mannheim in 1776, published in the same year a sort of manifesto of the principles which he taught there in a book entitled

Tonwissenschaft und Tonsetzkunst (La science de la musique

et de la composition), followed by a sort of commentary on these principles which appeared under the title Ghurpfalzlsche Tonschule. and a paper of

l*For a comparative and detailed analysis of the theories of Marpurg and Klrnberger, see Joyce Mekeel, "The Harmonic Theories of Klrnberger and Marpurg," JMT, IV (Nov., I960), 169-93•

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168 the progress of the school hy the new method entitled

Betrachtungen der

Hannhwimer Tonschule (Examens de 1* ecole de musique de Mannheim). The necessity of all these explanations does not give a favorable Idea of the system’s lucidityf the obscurity of the doctrine and the incoherence of Its elements were actually the faults for which the critics of the time reproached him,

Welsbecke , professor of law at Erland, attacked

this theory in writings published in 1783 and 17841 Knecht, pupil of Vogler's first school, was obliged to go to the defense of his master, and the Gazette musicale of Spire became the organ of a polemic on this subject.

Later Vogler reproduced his system in Copenhagen, writing In

Banish, and undertook a new demonstration of his principles in a school founded for this purpose.

Finally in 1800 he published in Prague, where

he made a course of his theory, a Handbuch zur Harmonielehre und fur den Generalbass (Manuel de la science de 1'harmonie at de la basse continue. d'apres les prlnclpes de 18ecole de Mannheim).

There he complains with

bitterness, in a long preface, of the attacks of which his works and his person have been the object, and of the accusations of charlatanism which were hurled in his face.

Whatever opinion one has of the doctrine

and writings of Vogler, one is deeply touched to see a man who has had the glory of training in his school at Darmstadt the two most eminent German musicians of the present time, Carl Maria von Weber and Meyerbeer, obliged to debate the legitimacy of his claims for the respect of artists, After this digression, which seemed necessary for those who do not know Vogler's works, I return to the analysis of his system. This system is taken, as far as its theoretical part, from Levons' book, and, as for the practical applications, from Vallotti's principles,

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169 of which I will speak presently.

Taking a string which he divided har­

monically on the one hand, and in an arithmetical progression on the other, he obtains the harmonic and diatonic intervals in the lower and middle notes, in accordance with the acoustical construction of the trumpet and horn, and the chromatic intervals in the higher notes. Like Levens, he establishes three [whole-3 tones whose proportions are different, namely, a major tone, b-flat to c, in the proportion 7:8| a middle tone, c to d, as 8:9I a ninor tone, 9*10 (Tonwelsscnschaft, pp. 122-23) , The arithmetical progression, extended to the 32 term, gives Vogler a chromatic scale, a major scale with the notes c, d, £, f , g, a, b, an enharmonic scale of c-sharp and d-flat. d-sharp and e-flat, e-sharp and f-natural, etc., and, finally, a minor scale. Vogler also obtains the perfect major chord (c-e-g) from the division of his string by the arithmetical progression, the perfect minor chord (g-b-flat-d), the chord of a minor third and minor fifth (e-g-b-flat), the chord of a minor seventh with a major third (c-c-gb-flat), the major ninth chord (c-e-g-b-flat-d), the chord of a minor seventh with a minor fifth (e-g-b-flat-d)i finally, all the harmonies, without excepting those whose intervals are generally designated by the name "chromatic."

Therefore, following the author of the system, it is

no longer a question of putting each of these chords on the degree where T*Vogler observes that the difference between each type of wholetone is equal to a comma, the very discrepancy in just intonation which led to the emergence of mean-tone temperament. Vogler also observes the existence of two kinds of half-stepsj the "major" half-step (15:16) occurs distonically between e-f and b-£, while the "minor" half-step (2^:25) occurs chromatically, as c-c-sharp. (Abbe [Georg Joseph] Vogler, Tonwissenschaft und Tonsetzkunst [Mannheim, 1776], pp. 12-13.) Fetis* page references are most curious— the treatise has only 86 pages!

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170 It is most correctly placed*

Undoubtedly this would be a radical

difficulty with respect to tonality, if Vogler accepted the formulas of tonality which might expressly define the place of each, because of certain functions of successions*

But he does not forget that the

arithmetical progression has given him not a scale, but a chromatic scale, and, faithful to his principle, he establishes that all possible chords, fundamental or derived, can be formed on each of the notes of this scale.

Although he may be obliged to comply with usage and establish

the keys of c, d, e-flat, f, etc,, he maintains that in these keys every note which does net seem to belong, every harmony which is foreign, may be put in place without resulting in real modulations, unless cadential action sets up the new key.

It is true that, in the formation of this

monstrous system, so completely contrary to every feeling of the most delicate part, the most sensitive of the art, he forgets a greater difficulty, namely, that the tendencies determined by the inequalities of the justness of the intervals under the terms of the arithmetical progression do not permit the harmonies to transfer to notes other than those fixed by these tendencies, without affecting the ear by the dis­ comfort which false intervals produce, because the ratios are no longer the same.

For example, the number seven, which gives the gravitational

dissonance of the fourth degree with the dominant, sets up a ratio which can only exist between these two notes, following their functions which consist of foreshadowing the return to tonic or the preparation of a necessary modulation.

Let us say

that a similar theory is the negation

of all exact theory, because it reduces the art and the science to a collection of disconnected facts.

The laws of harmonic succession are

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171 reduced to a similar maze of heterogeneous chordsf all the efforts of Khecht to establish these Ians? without refuting his master's system, In the hook conforming to this theory, which he published with the title

GemelnnStzliches elementarwerk der Harmonle and des Generalbasses

(Traite elementalre de 1 *harmonle et de la basse continue^- have been fruitless. Such was the last system of harmony which closed the eighteenth century in Germany, and which was not at all a success, in spite of the public teaching which its author did in several large cities.

As for

the treatises of harmony and basso continue of Albrechtsberger,^ Turk,3 Portmann,^-Kessel,-5 and several others, I do not believe I have to give an analysis of them because they contain more or less new views only in detail, and only in what concerns more or less easy methods of teaching. It is in this way that I have employed some of them for certain books of the same kind published in France in the second half of the eighteenth

^Justin Heinrich Khecht], Augsburgi Lotter, 1792, 94, 98j 2d edrj Munichj Falter, 1814, with 1 vol. of plates. ^Johann Georg Albrechtsberger], Kurzgefasste Methods, den Generalbass zu erlemen (Methods abregee pour appendre la basse continue) (Vienna und Mainz, 1792). Daniel Gottlob Turk], Anweisung zum Generalbassplelen (instruction sur 1 *accompagnement de la basse contlnueTT 2d ed.f Halle und Leipzig, 1800), ^Johann (Jottlisb Portmann], Lelchtes Lehrbuch der Harmonle, Kbmposltlon und des Generalbasses, zum Gebrauch fttr Liebhaber der Musik. angehende und fortschreitende Musiker und Komposltion mlt VorschlSgen elner neuen Bezifferung (Methode facile d'hannonle, de composition et de basse continue) (Darmstadt. 1789), ^Johann Christian Bertram Kessel], Unterricht lm Generalbasse zum Gebrauche fur Lehrer und Lemende (instruction sur l 1usage de l'harmunle pour les professeurs et les el&ves) (Leipzig1 Hertel, 1791JT

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172 century, e«g«, those of Bethizy^ and of Bemetzrieder,^ who belong to the same category Before dealing with ths reform of the harmonic system by Catel, who opened the nlnteenth century In France, It remains for me to cast a glance on what was done In Italy and England In the last part of the eighteenth century. Tartini's theory was unsuccessful in Italy because it contained only vague speculations which did not have any direct application in practice.

It was not the same with respect to a theory, both systematic

and practical, conceived by P. Vallotti, great Franciscan monk of the monastery at Padua, who formed a school whose doctrine, developed by the pupils of this savant musician, was very different from those of the other schools in some essential points.

Contemplated in the calm

of a cloister and during a long life, Vallotti's theory came to its point of maturity when the author decided to publish it, but he had then reached the age of 82 years, and death surprised him before he was able to bring it to light.

Only the first part was published under this

title i Della Scienza teorica e pratica della modema muslca (Padua t Stamperia del Seminario []G* Manfre]], 1779)I it is purely speculative* Ehe three other parts, unpublished to this day, were expected to deal

■*■[]Jean-Laurent de Bethlsy], Exposition de la theorle et de la Pratique de la muslque. suivant les nouvelles decouvertes {Paris: Lambert, 175^5* ^Anton Bemetnrieder]], Kouvel essai sur l'haraonie* suite de traite de musique (Psirisi Chez 1'auteur et Onfroy, 1779)• 3*The harmonic principles of Bethizy are discussed in Krehbiol, pp. 198-200.

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173 with some of the practical elements of music,, counterpoint, the rules of harmony and accompaniment. Today we would not know, except by tradition, the applications made by Vallotti of his speculative theory to the practice of the art and to the system of harmony, if his pupil and successor, Sabbatinl, had not made them known in his treatises of harmony3- and fugue*2

The first

of these works alone must occupy us here. Sabbatini's method was purely empirical| one must not search for a general view of systematic construction*

The facts are ascertained

by their existence, but without searching for their origin.

Thus

Sabbatini finds the perfect major chord on the tonic, the perfect minor chord on the sixth degree, and a progression of these chords by a series of bass movements descending a fifth and rising a fourth leads him to the minor third and minor fifth chord, which is made on the leading tone. As far as this last chord is concerned, he has shown more sagacity than all his predecessors} the latter considered it as a natural chord in the place which it occupies, while Sabbatini, or rather Vallotti, saw very well that this chord which responds to no tonal condition of the major or minor modes is affected only by analogy in a progression of perfect non-modulatory chords,-^* It is remarkable that the more modem harmonic ^Luigi Antonio Sabbatini], la vera idea delle musicali numeriche segnature dlretta al giovane studioso dell' armonla (Venicei Valle, 1799)• ^Trattato sopra le fughs musicali , * , corredato da copiosl saggi del e le opere del Padre P. A, Vallotl (Venice, ca. 1802). 3*Sabbatini classifies the third, sixth, perfect fourth, and fifth as consonant intervals. The leading tone triad, containing the dissonant interval of a diminished fifth, is classified as consonant harmony by analogy (armonle consonant! per rappresentanza)g (Sabbatini, La vera idea, p, 12,]

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174 theorists have shown themselves less advanced on this points As for the harmony derived from the fundamental , Vallotti and Sabbatini followed Rameau's doctrine * Considering the chromatic scale as a true scale, these authors do not present the augmented fifth, diminished third, nor the other modified intervals of consonant chords as the alterations of the natural intervals of the perfect major, minor, and diminished chords, but as an arbitrary employment of the intervals which are all assumed in this chromatic scale. Going ahead to dissonant chords, Sabbatini constructs them by the addition of intervals to the perfect major, minor, and diminished chords,^* Thus the addition of a major third above the perfect major chord on tonic gives him a major seventh chord (c-a-g-b) which he considers as the first in order.

Likewise, the addition of a minor third above the perfect minor

chord on the sixth degree creates a minor seventh chord (a-c-£-g). Prom these two fundamental chords he draws, by inversion, the six-five, sixfour-three, and second chords,3* Lastly, a major third added above the 1#

At no point does Sabbatini discuss "the chromatic scale as a true scale," nor is the use of the augmented triad and diminished third triad arbitrary. The augmented fifth and the diminished third triads or their inversions are, like the diminished fifth triad, consonant harmony by analogy, and are derived by chromatic alteration from a specific chord in minor. The augmented fifth triad occurs on the mediant and precedes tonic in a quasi dominant-like manner! the diminished third triad occurs on the raised subdominant and precedes the dominant, (ibid., pp. 15-19 *) 2* Sabbatini was obviously well-versed with Tartini1s Trattato, be­ cause his precept for the formation of dissonant chords from consonant chords (”. , . che non si dk, nk puo darsi posizione dissonants, se non fondata sopra la posizione consonante" £p. 20]) is almost verbatim what Tartini stated on page 77 of his treatise, (See p. 132, n. 2*.) 3*The complete figuring six-four-two is more commonly abbreviated four-two or two. Marpurg, in his discussion of the "chord of the second," describes it thus: " . . . the lower end of it, and accordingly the bass,

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175 minor third and minor fifth chord creates the leading-tone seventh chord (b-d-f-a). Sabbatini says next (to vera idea delle musical! numerlche segnature, art. v, p. 32) that there is another minor seventh chord which is made on the fifth of the principal note of the key, and which is com­ posed of a major thirdr just fifthr and minor seventh, as g-h-d-f.

That,

he says, differs from the others in that it does not need to be prepared, whereas the dissonance of the former always ought to be heard beforehand in the consonant state. We see that the absence of a good classification of the original chords at this point throws the author of this system into a great con­ fusion of ideas, and the logical order which we have seen with the authors of the most erroneous systems no longer appears here.

Because

what is this seventh chord which exists outside of the system of practical generation adopted by the author, which has different conditions for its use, and which only resembles them by the necessity to resolve the dissonance by descending?-*-* And how is it, that having found from practice that this dissonant chord does not need preparation, Vallotti and Sabbatini did not conclude from it that it was the constituent chord of tonality, as well as the perfect major and minor chords?

How, finally, is it that

the necessity of preparing the dissonances of the other chords of the

contains the dissonance, since the parent chord is here standing on its head." (Harpurg, Handbuch bey dem Generalbasse j^2d ed„j Berlin, 1762], par. 44, p. 66.) For the special priviledge accorded the major-minor seventh chord, Sabbatini takes refuge in the seventh partial of the overtone series which, while not consonant, is also not a "true dissonance." To support his theory he cites both Tartini (Trattato, pp. 126ff») and Vallotti (Della Sclenza teorlca, p. 115). (Sabbatini, to vera idea, pp. 32-34.7

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176 seventh did not make them see that these chords had an origin other than the additions of thirds to the perfect chords?

Many other imperfections

arise from this systemr hut I hasten to arrive at the singularities which made this system he rejected hy the purist schools of Xtalyr in deference to the practice. The addition of a minor third ahove the perfect diminished chord of the minor mode leads Vallotti and Sabbatini to the diminished seventh chord f the same addition to the same chord with a chromatic or diminished third produces the diminished seventh chord with a diminished third (d-sharp-f-a-c)

finally the addition of a minor third ahove the perfect

augmented chord gives rise to the major seventh chord with an augmented fifth (c-e-g-sharp-b).^* All the harmonies derived from these chords are formed by their inversion. As far as thatr if the theory is unsatisfactory, the practical examples of harmony of Sabbatinifs book conform to what is done in the modem school.

But here is a new, unusual part, where the ear is offended

by strange associations of sounds whose movements are not known to give the sensation of resolved dissonances, inasmuch as the notes on which the resolutions are made are always heard in the chord.

Thus, in the perfect

chord c-e-g-c, where he even doubles the intervals, Sabbatini says when one adds the ninth, so that the chord he presents is composed of c-e-g-

•*-*This seventh chord with a diminished third occurs on the raised fourth scale degree in minor, and resolves immediately to the dominant. Unlike the augmented triad, this mediant seventh chord in minor resolves to a dominant seventh chord.

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177 c-d-e-g-c,1* the first derived harmony is a seven-six-four chord (e-g-c-d), the second, a six-five-four chord (g-c-d-e^), and the complete inversion, a seven-four-two chord (d-e-g-c)» It is, moreover, in this way that Sabbatini, following Vallotti, adds a dissonance of the eleventh to the perfect major or minor chord whose intervals are redoubled* the chord thus composed occurs in this form*

£-£—g-£-£-f—g.

Its first derivative is the nine-eight-six-three

chord (e-g-c-e-f), the second, the seven-six-four chord (g-c-e-f), and the £third] inversion, the seven-five-two chord (f-g-c-e).^* Yet, who would believe that these harmonies, so harsh,-'

so devoid

of the means of good resolutions, were imagined by a savant musician— educated in the most perfect principles'— only by the spirit of the system, and because he had not understood the technique of suspension which delays the natural intervals of the chords;

If he had. understood the theory of

this technique, he would have seen by this alone, that if a note in a chord is delayed, it can not be heard at the same time as the suspension* consequently, instead of composing the ninth chord of c-e-g-c-d, he ought to form it of c-e-g-d, delaying c-e-g-c,

Prom that point he would have

^-*First, in each musical example the ninth is a suspended dissonance, a decorative pitch, and not a true harmonic member. Secondly, Fetis' criti­ cism here is wholly unwarranted, because in none of the musical examples is the note of resolution present with the suspension. While Sabbatini does have this "doubled" ninth chord, it does not occur in a musical context, but in a figure illustrating the dissonant interval (9) as it occurs between 8 and 10* neither this chord nor any other chord in a "figure" is resolved. O* In the musical examples, the ninths and elevenths are all dominant In function, 3*a germane observation by Fetis, because eleventh chords tend to lose their identity when the root is in an upper voice and occurs above the eleventh* the resultant clash between the major third and the "eleventh" is undoubtedly what Fetis means by "harsh."

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173 avoided all the harmonic abhorrences which he presents as derivatives of his primary harmony.

Likewise, the principle of suspension would have

proven to him that his so-called eleventh is only a fourthj this fourth delays the third, and consequently the third and the fourth can not he heard simultaneously.

Thus, instead of having a chord composed of

£-£-f-g-c which does not occur in any well-written piece of music, he should have had £-f-g-c, delaying c-e-g-c j its derived harmonies would have had the same regularity, I do not need to extend farther the examination of this bizarre theory in order to make clear that, when Vallotti'a students began to propagate it, those provoked against it were all distinguished composers in Italy,

It [[the theory^ was special in that it alone had the pretension

of reforming the art of writing, because all the other systems had limited themselves to giving more or less false explanations of the facts, more or less close to the truth, or creating some simple speculative hypotheses. We can not consider the Begole musicali per i princlpianti di cembalo, nel sonar coi numerl e per i princlpianti dl contrappunto (Naplesi Mazsala, 1795)^* of {jFcdele] Fenaroli as the expose of a harmonic theory} it is only a practical outline of the tradition of Duranters school, pure but outmoded tradition} it did not represent the actual state of the art. In the eighteenth century England did not have any harmonic theorists whose works are worthy of any notice.

Five well-known musicians, as a

matter of fact, published some harmonic treatises, but four of these musicians were Germans, and the fifth was Italian,

The first, Gottfried

^*lfce first edition appeared in 1775» not 1795»

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179

Keller, settled in london about 1702.

like all the predecessors of

Rameau, it was not his intent to make a system of harmony of his Methods complete pour apprendre a accompagner la basse continue, but to formulate some rules for accompaniment, as the title of his book indicates.-*- There is more analysis in Pepusch's work, which has for a title

A Treatise on

Harmony,^ but this musician, also German by birth, does not seem to have known the Tralte de 1'harmonle of the French theorist, and stuck to con­ sidering harmony in the art of composition, in lieu of presenting a system of generation and classification of chords.

The first systematic

book published in England dealing with harmony is that of Johann Fried­ rich Lampe, German by birth, who published a method of basso continuo based on Bameau*s principles of fundamental bass in 1737*^

A few years

later the celebrated violinist Geminianl published his Guide harmonique**' (production worse than mediocre), which can not be taken as a systematic treatise of harmony, because it is only a sort of dictionary of chordal successions and of modulations.^*

A Compleat Method for Attaining to Play a Thorough Bass upon either Organ, Harpsichord or Theorbo-Lute. . . . (Londoni Walsh and and Hare, n.d.). ^Johann Christoph (John Christopher) PepuschJ, A Treatise on Harmonyi Containing the Chief Rules for Composing in Two, Three, and Four Parts (Londoni W, Pearson, 173lT* ^A Plain and Compendious Method of Teaching Thorough-Bass (Londoni Wilcox, 17371:-------- ---------------------- ------- ----\Francesco Saverio Geminianii Gulda Armonlca o Dlzlonario Armonlco being a Sure Guide to Harmony and Modulation (London: Johnson, 1742), 5* The third part of Serre’s Observations sur les principles de 1'haraonis is a critique of Geminianl's treatise.

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180 Kollmann, the last of the writers mentioned above , had come from Germany about I782 to settle in London* Fourteen years later he published a book entitled

Basal sur 1*harmonle muslcale, sulvant la

nature de cette science et les prlnclpes des auteurs les plus celebres*! These principles are those of Kirnberger, whom Kollmann frequently limited himself to translating) but in endeavoring to supplement some gaps in it, he borrowed some ideas from Marpurg, not knowing the contra­ dictions which axe found between the doctrines of these two theorists* Later he noticed the anomaly of the two systems which he had tried to reconcile in this work, believed he found a more rational and homogeneous theory in Balliere's book, and published it in his Nouvelle Theorie de 1*harmonle muslcale*

The analysis which I have done of the principles

of Balliere exempts me from the examination of Kollmann's

b o o k . 3*

[^August Friedrich Christoph Kollmann (Augustus Frederic Christopher Kollman)^ r An Essay on Musical Harmony, according to the Nature of that Science and the Principles of the Greatest Musical AuthorsTLondon 1 W» Bulmer and Co., I796). ^Kollmann], A New Theory of Musical Harmony. According to a Com­ plete and Natural System of that Science (London1 V. Bulmer and Co*. 1805J. ' jsrwin H. Jacobi discusses Kollmann's writings in his article "Harmonic Theory in England after the Time of Rameau." JMT, I (Nov., 1957), 138-143.

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181 CHAPTER III THE NINETEENTH CENTURYi

THE DEVELOPMENT OF THE ART.

A COMPLETE

AND DEFINITIVE FORMATION OF THE THEORY OF HARMONY

After the discovery of the natural harmony of the dominant, which created modem tonality and the first means of actual modulation* the composers* placed under the influence of this tonality* devoted themselves to developing the immediate consequences* and more than 50 years elapsed before one felt the need for new means of effects.

It was

only towards the end of the seventeenth century that musicians began to introduce ascending or descending alterations of natural intervals into the consonant chords.

The first of these alterations consisted of

raising the sixth of a chord of the sixth on the sixth degree of the minor mode a semitone* whence resulted an attraction analogous to that of the leading tone.

For example* the chord of the sixth on the sixth degree of

the key of a minor (f-a-d) becomes an augmented sixth chord (f-a-d-sharp). This d-sharp, placed in the relationship of a tritone with a* determines a necessity for an ascending resolution.

But

this interval of a tritone

or major fourth can be thought of as a minor fifth* if one changes d-sharp to e-flat> the intonation in voices and Instruments of variable pitch does not differ except in a tiny amount* which determines a descending attrac­ tion.

Now let us suppose that the augmented sixth chord f-a-d-sharp is

spontaneously changed into a seventh chord f-a-e-f1st > an unexpected modu­ lation will follow it since* following the law of tonality* the seventh chord f-a-e-flat immediately defines the key of B-flat* being the dominant of this key.

Likewise* the harmony of the dominant seventh chord can be

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182 changed Into that of the augmented sixth and consequently define a modu­ lation from the key of B-flat to that of a minor« From this possibility of unexpected key changer the feeling of surprise results, a feeling which did not exist in music prior to the use of alteration in the natural intervals of chords.

In ay Philosophic

de la muslque, I called the category of harmonic occurrences resulting from this alteration the ordre transltonlque of music, and I have discovered the principle of the variable proportions of intervals} because of their resolution tendencies, the principle introduces some new numbers into the calculation of these intervals} finally,the principle, unknown until today to all theorists, whose unrecognized existence had been the cause of so much bad reasoning and so many vain disputes f~sic!. The alteration of the intervals of consonant chords was the expression of the composer's audacity for a long time} in the first part of the eighteenth cer^ury, some isolated occurrences of the alteration of dissonant chords, as a matter of fact, made them be noticed individually very infrequently, but as the results of accident and, as it were, unknown to the musicians who used them.

Mozart was the first who, observing the

expressive accent which resides in the alterations, increased the usage with a rare sagacity, and inserted them systematically into the dissonant chords.

From that time, he could not only insert a greater number of

dramatic accents into his songs, but also increase and vary the means of unexpected modulations, because we understand that a great many new attractions were bound to be b o m of these combinations which group together dissonances of different kinds} some of them are ascending In their capacity as leading tones, and others descending as ordinary

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183 dissonances.

For example, let us suppose that we Introduce an ascending

alteration in the third of the minor fifth and sixth chord (commonly called diminished fifth) of the key of Cj we will have a chord composed of b-d-sharp-f-g, which we will arrange in a more attractive manner for the ear. making it b-f-g-d-sharpj because d-sharp is the expressive accent, it should be found in the melody, namely, in the upper voice. will be the result of this chord?

Now what

A gravitational minor fifth between

b and f and a dissonance of the second between f and g will oblige f to descend} a diminished third or augmented sixth between d-sharp and f. giving d-sharp the instantaneous character of the leading tone, will oblige it fd-sharp! to risej lastly an augmented fifth or diminished fourth between g and d-sharp f sicl. Suppose

that to all of these attractions, some combinations of

substituted notes (which I will speak about in the examination of a definitive theory) and the delay of some one of the natural notes of the chords are joined! the attractions will multiply, and the means of modulation will be increased in the same proportion.

Beethoven,

Cherubini, Weber, and Rossini, having followed in the footsteps of Mozart, have extended the domain of the ordre transitonlque in intro­ ducing there, through the means that I have just indicated, lots of new occurrences.

1*

The last expression of this course is that where the

simple and multiple alterations are joined to all the combinations of varied tendencies which can be added there, we arrive at the solution of "I

Fetis, in his Traite d 1harmonle, explores the concept of common tone modulation, which he calls an "intuitive attraction," because ", . , musical sense compensates for this implied harmony at the moment of the tonal change." (Bk, III, ch. iii, par. 270, pp. 180-81.)

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184

this problem*

A note being given, find some combinations and some har­

monic formulas so that it can be resolved in all of the keys and in their different modes,, Therefore, having come to the ordre omnitoniquer the art will have no more harmonic discoveries to make, at least following the construction of our scale. When the Baris Conservatoire of Music was organized in 1796* the most renowned professors of each branch of the art who taught, each according to his ideas and his method, were brought together, because there had not been time to prepare a main doctrine for a uniform education. This is why Rudolph gave lessons according to his empirical method, stripped completely of the capacity of analysis^

why Rey made his courses according

to the system of fundamental bassi why Langle developed the results of the theory which we saw set forth previously, and why Berton, freed from every consideration of system, used the practical method with his students.

It

was only some years later that this celebrated composer conceived his family tree of chords sued the dictionary which is its outgrowth,

p

Still we soon became aware of the inconveniences of this diversity of method and system in a school where doctrinal unity ought to be the foundation of instruction.

A commission composed of Cherubini, Gossec,

Martini, LeSeur, Mehul, Catel, Lacepede, Frony, and the professors who have just been named^* was appointed to begin in 1801 with a view to ■^Johann (jean) Joseph Rudolph (Rudolphs)], Theorie d’aecompagnement et de composition, a l'usage des elSves de 1*ecole natlonale de musique "(Baris i Naderman, 1779)* ^Henrl-Montan Berton], Traite d*harmonic base sur l*arbre genealoglque des accords (Barisi Mme, Duhan, 1804)i Dlctlonnalre des accords (3 vols.i Baris* Mme, Duhan, 1804), 3*Andre Frederic ELer (1764-1821) and Nicholas Etienne Framery (1745-1810) were also members of the commission.

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185 discussing and laying down the foundations of a system of harmony.

That

of Rameau, especially, was the object of serious examination, because it still had many followers in France, Mehul, named chairman of the commission, expressed it in this way in the last meetingi In a conflict of contrary opinions, supported by the partisians or antagonists of the system of fundamental bass, the commission, not being able to distinguish the whole truth, sus­ pended its judgement when the work submitted for our confirma­ tion came to terminate all the discussions, by offering a corplete system, simple in its principles and clear in its developments,1* (Fetis* italics,) This system which unified so many advantages, according to Mehul, was the one that Catel published shortly afterwards under the title Tralte d*harmonic adopts par le Conservatoire, pour servir a 1*etude dans cet etabllssement (Paris» 1802),

The influence which the Con­

servatoire already exercised in this period^* soon confirmed beyond all question what the most celebrated musicians of France declared to be what was better;

this was the coup de grace given to Bameau's system,

and the destruction of the latter was proportionally more complete and rapid because the remaining sectaries were excluded from public teaching at this time.

1*Arretes relatifs a 1*adoption de Traite d*harmonle as quoted in Charles Simon Catel, Traits d*harmonle adopte par le Conservatoire, pour servir a 1'etude d£ns cet etabllssement (Leipzig: Peters, 1&0?), p, ii.

2*

The influence of the Conservatoire extended beyond the borders of France and Europe* 30 years after the initial publication of Catel*s treatise* an edition appeared in America, This edition was A Treatise on Harmony, Written and Composed for the use of the Pupils at the Royal Conservatoire of Music in Paris; s , » From the English Copy with Additional Kotss and Explanations by Lowell Mason (Boston; James Loring,

1832).

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186 What then was this theory, so satisfying that it was adopted unquestionably by the most skillful French musicians, and which acquired immediately a vogue which had been the reward of Bameau only after 30 years of works, struggles, and multiple publications?

Catel discloses

it in a single phrase i "In harmony there exists only a single chord in which all how is

of the others are contained" (p. 5)*^*

it formed?

What is thischord, and

Here is a resume of what Catel says in thatrespect.

If we take a string which is tuned to the lowest g and if

of the piano,

we divide it in half, we find its octave Ql] j its thirdpart gives

the octave of its fifth fd]f its fifth the double octave of its third fbji its seventh, the twenty-first interval, or the double octave of its seventh ff *1; finally, its ninth, its twenty-third £interval]], or the double octave of its ninth fa'"], thus, from this division.

A chord composed of g-b-d-f-a results,

This chord is that to which in practice the

name "dominant ninth" is given.

It contains the perfect major chord

(g-b-d), the perfect minor chord (d-f-a), the perfect diminished chord (b-d-f), the dominant seventh chord (g-b-d-f), etc., and the leading-tone seventh (b-d~f-a).

By continuing the operation of the division of the

chord to the third octave, ire., starting from the sound 1/8 [~grl« we find the sounds l/lO, l/l2, l/l4, and l/l7, which produce the dominant minor ninth chord (g-b-d-f-a-flat) and the diminished seventh chord (b-d-f-a-flat).

All of these chords are natural and fundamentalj

by inversion of the intervals we can obtain the natural chords like them,

Peters edition, pff ^ Fundamental means root position.

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_

187 and which, the same as the fundamental ones, are attacked without * 1* preparation, as arising from the formation of tonality. Notice that the geometrician de Prony, who joined the commission, had no difficulty admitting the sound l/7 as the true f of a scale, and the sound 1/9 as a, although these proportions are not those of geometri­ cians for those notes, with the exception of Euler, whose memoir on the first of these numbers appears to have heen unknown or unnoticed by mathematicians.

2*

Then with the consent of de Prony, it is verified by

these words of the minutes of adoption imprinted at the front of the Traite d1harmonle1 "The citizen Catel develops his system of harmony. After a mature deliberation, the general assembly adopts it unanimously. . .

(Fetis’ italics.) The natural chords being found, as we have just seen, Catel

establishes that all the harmonic combinations other than the former are formed by notes foreign to the chords, which we call "passing notes," by 1* All harmony having its origin in the first nine partials of the monochord or sonorous body— perfect triads, dominant seventh or ninth, and the half-diminished seventh— constitutes "natural or simple harmony"* the dissonance does not have to be prepared. ^*Euler was one of the first mathematicians to employ logarithms for the calculation of intervals, Baron Gaspard Biche de Prony (1744-1839) lamented the lack of influence engendered by Euler’s Tentamen novae theorlae musicae, which was either unknown or scarcely known in France. ("Du rapport fait a l ’academie des sciences sur cet ovrage," in Baron Blein's Princlpes de melodie et d*harmonle dedults de la theorle des vibrations £paris, l§38j, p. xTT Following Euler, de Prony UBed logarithms to determine intervals, and in Instruction elementaire sur les moyens de calculer les intervalles muslcaux (Parisi~ Bidot, 1822) pointed out the advantage obtained by utilizing logarithms based on two for determining intervals. Moreover, de Prony also calculated a table of logarithms based on the twelfth root of two, a table used for equal-tempered timing, (larousse ds la muslque Cl957j* Ir 555*) 3*Arretes relatlfs a 1’adoption de Traite d*harmonle. p. ii.

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188 prolongations which suspend or delay the natural Intervals of the chords* or finally by the alterations of these same intervals,"*'*

As for the

substitution which I will discuss later, Catel did not see it at all, but he has a sort of intuition about it when he said, in reference to the analogy of the use of the diminished fifth chord (fifth and sixth minor) and of those of the leading-tone seventh and diminished seventhi

The similarity^* which exists between these chords proves their identityf and clearly demonstrates that they have the same origin (p. lif), Everything which has been said about the leading-tone seventh, as far as its relation to the dominant seventh, applies to the diminished seventh (p» 16)»

It is an idea, the results of which are fertile, since that of the search for the origin of the harmonies and of their analogy in the destination which they have conform to the order of tonal succession. If Catel had delved further into this consideration, he would have left nothing for his successors to do, because he would have found the complete system of which he has only denoted some parts. As far as prolongation, although he had not formulated the theory in a general manner, and although he had paid too little attention to some special cases, he knew the technique well in what concerns the consonant chords and some of the dissonant chords.

But the obstacle

against which some of the preceding theories had run aground still recurs in Catel’s and leads to a similar ruin.

This obstacle is, as we quite

1♦ chords

These harmonic combinations which are founded on the natural Catel calls "artificial or composed harmony,”

The two chords, the dominant seventh and the half-diminished seventh, have three common tones, and both of the chords resolve to tonic.

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189

expect, the minor seventh chord on the second degree and the harmonies which derive from it*

We have to recall that In the dominant major ninth

chord, produced by the division of the string, he found the perfect minor chord, d-f-aj therefore this chord exists, for him, on the second degree of the scale, although this Is not the one that Is located there In the determination of tonality*

According to him, in the succession of this

perfect chord to that of the tonic, if this tonic Is extended, it produces the seventh chord which it concerns.

But more difficulties arise here:

(l) the perfect minor chord on the second degree does not belong to the tonality, while the prolongation which produces the seventh chord is tonal*

(2) Catel can only show this alleged origin of the seventh chord

by writing in five-parts,^* in order to have it complete, which is an exception contrary to the principle of unity on which the whole veritable theory ought to rest.

(3) Lastly, the principle of the artificial con­

struction of chords by prolongation maintains that the prolongation coming to an end- the delayed chord occurs immediately.

Now, every prolongation

which produces a dissonance before inevitably resolves by descending a degree| the application of this fundamental rule can not find its place here, for if c (the seventh of d-f-a-cj descended to b, we would have a new dissonant six-five chord, d-f-a-b, which does not belong to the key, and which would be that of the fourth degree of the relative minor key. Catel understood this difficulty well, but not knowing how to get out of it, and not having boon able to find the true origin of the chord, he had recourse to this arbitrary rule whose falseness arises from itself, and which he expresses thus: l*Catel uses five-parts only for the root position seventh chord, and not for any of the inversions.

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190 The prolongation can he made, as well, on an already complete chord in which the prolonged note will not have any resolution! hut it should resolve, of course, in the following chord hy descending a degree.*1'* If Catel’s views had heen more general, and if he had known the technique of substitution and the combinations of the collective modification of natural chords, he might have avoided the stumblingblock against which a part of his system has shattered. If we search for what is original in Catel's theory and what was borrowed from his precursors, or at least what was said only after them, we will see that Sorge was the first to consider (in 17^5) consonant harmony and the dominant seventh harmony as forming the class of natural chords, but that the latter had been mistaken in classifying the minor seventh chord on the second degree in the same class, while Catel had seen quite well that he created an artificial harmony, although he did not discover the nature of the device.

Sorge also saw, however, that

some chords, notably those of the eleventh, of Rameau and Marpurg were only the products of the prolongation which formed artificial chords, but Schroter (in 1772) is the first who saw the seventh chord on the second degree is one of the chords of this class, although he could not say how the prolongation was effected.

Finally, Schroter was the first

who analyzed clearly the facts of the alteration of the intervals of the natural chords and the new aspects which they give to these chords.

If

Catel had had no knowledge of these author's books, he at least had only to renew what they had already published.

But what appertains to him in

■^*Catel, ^Peters edition], p. 16.

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191 particular is the view of the analogy of the major and minor dominant ninth chords, leading-tone seventh, and diminished seventh with the domi­ nant seventh chord.

Also, it is the order which he had put in various

parts of the system and, lastly, the analysis of the facts of practice where he demonstrated the skillfulness of a great musician.

Consequently, we are

not astonished at the general success which his theory achieved in France during the first 15 years of the nineteenth century* notice, on the con­ trary, how many motives seemed to he compelled to stand in opposition to a backward step that Reicha and some other harmonists have tried to have done to the science for 25 years. The great reputation which Reicha enjoyed in France and even in Germany in these 25 years obliges me to enter into more details than I would like on his system, which is, in a way, only the reproduction of old theories outmoded by the discoveries of Sorge, Schroter, Kimberger, and, above all, by Catel's method, Reicha, setting aside the consideration of the succession of chords which had caused such great strides to be made in the science since Sorge, and consequently of the phenomena of harmonic construction resulting from prolongation, returns to the system of isolated chords, of which he forms an arbitrary classification, following certain considerations which are peculiar to him.

His theoretic foundation is composed of 13 consonant and

dissonant chords, among which a certain number are primary, and the others the products of alteration.^-* From the first few facts by Reicha in the -*-*The 13 chords are grouped into three classifications* (l) triads— major, minor, and diminished* (2) seventh chords* (3) ninth chords. Those which are the products of alteration constitute a fourth group, and include +6, +6, 7r +5. Only the major and the minor triads are classified by

5

4+53

3 3 Reicha as consonant chords.

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192 exposition of his principles,1 we perceive an unquestionable confusion in the fundamental ideas, which throws it into a maze of a multitude of pecul­ iar facts, a fault quite singular with a man who had followed courses in philosophy, jurisprudence, and mathematics in Germany. The two primary chords of Reicha*s classification sure the perfect major and minor chords? the third is the perfect diminished chord (third and fifth minor), which he makes a dissonant chord.

In this, by the

classification of isolated chords, he differs from the other authors of systems of harmony which recognized as dissonances only the sounds which clash by seconds or, in their inversions and doublings, of the seventh and ninth.

What causes Reicha to rank this chord among the dissonant

ones is that, by the effect of the very construction of the diminished (minor) f-fth interval, there is a sort of attraction between the two 2* sounds which compose this interval. But he ought to have seen that this attraction is not so overbearing that it does not disappear in a succeeding modulation to this chord, which has no place with respect to the true dissonance, unless it takes the character of the leading tone through enharmonics.

The fourth chord of the classification is that of

the augmented fifth? but here the confusion of the author's ideas of the system come out already, because in the chapter where he deals with this chord, he admits this is only a perfect major chord altered in its

[[Aston 3oseph Reicha]), Cours de composition muslcale, ou Traite complet et raisonne d*harmonie pratique (jarisi Gambaro, n.dT * ----2*Since Reicha defines consonances as ", , . the intervals which produce for us an agreeable and sweet sensation, the effect of which leaves nothing to be desired," the diminished fifth has to be classified as a dissonance, (Reicha, Cours de composition [[Viennat Diabelli, n.d.J, p. 10.)

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193 fifth.1* The fifth chord is that of the dominant seventh, which he calls the "first kind"; then comes the sixth chord, which is this chord of the minor seventh with the minor third, object of so many errors for all the harmonists.

Reicha gives it the name of the seventh of the "second

kind," and limits himself to saying that this chord ", . .is employed mainly on the second degree of a major scale" (p. 36), without further worry about its original formation, than of that of other chords.2* The seventh with a minor fifth, called "third kind" by Reicha, and the major seventh or the fourth kind,3* the major ninth, and the minor ninth are equally considered by him as primary chords of the same rank, and although chords 11, 12, and 13 are only alterations of the chords derived from the augmented sixth, with a fifth and with a fourth, and the dominant seventh with an augmented fifth, he places them, nevertheless, in his fundamental classification. Such is the system which was very popular amongst some artists in Paris, because the professor who invented it lived down its shortcomings in the explanations and practical applications which he gave to his students.

But that is, all the same, the least rational theory which it

Fetis has distorted Reicha's statement. Reicha said that the alteration of the fifth made the chord dissonant; the alteration and resultant dissonance automatically preclude its inclusion in the first classifi cation, 2*

Reicha adds a footnote to this sentencei the third and sixth degree of the same scale."

"It is also found on

3*0nly the seventh chords of the second, third, and fourth kind need preparation; the first kind (dominant seventh) may be taken without preparation because it determines tonic and is ", . . the most pleasant of the dissonant chords after the diminished chord." (Reicha, [^Diabelli edition], p. 41.)

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19^ might he possible to conceive, and the most deplorable return to the flagrant empiricism of the early methods at the beginning of the eighteenth century*

This system annuls the good that Catel's method had done in

France, and opens the door again to a multitude of false theories which had been produced in this country and elsewhere for some years.

Propor­

tionally more dangerous because it was supported by a name justly esteemed in other parts of the art, it calls into question again what was decided by the authority of intellect and experience, and formed partisans who declared it a genius' conception, while in reality it could have led to the destruction of the science, if it had not found in its path a theory both scientific and experimental, of which I will speak shortly, and which averted the harm which Reicha's false system could have produced. The systems of Catel and Reicha first attracted my attention amid those which France had seen bora in the firs'. 20 years of the nineteenth century, because these are the ones which were most successfulj it remains for me to speak of some tentative facts, about the same time, in order to challenge the adoption of other systems which, although announced with more ambitious pretentions, have not had the same repercussion. The first of these systems, in order of dates, is what its author, de Momigny, set forth in a book entitled

Gours complet d'harmonie et de

composition, d'apres une theorie neuve et generale de la muslque, Basee stir des princlpes incontestables, pulses dans la nature, d*accord avec tous les bons ouvrages-pratiques, anciens et moderaes, et mis, par leur clarte, a la portee de tout le monde (3 vols.j Paris* 1806),

By the author,

Since then, and until 383^, de Momigny reproduced or expounded

upon his system in polemic writings where he treats his adversaries

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195

with haughtiness, and in various works'** which have not heen popularly received. Starting from the point of view of Levans, Balliere, Jamard and Sorge for the pursuit of the bases of construction of the scale, de Momigny, following the arithmetical progression, finds them in the divisions of a sonorous string which gives the resultant scalei £, d, e, f, g, a, b-flat;^* but inasmuch as this scale does not conform to that of the music of modem Europeans, and that the b-natural is found only in the fifteenth division of the string, de Momigny, in lieu of adopting, as Levens and his imitators, an eight" note scale with b-flat and b-natural,

does not think to consider the

string as tonic, but as dominant; hence his scale is g, a, b, c, d, e, f. He enumerates at length the advantages which result from the position of the tonic in the middle of the scale, ", . , as the sun at the center of our planetary system."*** For example, find Jerome Joseph de Momigny, La seule vrale theorle de la muslque, utile ei ceux qul excellent dans cet art, comme ^ ceux qui en sont aux premiers elements, » , . (Paris* The author, 1821);EncyclopSdle mgthodlque, Muslque publlee par MM, Framery, Ginguere et de Momigny (2 vols.j Paris, 1791-1818)} Cours gfenfaal de muslque de piano, drharmonie et de composition, depuls A .jusqu*a Z, pour les feleves, quelle que solt leur InfSriorlte et pour tous les musiciens du monde, quelle que solt leur superior!t6 reelle (Parisi The author, 1834). ^*Momigny, following the arithmetical progression, begins on GG and extends the progression to the fourteenth term, f ''*t the resultant scale, "the true scale," extends from g ’ to f ’*. This is "the true scale," because not only is it given by nature, but the two tetrachords which compose it are regular and symmetrical* g* to c’’ and c** to f ’', and tonic is in the center. (Momigny, Cours complet d'harmonie. p» 26,) Momigny explicitly rejects an eight-note scale with a b-flat and a b-natural, because . this would demand that we admit a part of the chromatic genre in the diatonic genre" (p. 27), ***Ibid., p, 26.

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196

the two semitones in the Beven notes without resorting to the repetition of the first at the octave, divide the scale into t~'o just fourthsr and have the semitones in the same place in these fourths» because one of the most severe objections of de Momigny against the scale form com­ mencing on tonic bears on the major fourth or tritone, which the fourth and the seventh note create between them, not noticing that it is precisely this relationship which is basic to tonality, and which leads to the final conclusion of all melody and all harmony. The division of a string, considered as a dominant, led. de Momigny, in what concerns harmony, to the same results that Catel had obtained by i V same means, but,whatever his pretensions are in this respect, he exposes them with a great deal less clarity.

In this way,

like Catel, he arrives at the formation of perfect chords and of those of the dominant seventh and leading-tone seventh, which he views as the sole natural chords.

But as for the others, in lieu of explaining

what devices create them, he declares them "chords which are not," and names them "discords in major" and "discords in minor," in such a way that the exact analysis of a harmony devised from many prolongations jo^ued with alterations of different kinds would become impossible for anyone who had read only the fastidious explanations of de Momigny, Moreover, nearly all of the examples which he gives of the use of chords are poorly written, and prove that this author had only confused notions of harmonic usage. If one duly examines de Momigny*s pretensions at originality, he will admit that he borrowed the arithmetical progression from Levans, Balliere, and Jamard, the transfer of his fundamental tone of the scale

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197

to the dominant from Sorge, the division of the string from Catel, in order to arrive at the primary harmony* the combinations of thirds in order to form the natural chords from bangle, and the progressions of fourths and fifths for the formation of scales from Abbe Roussier* Really, everything belongs to the one who raised his voice so high for

30 years in favor of a system rejected by the musicians! there are some who did not fall to notice the justness concerning measure and rhythm, and a pretentious phraseology arising from neologism, I do not think X have to examine, at this point, the efforts made by

Rey,l

an opera musician, and his n a m e s a k e t o reestablish the system

of fundamental bass some years after the publication of Catel*s bookf barren efforts could not revive a theory whose mission had ceased*

I will

observe the same silence towards a pretended Theorie muslcale Imagined by Emy-de-Lylette,3 an ill-digested extract of Lirou's ideas, and which is no less defective in its applications than in its systematic conception* There is more merit in the book which G* L* Chretien published under the title

la muslque etudiee comme science naturelle. certalne. et comme

art, ou Grammalre et dlctlonnalre musical (Paris*

1811),^ This book is

^Jean-Baptiste Hey], Exposition elementaire de l*harmonle* Theorie generals des accords d*apres la basse fondamentale (Paris* 1*auteur et Nadermann- I8O7)-

Chez

^ V * F, S* Reyj, L rart da la muslque theorl-physio-practique general et elementaire* ou exposition das bases et des developpements de systems de la muslque (Paris* Godefroy, 1806)*

3[Antoine-Ferdinand Esy-de-Lylette], Bieorie muslcale* contenant la demonstration methodique de la muslque, a partir des premiers elements- de cet art jusques et comprls la science de l'harmonia (Paris, 1810). volume of text, and a cahier of plates with a mobile clavier*

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198

a sort of regeneration of Rameau’s systemr but considered from a new point of view*

Following the ideas of Chretien, all music resides in the

phenomenon which expresses the perfect major chord in the resonance of certain sonorous bodies*

All the theories based on the divisions of

the monochord and on the calculations of geometricians for the ratios of intervals are false, he says, because the mono chord and the calculations can only have a force of inertia* and not a generant force. The monochord and the calculations can serve to measure and verify the justness of the intervals, but they would not know how to generate either a scale or a harmony (p. 42),

Observe this thought whose correctness is incontestable,

and one can ccnsider [it] as new, although it is only a positive expression of the vague theory of the Aristoxenians.

Unfortunately, it does not lead

Chretien towards the consideration of the metaphysical origin of this scale and this harmony, object of his quests} his enthusiasm for the phenomenon of harmonic resonance does not allow him to see that this phenomenon can not be more than the division of a string, the principle generator of tonality.

Moreover, the way this theoretician pursues an

argument is not very demonstrative, because he almost always is content with some affirmations. Re says, I affirm that from this point of view [that of his theory], quantity of precepts and doctrines which seemed 'unintelligible will acquire a lustre which was largely found overshadowed (p, 9)* They [the geometric theoreticians] have meant the truth} I love to believe it, but many have deceived themselves, I affirm it (p. 13)* I affirm that this natural way and inspirator which he [BameauJ called the resonance of the sonorous body, and which I call the phenomenon of harmony, was at all times the unseen cause of the music properly said, and that it is the only foundation to which musicians ought to turn their attention (p. 40). (Fetis*italics.)

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199 Let us examine what results from all of these affirmations of Chretien, There is only one chord, the perfect major chord, produced by the harmonic phenomenon.

The -perfect minor la created by analogy with this

unique chord by lowering, with the consent of the ear, the third a semitone, The perfect major chord, dissonant from the seventh, produces the dominant seventh. The perfect minor chord, dissonant from the seventh, generates the seventh on the second degree. The perfect major chord, dissonant from the sixth, produces the six-five chord on the fourth degree (fundamental following Rameau's system). The perfect minor chord, dissonant from the sixth, produces the six-five chord of the minor mode. Every interval of a semitone holds the place of a tone and represents it.

It is thus that the dominant seventh, arising from the dissonance

added to the perfect major chord, being altered in its lower note, gives birth to the diminished seventh chord which represents it.

It is thus

that all the harmonies of the six preceding chords are formed, regardless of the aspect in which they appear. It is regrettable that in his love for the harmony given by nature, Chretien did not learn that there are some sonorous bodies which utter, in the midst of all their resonances, the perfect minor chord} that there are others which give the constitutive interval of the dominant harmony, and even the altered harmonies,

Vith this help, he might have greatly

simplified this theory.

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200

However that may he* in the state where he left it, there is only the perfect major chord in nature.

But the perfect major chord, c-e-g,

gives only three notes of the scale, and Chretien taught us that we have to find the diatonic scale in the means which nature offers us*

This

difficulty did not stop him for long, because^ having need of the perfect major chord of the tonic, fourth degree, and dominant in order to form his consonant and dissonant harmonies, he simply takes three sonorous bodies of which one gives £-£-£, Mother f-a-c, and the third g-b-df volla, £, d, e, f, g, a, b, c.^* After so simple a thing, he only has to "affirm" that a minor third is included in the resonance of the perfect major chord, since in f-a-c the third, a-c, is foundj it is only necessary to add the fifth, e, originating from the perfect chord c-e-g, to have the perfect minor chord, and consequently in order to find in these combinations the types of modes, major and minor.

Such are the

consequences where one can arrive with a strong faith in any one fact, proven or only perceived, Ghoron is one of the French musicians who, with the most perseverance, turned his attention to the theory of harmonyi yet I can not attribute any particular theory to him, because his opinions have been in incessant fluctuation from the publication of bis Jftrlnclpes

Chretien does not "simply take three sonorous bodies" t like de Lirou and Hameau, he generates them from a fifth, ", « • the basic Interval of perfect chords" (p, 76). Thus the fundamental note of the perfect major chord c-e-g is the generator of the perfect chord g-b-d, and the fifth of the perfect chord f-a-c, (Chretien, pp» 76-79*7 With this manipulation of sounds, Hameau, de Lirou, and Chretien destroy c as tonict when a perfect fifth (f) is added below tonic (c), it becomes the generator of the series of fifths.

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201 /

|

d*accompagaement des ecoles d1Italle in the space of 30 years.

p

to his Manuel de Muslque,

namely,

Cancelling in his Princlpes de composition des

ecoles d*Italle^ what he had done in the preceding work, establishing an eclectic doctrine which seems to he his last word, then adopting Alhrechtsherger' s method in which he criticized harshly enough, however, the details in the notes of his translationj next getting enthusiastic for a new theory which he thought he had discovered, giving it up for a had job before the printing of his work was complete and stopping publica­ tion! lastly, returning to Marpurg* c false theory towards the end of his life and making it the basis for his Manuel de muslque. although he also made a stinging criticism of the details of this doctrine in multiple and extended notes. harmony.

This is what was Choron’s career in the theory of

Doubt had thus tormented his mind with regard to the existence

of a complete and rational system of harmony, and his works are, in a way, cancelled in this history of this science. With respect to Germany, since the commencement of the nineteenth century, we find a multitude of methods and writings relative to the theory of harmony, but in -Lie midst of so many volumes published on this matter, there are few which merit a place in the history of progress and divergence of this science, so difficult to coordinate in all its parts, according to a clear principle*

Among those which have found some

-'-Parisi Imbauit, 1804. ^Etienne Alexandre Choron], Nouveau manuel conplet de muslque vocale et instrumentale, ou Encyclopedia muslcale (6 vols.f Parisi Roret, 1836-39). ■^Parisi

le Due, 1808.

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202

followers, I will cite only

(l) Princlpes d’harmonle^ of Schieht,

director of the Thomasschule in Leipzig, in which the dominant of the key is considered the foundation of the perfect major chord* g-h-dj the seventh chord, g-b-d-f, from which he extracts the perfect diminished chord, (b-d-f)| the ninth chord, g-b-d-f-a, from which he obtains the lead*ng-tone seventh (b-d-f-a) and the perfect minor chord (d-f-a)j the eleventh chord, g-b-d-f-a-c, from which he extracts the minor seventh chord (d-f-a-c)j and finally, the thirteenth chord, g-b-d-f-a-c-e, from which he obtains the major seventh chord (f-a-c-£). The chromatic alteration of the intervals of these chords completes the empirical system of harmony conceived by Schicht» (2) Still more strange to the conception of a veritable theory of harmony are the books of Preindl,^ cantor of St. Stephens in Vienna, and of GSroldt,3 director of music at Quenlinburg.

These books, where the

chords are all considered individually, can only be ranked in the class of practical manuals. (3 ) It is not the same for Traite elementaire d'harmonie^ of Friedrich Schneider, today Kappellmelster in Dessau and distinguished

•^Johann Gottfried Schicht}, Grundregeln der Harmonle und dem Verwechslungssystens (Leipzig* Breitkopf und Hartel, 1812). ^Joseph Preindl}, Wiener Tonschule. Oder Blementarbuch zum Studlum des Generalbasse. des Contrapunkts, der Harmonle-und Fugenlehre (2d ed.j Viennae Haslinger, I832). 3[johann Heinrich Goroldt}, Grundllcher Untsrricht im Generalbasse und in der Composition oder. deutliche ErklSrung von den T^ggn? Tonarten. Intervallen. Accorden, Harnonien und Melodien (2 vols.i Quendlinburg und Leipzig* Ernest, I832). \ Johann Christian Friedrich Schneider}, Blementarbuch der Harmenie und Tonsetzkunst (Leipzig! Peters, 1820; 2d ed,j Leipzig! Peters, 1827).

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203 composer*

Now* there is a theoryr a false theory to tell the truths whose

origin is found in the hooks of Vogler and which Gottfried Weber had previously developed in the work which will he spoken of in a little while.

According to the fundamental principle of this doctriner the

perfect chord and the seventh chord are made on each of the notes of the scale where they are present! as far as the nature of their intervals* they conform to the construction of the key and mode* having* because of the note where they are placed, either a major or minor third* either a just or diminished (minor) fifth, either a major or minor seventh.

It is

the same for the ninth chord* and it is only a question* in order to complete the nomenclature of the chords* of altering its various intervals* I have just mentioned the name Gottfried Veher* whose system caused considerable stir in Germany for about 15 years, and has been nearly abandoned today.

The work* which he published in 1817, has Essal d*une

theorie systematlque de la composition^ for a title* and the success was such that it was necessary to make other editions in a few years*

We have

just seen from what angle the chords were considered* but what distin­ guishes Weber’s book from all those of the same kind is the care which the author himself takes to damage confidence in this theory,

*.n every

other one* declaring that he does not believe in the existence of a system with which all the facts of the harmonic experience would agree*^# so

Mainz:

^Versuch einer geordneten Theorie der Tonsetzkunst (3 vols*! Schott, 1817-21)*

Weber believes that the musical art is not suited for a systematic establishment! he labels his theory geordneten to avoid ", * . the pompous title of system” (p, xiii), Furthermore Weber contends that any "system" which resorts to categories called "exceptions* licences* ellipses* etc." to explain incongruent phenomena is very pretentious and rendered vulner­ able. (Weber, Versuch einer geordneten Theorie der Tonsetzkunst [[4 vols., 2d ed*f Mainz? Schott. 1824_J. I. xiii*)

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204 thatr according to him, the hest work concerning harmony is that one which contains the greatest number of these facts in the analysis! thus it was to extend this analysis of which he had made a point as far as it was possible in his book.

This erudite was not aware that his thesis

took the science back to whet it was in the time of Heinichen and Mattheson, and that by thus undermining the faith of his readers in the possibility of a principle of the science, he was destroying the science itselfi for what would a science be, if it were composed only of isolated facts from which it would be impossible to establish methodical concate­ nation?

Unquestionably it is to Weber*s ambition to deny the possibility

of a rational theory which brought about the premature renunciation of his own, and the swift reaction which took the Germans from admiration to indifference for his system,1* Now this long analysis of what has been done since the commencement of the sixteenth century for the creation of a science of harmony, and especially since Hameau laid the foundation, should end.

In summarizing

it, we find that all the systems have had one of the six methods from the following facts for a principle*

(l) the harmonic resonance of

sonorous bodies or, more generally, acoustical phenomena of different types! (2) the arithmetical progression determined by the harmonic series of the horn or the trumpet! (3) the triple progression! (4) the division of the dominant monochord according to arithmetical progression! (5) the arbitrary construction of chords by the addition and subtraction of thirds! (6) lastly, the arbitrary placement of certain model chords on

•^Weber’s main contribution was a syste.-a of analytical symbols, (Mitchell, p, 128.)

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205 all the degreeb of the scale*

Therefore, it is evident that all these

systems more or less derive from sources which are not tied Intimately to the music itself, i«e*r to the art as It appears In Its Immediate consequences, and that In all [[the systems] It has been necessary to a certain point to adjust this art to the strange principle which was given to it* The only thing which we have not thought of directly is the quest for the principle of harmony in the music itself, i.e., in tonality. But what is "tonality"?

However foolish this question may seemingly be,

it is, nevertheless, certain that few musicians would be able to answer it satisfactorily.

For me, I will say that uCuolity resides in the

order in which the sounds of the scale are placed, in their respective distances, and in their harmonic relations.

The composition of chords,

the circumstances which modify theJi, and the laws of their succession are the necessary results of this tonality.

Change the order of the

sounds, or change their distances, and most of these harmonic relation­ ships will be destroyed.

For example, try to apply our harmony to the

major scale of the Chinese, f, g, a, b, c, d, ej to the minor scale of ancient Irish music, a, b, c, d, e, f-sharp > or to the incomplete scale of the Scottish Montagnards— the successions of this harmony will become unworkable in these tonalities.

Indeed,what makes a combined harmony as

that of our music in a major scale whose fourth degree is a semitone higher than in our scale of the same genre, and is only separated from the fifth note by a semitone, so that the attraction which exists between the fourth note and the seventh of our harmonic scale and constitutes the dominant harmony

here is between the tonic and the fourth degree, and

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206

consequently makes every final cadence impossible?

What makes a harmony

slntilar to that of our minor mode in a minor scale whose sixth degree is higher than ours by a semitone, and which does not have the seventh note? It is evident that these things are not at allmade to go together,

The

Irish airs which have been published in the anthologies of national airs are in the major mode, or belong to modern times, which

has

allowedthem

to be harmonized as best they could | it is thesame for the Scottish airs and those of the Gaelic countries, whi^h- moreover, are often accompanied at the octave or with a pedal, because their tonal character does not allow the cadential action of our harmony to be used.

The strange

character that we notice in these airs does not result from the whim of their composers, but from the scale which they have used. What I call tonality, therefore, is the succession of melodic and harmonic facts which results from the disposition of the distances of the sounds in our major and minor scales| if only one of these distances was inverted, the tonality would assume another character, and all the differ­ ent occurrences would be manifest in the harmony. quences of this tonality are

The immediate conse­

(l) to give to certain notes a character

of repose which does not exist at all in the others, and to designate these notes as the terminal points of cadences, i.e», the perfect chordj such are the tonic, fourth, fifth, and sixth degrees.

This deprives the

third and seventh degree of this character of repose, and consequently excludes the perfect chord from them.

(2) It assigns to the relation of

the fourth and seventh degree a resolutory attraction which gives the dissonant harmony of the dominant its own character, and obliges it to be resolved by a perfect or imperfect cadence, or to be followed by a

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207 modulation! for there

1

b

no middle course for the harmony of the dominant-—

it must resolve either in the cadence or in modulation.

(3) The rules

which forbid the immediate successions of fifths and major thirds are also the results of tonality* because two successions of fifths* ascend­ ing or descending, and two major thirds have the disadvantage of putting two tones which have no analogy between them in immediate touch.

All of

this, I repeat, derives necessarily from the form of the major and minor scales, and constitutes what we call the laws of tonality. But, we ask, what is the principle of these scales and what has dictated the succession of their sounds, if not the acoustical phenomena and the laws of calculations?

I reply that this principle is purely

metaphysical! we conceive this order and the melodic and harmonic phenomena which flew from it through a consequence of our conformation and our education.

It is a fact which exists for us by itself and

independently of every cause extraneous to us.

Well then, we would not

want to concede that it satisfies our instinct to join with experience in order to place in a scale the bases of the pleasure intended for our intelligence* and we will search in some ignored accoustical phenomenon for the secret cause of this organization of tonality made for our use! Notice that these acoustical facts, poorly analyzed at first* do not have the import that one thoughtlessly accords them.

For example* the

production of the harmony of the perfect chord, which we observe in the resonance of certain sonorous bodies, is accompanied by many other resonances.

It is the same with respect to certain other bodies which

produce other harmonies.

Moreover* experience has proven that different

modes of vibration accorded to the same bodies give rise to diverse

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208

phenomena*

Mr. Troupenas has shown (Revue muslcale„ XII, 125) that the

interval of a trltone discovered by Baron Blein in the resonance of a square metal plats struck on one of its comers is no different from the result of the vibration of this plate in the direction of its diagonal, whereas the vibration in the direction of one of the sides of this plate gives rise to other phenomena.

Let us supposer in order to give the

greatest possible extension to the alleged natural bases of harmony, that in the course of time we discover some acoustical phenomena which give all, the possible harmonies to our systemj are we to conclude that these ignored phenomena are the origin of these harmonies found a priori by great musicians?

Truly this would be an odd encroachment of the action

of hidden causes supported by certain sophists upon our determinations, and this would be a rude blow dealt to our philosophic liberty*

To be

5*0X6 , when Monteverdi found the dominant harmony which changed the character of music and constituted our tonality in major and minor modes, always uniform, whatever the key may be, the existence of the diagonal vibration of the plate was nought for him and was determined only by his instinct and by certain analogous observations.

His audacious thought

did not create the fact, but discovered it, and the principle which guided him is absolutely metaphysical. Shall I speak of the acoustical phenomenon of the harmonic series of the horn and the trumpet, which coincides with the arithmetical pro­ gression?

It furnishes, it is true, the elements of a false scale which

is not ours, and we have seen what Levens, Balliere, and Jamard were able to do with it.

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209

Shall I speak of the division of the monochord, by introducing there, at the seventh term, the number acknowledged necessary by Euler, as Catel and de Momigny have done?

It contains the harmony of the natural

chords, but by stopping at the latter, one has neither all the tones of the scale, nor the elements of a tonality.

In order to attain these, it

would be necessary to extend this division to all the sounds, g-b-d-f-ac-e, as Schicht has done| but then the natural and artificial chords will be confused, and the rational classification of these chords will no longer exist.

Shall I speak of the purely harmonic progression?

It

provides the exact measure of the unchanging intervals of the tonality of plainchant, where no talented interval of attraction exists, but it can not lead to the formation of a scale.

Moreover, had the acoustical

phenomena and the calculation given the elements of our tonality, they would not at all provide the order in which they must be ranked in order to compose this tonality, and we have seen that this is where the radical difficulty resides. If it is recognized that these foundations of the system are deceptive, that they have misled all those who have taken them as a point of departure, and that they are powerless to support the edifice of tonality, it is evident that there remains no other principle for th~, construction of the scale and the tonality than themetaphysical principle f a principle both objective and subjective, a necessary result of the fsel“ ing which perceives the relationships of sounds, and of the intelligence which measures them and deduces their results.

After so many centuries

of study done in absolutely opposite directions, we manage to recognize that the Pythagoreans were mistaken in attributing to numbers a basis for

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210

tonal construction which does not belong to them, and that the Aristoxenians were no less mistaken in attributing to the ear a faculty of comparison which it does not have*

The ear perceives the sounds) the

mind compares their relationships, measures them, and determines the melodic and harmonic conditions of tonality. This laid down, the science of harmony is all done, because this science is nothing different from the systematic expose of the art. Tonic appears through the absolute feeling of repose which is felt there, and the dissonant harmony of the dominant finishes by giving it this character by its attractive resolution on the consonant harmony of this tonic. The fourth degree of the scale, the fifth, and the sixth are also recognized as notes of repose by the faculty of the determination of the subordinate cadences with which these notes are provided) the consonant harmony, i.e., the perfect chord, then also forms a part of them.

These

harmonies conforming to the key and the mode are major or minor because of the natural state of the notes.

The third and the seventh degree,

which are separated only by a semitone from their upper notes and because of that have attractive tendencies, can neither be considered as notes o2 repose, nor consequently support the harmony of a perfect chord which has a conclusive character.

Following tonal order, they can only, therefore,

be accompanied by derived harmonies.

The second degree of the scale,

able to be the conclusion of cadential activity only in a progression,

1*

has only a character of equivocal repose; thus it happens that the harmony of the perfect chord does not belong to it in the ascending and descending 1* A "progression" is a sequential pattern.

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211 harmonic series of the scale, and that this note is accompanied in the same formulas only hy a derived harmony. Of the natural fundamental chords, there are only the perfect chord and that of the dominant seventh.

Following Rameau’s beautiful

discovery, admitted into all the systems of harmony, the other natural harmonies derive from the former through the inversion of the intervals of the fundamental chords. With the natural fundamental and derived harmony, all the harmonic tonality is established, and the faculty of modulation exists.

All the

other harmonic groups which can affect the ear are only the modifications of these natural chords.

On the one hand, these modifications have

variety of sensations for their aimj on the other hand, they establish a greater number of relationships between the various keys and modes.

The

modifications of the chords consist in the substitution of one note for another| in the prolongation of one note, which delays an interval of the chordj in the ascending and descending alteration of the natural notes of the chords| in substitution coupled with prolongation, in alteration coupled with prolongation, in collective ascending and descending altera­ tions, in the anticipation, and in passing notes. Substitution occurs only in the dominant seventh chord and in its derivatives,-1-* The substituted note is always the sixth degree, which takes the place of the dominant,

Thus, when the seventh chord is written

in five parts, namely, £-b-d-f-g, if one substitutes a for g in the upper part, i.e., the sixth degree for the dominant, one has the dominant ninth

-1-*For details, see Fetis, Traits d*harmonle, Bk. II, chs v, pp. 46-58.

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212

1#

chord, which,conforming to the mod*, is major or minor.

If a similar

substitution is made in the first inversion, b-d-f-g, one has, in the major mode, the leadlng-tone seventh (b-d-f-a) and the diminished seventh chord (b-d-f-a-flat) in the minor mode. derivatives.

It is the same with all the other

What shows the analogy of the chords and the origin of their

formation is the identity of their use and of their tonal resolutions, Catel indeed saw this identity and ascertained the facts of the substitu­ tion of a chord in his analogy, but he did not know the technique of the substituted note.

This technique is very important, since it leads to

the demonstration of the origin of certain other chords which have been the stumbling block of all th6 theories. In the succession of two chords, every note ascending or descending a step can be prolonged into the following chord, where it delays the normal construction.^*

If the prolongation produces a dissonance, it

ought to be resolved by descending, like every dissonance which is not a leading tone; if it is a consonance, it effects its movement by ascending. It is thus that a prolongation which delays the octave in a perfect chord produces a nine-five-three chord; that that one which delays the third produces a five-four chord; that the retardation of the sixth in the first inversion of a perfect chord produces the seven-three chord; that that of the sixth in six-four chord produces a seven-four.

It Is, more­

over, in this way that the retardation of the third of a seventh chord

-*-*Fetis is vulnerable to the same objection he raised against Catel*s origin of the supertonic seventh: he can show its alleged origin only by writing a five-voice chord. ^*For details, see Fetis, Tralte d'harmonle, Bk, II, ch. vi, pp. 59-76,

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213 produces a seven-five-four chord! that the retardation of the bass note in the first derivative of this chord produces a five-four-two chordf that the retardation of the sixth in the second derivative creates a seven-four-three chord! lastly, that the retardation of the major fourth in the last derivative of the seventh chord produces a six-five-two chord. Notice that in the seventh chord and its derivatives, it is always the tonic which delays the seventh note.

Except for substitution, retardation

does not change the destination of the natural chords, and the use of the latter remains Identically the same after the prolongation is resolved. If the circumstances of the substitution are coupled to those of prolongation,-*-* one has a nine-seven-four chord for a combined modifica­ tion from the dominant seventh chord,

[[See Example 1, p. 214,3

Fbr that

of the first derivative, a six-four-two chord! for that of the second derivative, a chord of a minor third, fifth, and minor seventh! finally, for that of the last derivative, a six-five-three chord.

These combined

modifications do not alter the destination of the natural chords! this destination remains the same as the resolution of the modifications. Thus, such is the origin of these seventh chords of the second degree, of the fifth and sixth, etc., an origin which has been the stumbling block of all the theories of harmony, because their authors did not know the technique of substitution and of the collective modifications of chords.

1* Ibid,, ch, vii, pp- 78-89.

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214 Example 1* 1,

Substitution and Prolongation,

Natural harmony

2,

Substitution

3*

Prolongation

'Tf HE 4,

£

Union of substitution and prolongation

iS T

Each ascending or descending note of the interval of a whole-tone in the succession of two chords can be altered a semitone.

The ascending

alterations can be effected by the addition of a sharp or by the cancella­ tion of a flat, each descending alteration by the addition of a flat or by the cancellation of a sharp.

Each note affected by an ascending alter­

ation takes on the character of an accidental leading-tone v and is resolved inevitably by rising. The alterations give birth to an immense quantity of alterations of natural chords, and are combined with simple substitution, prolongation, and substitution coupled with prolongation,

2*

i#Pctis, Bk, II, ch, vii, p, 77* 2* For details, see Fetis, Bk, II, ch, viii, pp. 89-104,

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215 The ascending and descending alterations can be prolonged in the5 succession of two chords.^-*

When the prolongation is that of an ascending

alteration, it ought to be resolved by ascending, although dissonant, because the character of the attraction resulting from that of the accidental leading tone absorbs that of the dissonance. £See Example 2 J

Example 2.

Prolongation of altered notes.*'

Prom these complex modifications of the natural chords, some multiple affinities result, which put all the keys and their modes in touch, fulfill the last period of the development of harmony which I have just indicated by the name ordre omnitonlque. and furnish the solution to this problem:

A note being given, find the combinations

and the harmonic-formulas such that it can be resolved into every kejr and into their various modes. They also generate a great number of new chords not yet employed by composers, and whose form, destination, and use I have determined a priori by analysis.

1*Ibld., ch, ix, pp. 104-12, 2* m d ». Bk II, ch. ix, p. 105.

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216

The anticipation Is a device by which one hears In a chord one of the notes of the chord which ought to follow it} this device is always melodic, because it Is the melodious part which uses it, Passing notes are those which, too rapid or of very little meaning in the shape of the melody or of the accompaniment for each to have a particular harmony, are nevertheless necessary for the completion of these shapes.

The ear accepts the usage of these particular expletives

of harmony, provided their movement takes place by conjunct pitches whan they are foreign to the chords. There are harmonic formulas called "progressions*' or "sequences,"" because the bass makes a series of similar movements, such as to rise a second and descend a third, to rise a fourth and descend a fifth, etc. In these progressions, one places on each completed movement of bass notes the same chords which accompanied the first.

There are some of these

progressions which modulate with each completed movement £real or exact sequence]]} there are others which do not modulate []tonal sequence]]*

In

these last, the mind suspends any idea of tonality and of conclusion until the final cadence, so that the scale degrees lose

their tonal character,

the ear being preoccupied only with the analogy of movement.

It follows

that, in these non-modulatory progressions, any one of the chords can be placed on any one of the notes.

Thus in a progression which rises a

second and descends a third, one will alternately put the perfect chord and the chord of the sixth on each of the notes} whence it will happen that the perfect choixL, being placed on the seventh degree, will have a

For "sequence" the French use the term marches de basse} each recurrence of the sequential pattern (modele) is a progression.

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217 minor fifth,

In this way, moreover* in a progression rising a fourth and

descending a fifth (beginning with the seventh chord on the dominant), one will place the seventh chord on each of the degrees, and it will happen from this similarity of movements and of harmony, that the chord which it concerns will be composed of a major third, just fifth, and major seventh on the fourth degree and on the tonic| of a minor third, just fifth, and minor seventh on the third degree and on the sixth.

Such

is the origin of Vogler's, Weber's, and Schneider's theory which places the perfect chord and the seventh chord on each note of the scale, although actually such a use of these chords would destroy any sense of tonality if it were done somewhere else than in the non-modulating pro­ gression, where tonality is actually destroyed until the final cadence. Having reached this point, the theory of harmony is the last expression of the art and the science) it is complete and nothing can be added to it.

It is this theory of which I have given a resume in my

Methode elementaire d*harmonle et d 'accompagnement,^ and of which my large Tralte d'harmonle^ contains the developments,

Hameau, Sorge,

SchrSter, Kirnberger, and Catel found the first elements, and I have completed it by putting it on a solid base,

What shows its excellence

invincibly is at the same time the history of the progress of the art and the best analysis of the art of composition.

1Faris, 1823) (2d ed,) Paris*

V. Lemoine, 1840).

^Now in press.

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APPENDIX

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219

Appendix A A Note in Reply to That of F6tis In the intention of its author, this note [Fetis*] was destined to refute the ideas which I have expressed on the origin of our tonality in a conference held some months ago by the Society of the Composers of Music, and of which the resumes have appeared in the Menestrel (issues of 8, 15, and 22 November 1868),

As a matter offact, my learned contradictor

restricts himself to repeating once more his favorite thesis* Invention of the Dominant Seventh Chord by Monteverdi."

"The

Although this

is only one of the small aspects of the problem which I have dealt with, I accept the debate on the narrow ground that it has pleased Fetis to choose, and I am going to furnish some new proofs in support of the arguments which I am producing. These arguments, according to what Fetis says, are not new) I know nothing of them.

Certainly his are no longer new, :nd I do not know that

he has added an iota to them since 1835* May I be so bold

as to state precisely once more the point under

discussion?

Monteverdi, yes or no, is the oldest composer with whom one finds the dominant seventh in a form other than that of syncopation? yes. Other more or less competent musicians say no . . .,

Fetis says

and to the

affirmation of Fetis, I offer the following facts * First objection. The constant use, from the twelfth to the seven­ teenth century, of the second inversion of the chord in question 1* Illegible printing.

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220 (d-f-£g[]-b)f with suppression of the fourth (fundamental of the chord)«, This assimilation is not admitted "by my opponent, who sees in the aggre­ gation d-f-b only a simple sixth chord, insignificant from the tonal viewpoint. r . . WhatI Insignificant from the tonal viewpoint, the chord which contains the tritone, the famous dlaholus in muslca of the middle ages5 Is it not Fetis who writes in the Traite d ,harmonie (par, 25, p, 9)t It is noteworthy that these intervals (~f-b, b-f[] charac­ terize modern tonality through the energetic tendencies of their two constitutive notest the leading tone calling after it the tonic, and the fourth degree followed generally by the third. Although it may be there, Fetis finds my argument absolutely absurd, and, in an appeal a little tneatrical for the occasion, he calls the musicians of all countries to witness my ignorance!

I

suppose that he excludes from his appeal the greater part of harmonic theorists who, on this point, completely agree with me.

Among those

which I have at hand, I shall limit myself to citing two Germansx Reicha (Cours de composition musicale, p, 33) and Marx (Die Lehre von der musikallschen Composition, I, l4l)j two Frenchmeni

Barbereau

(Tralte de composition. I, 41, 152) and Durutte (Technle, or Lois generales du systlme harmonlque, p. 128)j finally, a Spaniardi

Eslava

(Escuela de composicion, Tratado prlaero [[Madrid, I86l], p, 43).

I do

not own any Italian, English, or Russian harmonic treatises, unfortunately, but it is not unreasonable to suppose that I would find in them still more evidence in favor of my opinion. Second objection. The existence of a complete dominant seventh chord among contrapuntists, and notably Palestrina in his famous

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221 motet "Adoramus te Christe". . « . This is how Fetis twists this argu­ ment x "Like all his predecessors, he ^Gevaert] offers me some passages (?) from Palestrina which I have accounted for 20 times, notably in my Traite d'harmonic."

I have always eagerly read the writings of Fetis*

I know particularly well his Traite d'harmonle, but I have been unable to find in any part a commentary on the passage in question.

I would be

curious, I admit, to see this succession explained with only the technique of the syncope or ligature in the meaning of early contrapuntists. Palestrina's example is the only one, moreover, that I have cited for the period of classic counterpoint.

I could have added some othersf

I even could have cited a number of examples where this famous chord is complicated (following Ffitis* theory) by "substitution" and "prolongation." But it was not part of my plan to examine all the compositions of the middle ages in order to verify a fact of very secondary importance in my estimation. I am arriving at my third and last objection, which I am repro­ ducing verbatim to avoid all ambiguity. If one raised the objection that it is the use of the dominant seventh for the condition directly and immediately preceding the tonal repose, we would respond that, even in this respect, Monteverdi can not lay claim to the priority. Actually, this example occurs in the majority of pieces of the Nuove Musiche (I have it in more than 25 places), and notably in one of the oldest monodies of Caccini, the madrigal "Dovro dunque morlre," indubitably composed before 1598. Fetis absolutely does not contest my date (I have established it in my "Introduction historique," which opens the first volume of Glories de 1*Italic), but he finds the argument worthless for the example in question, because, he says,

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222 The work from which the madrigal is taken only appeared in 1601, two years after the fifth book of Monteverdi's Madrigals. It is true that Caccini says, in his preface to the reader, that he heard these pieces several years earlier in Florence and Borne at the home of friendsj but none of these were rigging out prior to the publication of Nuove Muslche. We have reason to be astonished by a statement expressed in such absolute terms,

What J of the works performed before an audience composed

of people such as Galilei, Mei, Rinuccini, Peri, Cavalier!1 repeated next before all the village of Florence, the intellectual and artistic center of It-^xy, would these works have remained unknown to Monteverdi?

Did not

Caccini say explicitly that a long time prior to the publication of his work, his airs and madrigals were performed continually by the famous Italian singers and cantatrices, and that all the composers had adopted his style?

And the publication of Nuove Muslche, was it not precisely

for the purpose of giving an exact version of these songs which had become altered in passing from hand to hand? If Fetis were to doubt the influence exercised on Monteverdi by the Florentine monodists, then he should take the trouble to re-read the famous letter of Pietro della Valle and some passages from Doni (among others chapter ix of Trattato della muslca scenica)j he will see what was the opinion of some contemporaries in this respect.

And not only in Italy

did the compositions of Caccini cause a sensation 1 they penetrated into the north of Europe.

The pretty melody "Amarilli mia bella" had become

popular in Holland in the time of the venerable poet Catsj so that one could be convinced of this, examine one of his songs composed about I63O (Alle de Wercken van Jacob Cats [^Zwolle, 1862], I, 629)*

The same melody

is shown in another Dutch songbook from the same period1 Eruls,

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223 Miane-Spieghel de deughen (Amsterdam, 1640), pp. 5 ajid If Caccini's novelties did not provoke the censures of some theoreticians at their appearance, they hold in the midst in which they occurred.

The academicians of Florence cared very little for counter­

point, the destroyer of poetry (laceramento della poesia), according to them | on the other hand, monody was a genre unduly disdained hy Artusi because they took the trouble to remove the infractions of the established rules. . . . Let there be no misunderstanding about the sense of my words.

I

do not wish to raise altar against altar, to set Caccini against Monte­ verdi, to disparage the latter In order to exalt the former,

I shall

not imitate Fetis when he says of this hero "that he [[Monteverdi] was the only genius from the end of the sixteenth century to the first half of the seventeenth century."

At worst, It is a simple affair of

individual tastef but yet, Fetis ought to explain to us why he suddenly has become so severe towards Cavalieri, Frescobaldi, Giovanni Gabrieli, Grumpelzhaimer, Hans Leo Hassler, Peri) and why he takes away from these illustrious contemporaries of Monteverdi the distinction of "genius," which he ascribed so generously in the second edition of his Blographle universelle.

How is Pure Gold Changed into Base Lead? Farther on, we read that "it is to Monteverdi that we owe the first opera performed in the theatres of Venice,"

Fetis, however, can not be

unaware that this glory comes more legitimately to Manelli, Sacrati,

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22k Cavalli, and Ferrari, whose operas had preceded those of Monteverdi on the Venetian stages (see the catalogues of Ivanovitch and Groppo). Now that I have replied nearly line by line to Fetis' note, let him permit me to address a question for him.

Why has he limited himself

to an incidental point of my study and said nothing about my general conclusions, sfhich contradict much more severly his historic plan than my opinions about the dominant seventh?

This silence surprised me all

the more because I have carefully repeated these conclusions in the last part of my work. Well then! would Fetis find nothing to object to in these propositions?

But then, what happens to the question of Monteverdi? . . .

A simple curiosity of learning, a small chronological problem. If, on the contrary, since it is easy to suppose, Fetis does not abandon any of his convictions, when he rigorously defines what he calls the "tonality cf plainchant" as the harmonic and melodic principles on which it is basedf when he demonstrates that this tonality has not altered in its essential elements from St. Gregory until the eve of the publication of these famous madrigals, and that the changes which I observe, century by century, are a pure illusionj when he explains (always by the tenets of his system) the gradual fusion of the two major modes of antiquity, the appearance of the leading tone in the dorian, the gradual extinction of the Phrygian, and the existence of a unique major ^mode] at the beginning of the sixteenth century fsicl. Finally, at the end of the sixteenth century, when he makes us see clearly the destructive action of the dominant seventh on this tonality 10 times secular, particularly when he does not forget to explain to us

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225 how this chord, according to him, the cause, the direct agent of the revolution, how, I say, this chord hardly foreseen, disappears almost completely for a half a century, just at the moment when its presence is most necessary f slcl,

When nothing more remains of everything which I have

set forth| when my examples will he recognized apocryphals, my assertions poorly founded, my conclusions incorrectf when it will he indeed proven that our tonality, this musical atmosphere which we hreathe, this mold which was sufficient to contain the thought of a Bach or a Beethovenf when this essentially impersonal thing is the work of a single man, then Fetis can he justifiably proud at having revealed to the world a unique fact, and one without analogies in the annals of the human mind.

Then,

hut then alone, it will he interesting to know if humanity ought to salute this new Prometheus in the name Monteverdi or Caccini. Until then, let Fetis he reconciled to see his historic doctrines dehated, his assertions controlled, and his hypotheses reduced to their just value. Far he it from me to pretend to give Fltis a history lesson, or to aspire for myself the title of historian. humble role of popularizer.

I aspire only to the more

If I sometimes have exerted myself to deal

with, in modest connections, some subject having a hearing on the past of our art, my goal has been only to Inform my colleagues of the positive results of the science of today, ind to focus their attention on some works too little knownf never have I drawn publicity to these attempts, still less have I endeavored to provoke debates. What I have always had my heart set on is to extricate myself from all dispute, set forth only the facts which I have examined myself| to

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226 admit my ignorance on the parts which I have not studied sufficiently, to hold in esteem the opinions of others; and above all, always never to assume with my contradictors a motive other than the simple love of the truth.

F*-A. Gevaert

Parisi

10 December 1868

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Appendix B Example 1.

da la Halle, Rondeau Ho. 15. "Taut ecu je vivrai," mm. 9 ~ H

1 w

T~

l

m

l’Escurel, "A vousr douce defronnalre," mm. 1-2, 15-16, 6

-♦i

Example 2.

lii

ij ...

n1 CJ

i—

— i----------

=l== p nr~.. = H = F #

rf ,i

11

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-f j ,j

228 Example 3*

Landini* "Non arra ma* pieta," mm, 2-4

/

f 1

j ' r ...■ ...

—

....

— ~

jymm =J=*=

fj

----------- r------

_F

1

.... -

r

Dufayt Mlssa 1*Homme Armet "Kyrle" (trans, Klesewetter)

Example 4,

Dt n i g p U

“H ---- s r C -----

T— -a-°~

lr-"t'"w-----m f f P a

- - r f

* |"P""g

a---

. . . - f -

rJ j -=—

n : L

■

? -------- p ---- r-----------* p-— b — “7*—

- ------ ■

■

-e—

1

a ----------a

—

rfr

J

j }

|

—

*

■H -i=j ■ —. ■h— -fl-

^ ... — -----l a ------------------

—

Hr. W

0—

4P \H

a ----- f- — -y—-----------b ------------------

t r iF = " — .— i 0-.. ti- = * = = = = *

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1

229

|ju W* JI c ... flj

1 i 1^ i= -■ = i "1--- a---4-u--- ----d — a— i —-8--------- ----■■—

=

^— p-‘flan— —---ar ~d----------- p-“H n*-----------rr ■=$=---------- f L

"H------- a---- ~f' .1^....... W A ... , -ft--... -j -=---P------- _a----------y------la -----JLb----- :---------v " (

• • ----A

ll, I 11 i p4= ------ r a— 4pftr-d:..J p, (»a.-..zf=c=zr b> r ^

jjp- * =0^-— -■=

r

m

ri

hi— ■----------H &B---- 5 ------ :---- ^------ a-------- ■■■ '■■■'■..... ------- ■-------16 -B— - a.. W ~ ----fftT ~ f - : ~ i=i==£E=£ *—

s

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vft u O '— jrv

—

y r rra ^

-4—

P t ----- B--------=-----

m■ : #

f

W

4*

... •

""4

.j

i —,—" ---------- „£ :—4.

i

o

iff

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-* Dufay, Guillaume* Opera Omnia* Edited by Heinrich Eecseler, Rome* American Institute of Musicology, 1951*

Vol. III.

Euler, Leonhard* Tentamen novae theorlae muslcae ex certlssimis principlis dilucldae expositae. St. Petersburg, 1739* ______ . "Tentamen novae theorlae muslcae«" Translation and commentary by Charles Samuel Smith* Unpublished Ph.D. dissertation, Indiana University, I960, Fetis, Frangois-Joseph, 2nd, ed. Paris*

Biographic universelle des musicians* 8 vols, Didot Freres, Fils et c£e* 1860-65*

______ * Esqulsse de 1'historic de l'harmonie, cons.ideree comme art et comme science systematique. Paris* Bourgogue et Martinet, X85o, _______ . Resume phllosophlque de l'historle de la muslque* Blographie universelle des muslclens, Vol. I. Paris* Fournier, 1835*

______ * Traite complet de la theorie et de la pratique de l'harmonie* contenant la doctrine de la science et de l'art. IJth ed, Paris* Brandus et Cie. 1849* Forkel, Johann Nikolaus, Leipzig, 1801.

Allgemelne Geschichte de Musik. Vol. II.

Gasparini, Francesco. The Practical Harmonist at the Harpsichord. Translated by Frank S. Stillings and edited by David C, Burrows. New Haven* Yale University Press, 1968. Gerbert, Martin. Scrlptores occleslastlcl de musica. 3 vols. rpt. Milan* Bollettino bibliografico musicale, 1931 ■

1784?

Gevaert, Francois-Auguste. Reponse a M» Fetis, sur l'orlglne dc la tonalito moderne* Paris* Eugelmann, 1868, Girdlestone, Cuthbert M. Jean-Phlllppe Rameau* His life and Work. London* Cassell and Co., Ltd., 1957* Halle, Adame de la. The lyric Works of Adam de la Halle (Chansons* Jeux-Partis, Rondeaux, Motets). Transcribed and edited by Nigel Wilkins* [RomeJ1 American Institute of Musicology, 1967* Helmholtz, Hermann. On the Sensations of Tone. Translated by Alexander J, Ellis* 2nd* ed. New York* Dover Publications, Inc., 1954.

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234 Hindemith, Paul,

Mendel. 19*5*

Graft of Musical Composition.

Book I, ~5th ed.

Hew York*

Translated by Arthur

Associated Music Publishers,

Horsley, Imogens. Fugue* History and Practice, New York* Press, 1966.

The Free

Jamard, Canon. Reeherches sur la theorle de la muslque. Paris* Jombert, 1769* Keane, Mlchaela Maria, Sister. The Theoretical Writings of Jean-Phlllppe Rameau. Washington, B.C.* The Catholic University of America Press, 1961. Kiesewetter, Raphael G, Geschlchte der europfilsch-abendlandlschen oder unsrer nsutlgen Musik. Leipzig* Breitkopf und HSrtel, 1834. Kimberger, Johann Philippe, Die wahren grunds&tze zum gebrauchen der harraonle. Berlin und KSnigsberg* Decker und Hartung, 1773*

Krehbiel, James Woodrow. "Harmonic Principles of Jean-Philippe Rameau and his Contemporaries." Unpublished Ph.D. dissertation, Indiana University, 1964. Langle, Honore Francois-Marie. Traite d*harmonle et de modulation. Paris* Cachet, I785. Lowinsky, Edward E. Tonality and Atonality in Sixteenth-Century Music. Berkeley and Los Angeles* University of California Press, 1961. Lescurel, Jehannot de. The Works of Jehan de Lescurel. Edited from the manuscript Paris, B.N., f. fr. lk6 by Nigel Wilkins, n.p., American Institute of Musicology, 1966. Levens,

Abrege des regies de l'harmonie pour apprendre la Composition, Bordeaux* Chapiers, 1743.

Lirou, [jean-Franjcis Espic] de. Explication du systemo de l'harmonie. pour abreger 1*etude de la composition, et accorder la pratique avec la theorle. Paris* Londres, 1785. Maret, Hugues.

Elogue historlque de M£ Rameau. Dijon*

Mattheson, Jchann, 1735*

KLeine general-bass-schule. Hamburg*

Causse, 1766, J. C. Kissner,

Mitchell, John William, "A History of Theories of Functional Harmonic Progression," Unpublished Ph.D. dissertation, Indiana University,

1963* Mercadier, Jean-Baptiste, Nouveau systeme de muslque theorique et pratique. Paris* Valade, 1776.

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235 Momignyr Jerome-Joseph do, Cours complet d*harmonic et de composition, d'aprea une theorle neuve et gsn^rale de la muslquer , * ■» 3 vols. Parisi The author, 1806, Monteverdi, Claudio, Tutte le opere. Edited by G. Francesco Halipiero. Vol. Ill, Vienna* Universal, 1926-h2. Moritz, Robert E, Memorabilia Mathematics or The Philomath's QuotationBook, New York* MacMillian Co„ 1915, Niedt, Friedrich Erhardt. Musicallsche Handleltung . , , Erster thell, Hamburg, 1710, Osborne, Richard Dale, "The Theoretical Works of Abbe Pierre-Joseph Roussier," Unpublished Ph,D. dissertation, Ohio State University, 1966, Parrish, Carl, The Notation of Medieval Music, New York* Norton and Co., Inc,, 1957*

W, W,

Pease, Edward Joseph, "An Edition of the Pixereeourfc Manuscript* Paris, Bibliotheque Nationals, Fonds Fr, 15123," 3 vols. Unpublished Ph.D. dissertation. Indiana University, 1959* Rameau, Jean-Philippe. Demonstration du principe de l'harmonie, servant de base a tout I'art musical theorique et pratique, Paris* Durand, 1750, ______ , Nouveau system de muslque theoriquer ou l'on decouvre le principe de toutes les rfgles necessalres~a la pratique, pour servlr d 1Introduction au Tralt^ de l'harmonie, IterlBi Ballard, 1726, ______ , Nouvelles Reflexions sur le principe sonore. 1760* rpt. n.p,, American Institute of Musicology, 19^9» ______ • Traite de l'harmonie redulte a ses prlncipes naturels. divise en quatre llvres. Paris* Ballard, 1722. Reese, Gustave, Music in the Middle Ages. New York* Co ,, Inc .^"^9^0 , ______ . Music in the Renaissance. Rev, ed. Co., Inc., 1959.

W» W. Norton and

New York*

W, W, Norton and

Reicha, Anion Joseph. Vollstindiges Lenrubuch der musikalischen Composition. , , , Aus dem franzSslschen ins deutsche ubersetzt und mit Anmerkungen versehen von Carl Czerny, k vols, Vienna* A. Diabelli und Comp., j_184{f]. Riemann, Hugo. History of Music Theory. Books I and II* Polyphonic Theory to the Sixteenth Century. Translated with a preface, commen­ tary, and notes by Raymond H, Haggh. Lincoln* University of Nebraska Press, I962.

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236 Roussier, Pierre-Joseph, Traite des accords, et de leur succession, selon le system de la basse-fondamentale. Paris t Duchesne, 1764* Sabbatini, Luigi Antonio, La vara idea delle musicale numeriche signature diretta al glovane studioso dell* armonica, Venice* Valle, 1799* Schneider, Johann Christian Friedrich, Elementarbuch der Harmonic und Tonsetgfcunstir 2nd ed, Leipzig* Peters, 1627. Schrade, Leo. Polyphonic Music of the Fourteenth Century* Vol. IV, Monaco* Editions de 11Oiseau-Lyre, 195^-58• SchrSter, Christoph Gottlieb, Deutllche Aawelsung zum General-bass* in bsstandlger Veranderung des uns angebohrnen harmonischen Dreyklanges, mit zulangllchen Bxenpeln, Halberstadt* Gross, 1722, Serre, Jean-Adam, Observations sur les prindpes de 1 *harmonic, occaslonnees par quelques Ecrits modernes sur le sujet. , . • Geneva* H, A. Gosse and J, Gosse, 1763* Shirlaw, Matthew. The Theory of Harmony* An Inquiry into the Natural Principles of Harmony, with an Examination of the Chief Systems from Rameau to the Present Day. 2nd ed. DeKalb, Illinois* B. Coar, 1955* Sorge, Georg Andreas, Vorgemach der muslcallschen Composition oder* Ausfflhrllche * ordentllche und vorheutlge Praxin hlnlSngliche Anwelsung sum General-Bass. Lobenstein, 17^5-^?» Stilllngileet, Benjamin. Principles and Power of Harmony. London* J, and H. Hughs, 1771* Strunk, Oliver, ed. Source Readings in Music History. New York* W. W. Norton and Co,6 Inc,, 1950. Tartinl, Giuseppe* Trattato di muslca secondo la vera scienza dell* armonla. Padua* Stamperio del seminario, 1754. ______ , Traktat uber die Muslk gemass der wahren Wlssenschaft von der Harmonie. ttbersetz un erlautert von Alfred Rubeli* DSsseldorf* Gesellschaft sur FSrderung der systematischen Musikwissenchaft e. V., 1966. Thomson, William Ennis. "A Clarification of the Tonality Concept," Unpublished Ph.D. dissertation, Indiana University, 1952. Tinctoris, Jean, The Art of Counterpoint. Edited and translated by Albert Seay. Rome* American Institute of Musicology, 1961. Vogler, Georg Joseph. Tonwissenschaft und Tonsetzkunst. Mannheim* Kuhrfurstlich Hofdruckerei, 1776.

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237 Wangermee, Robert. Fran^ols-Joseph Fetis, Musicologue et compositeur. Contribution a 1*etude de gout musical au XIX* siecle. Brusselsi Acaddmie royale de Belgique , 1951* Weber, Gottfried. Versuch einer geordneten Theorle der Tonsetzkunst. 4 vols, 2nd ed, Mainz i Schott, 182^K 3arlino, Gloseffo. The Art of Counterpoint. Bart III of Le Istitutloni harmonlche, 1556. Translated Guy A. Marco and Claude V, Balischa, New Haveni Yale University Press, 1968.

Articles jsuler, Leonhard. "Conjecture sur la raison de quelques dissonances generalement recues dans la muslque," Memories de 1'Aoademie Berlin, XX (1765-), 165-73* , "Du veritable caract&re de la muslque moderne," 1*Academic Berlin, XX (1764), 174-99*

Memories de

Fetis, Frangois-Joseph. "Decouverte de plusieurs Manuscripts interessans pour l*historie de la musique." Revue musicale, I (1827)» 3-11, 106-13* ______ . "Historie de la Musique«" 265-70.

Revue musicale, VI, No. 34 (I832),

______ , Letter to Edmond de Coussemaker. 19 April 1841, du Conservatoire Royal de Musique, Brussels. Hindemith, Paul. "Methods of Music Theory." (1944), 20-28.

Bibliotheque

Musical Quarterly, XXX

Jacobi, Ewln, "Harmonic Theory in England after the Time of Rameau." Journal of Music Theory. I (1957), 126-46. Mekeel, Joyce. "The Harmonic Theories of Kimberger and Marpurg." Journal of Music Theory, IV (i960), 169-93* Pepusch, John Christoph. "Of the various Genera and Species of Music among the Ancients, with some Observations concerning their Scale." Philosophical Transactions of the Royal Society of London, XLIV (1746), 266-74, Plamenac, Bragan, "An Unknown Composition by Dufay?" XL (1954), 190-200,

Musical Quarterly,

PLanchart, Alejandro, "A Study of the Theories ofGuiseppeTartini," Journal of Music Theory, IV (i960), 32-61.

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238 Plattr Peter. "Dering’s Life and Training," No, 1 (January 1952), ^1-^9,

Music and Letters,XXXIII,

Prony, Baron Gaspard Riche de, "Du rapport fait a 1 ’academic des sciences sur cet ovrage," Prlnclpes de milodle et d*harmonic deduits de la theorle des vibrations, Baron Blein, Paris, 1838,

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INDEX

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240 INDEX Added sixth, 97 Aliquot parts, 100 Alteration, 154, 163, 165r 180-83, 188, 190, 196, 211, 214-15 Anticipation, 36, 37-38, 164, 216 Arnold, F« T,, The Art of Accompa­ niment from a Thorough-Bass, 74, 77n, l65n Augmented sixth, 142, 181, 193 Baini, Abbot, 76-79 Balliire, Charles, Traitfe de la muslque, 136-39» 141, 151, -33o, 195# 196, 208 Basso continuo, 72-79r 80-81, 82, 83, 86, 167, 171r 178 Belestat, Mercadlere, See Mercadier, Jean-Baptiste Berton, Henri-Montan, 184 Boyvin, Jacques, Traite abregS de 1* Accompagnement, 86-87 Burney, Charles, A General History of Music, 53, 65r 75n Canon, 42, 43n, 50-54, 57 Catel, Charles-Simon. Traite d'harmonie, 3 , 18, 172, 186-91, 194, 196, 209, 212, 217 Cavalieri, Emilo, Preface to La Rappresentazlone, 73^, 77 Chords by added thirds, 92-94, 132, 150, 159, 176, 204 Choron, Etienne-Alexandre, 200-01 Chretien, Gllles-Louis, La musique ett'dlae comme science naturelle, 197-200 Concord, 26, 29-30, 42 Consonance, 8, 30, 31, 33, 39 , 40, 41, 106-07, 212 Consonant chordsj 41, 44, 48-49, 50, 60, 74, l44n, 165, 180, 182 Chretien, 200 Cruger, 81 Fetis, 210 Heinichen, 63 Mercadier, 148 Reicha, 191n Sorge, 156-57

Counterpoint, 43, 50, 55, 56, 65, 66-69, 77, 82, 106 Coussemaker, Edr de Scrlptorum, 25, 26n, 47n, 48 L*Art harmonlque aux xile et xlll6 sleclesT 25n, 32n Crflger, Johann, Synopsis auslca. 75-76, 77, 80-61, 157n Daube, Johann Fr«, Generalbass in drey Accorden, 161-63 Deering, Richard, Cantiones sacra, 75 Descartes, Rene, Compendium muslcae, 91, H 7 Diaphony, 25, 33, H 6 Discant, 27-29 Discord, 26, 30 Dissonance, 8, 16, 31, 33, 38, 106 Artificial, 45, 48, 212 Natural* 8, 157 Dissonant chords, 33, 50, 64, 65, 69, 74, 182 Catel, 188, 189 Chretien, 199 Cruger, 81 Euler, 106-07 Heinichen, 83 Mattheson, 85 Marpurg, 159 Neidt. 81-82 Rameau, 95-98 Reicha, 192-93 Sabbatini, 174-75 Sorge, 157-58 Tartini, 132 Divisions, harmonic, 122, 186 Dominant seventh, 13-14, 15-16, 61-62, 117, 119, 158, 159, 160, 165, 187, 193, 210 Double emploi, 97, 100, 143, 150n, 160, 162 Euler, Leonhard, Tentamen novae theorlae musicae, 103-121, 137, 136, 155, 165, 187, 209 Exponent, 110, 111, 112n, 115, 119

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24l

Fetis, F,-J,r Blographle universelle, 2, Jnt 4n, 5n, 25r 34n, 80n, 124n, I4?n, 158, 166 Traite de contre-polnt, 4,

66-6? Traite d'harmonie, In, 5, 6-17, 50n, 6ln, 87n, 88n, 133, 159n, l83n, 211n, 212n, 2l4n, 215n, 217 Figured bass. See Basso continuo Franco, Ars cantus menBurabillB, 25-27, 30, 31, 33 Fugue, 50, 53, 57, 65-69 Fundamental bass, 96-100, 101, 102, 133, 139, 140, 150, 151, 163 Gaevert, F.-A., 13, 219-26 Gasparini, Grancesco, L'Armonico Practlco, 79-8O Geminani, Francesco, 179 Gerbsrt, Abbot, Scrlptores, 27, 30n, 39, 40n, 42n Halle, Adam de la, 25n, 29, 32 Haupt-Accord, 157 Heinichen, Johann Neu erfundene und Grundllche Anwelsung. 82-84, 204 Der General-Bass in der Composition, 84, 155, 161 Helmholtz, Hermann, 121n Hindemith, Paul, 7n, l44n, l48n Inversion, 92, 93, 102, 121, 150,

157, 162 Jamard, Canon, Recherches sur la Theorique de la musique, 139-4l, 195, 196, 208 Johannes de Muris. See Hurls, Johannes de Keller, Gottfried, 178-79 Kellner, David, Treulichner Unterrlcht im GeneralBass, 155 Kirnberger, Johann Philippe, Die wahren GrundsStze zum Gebrauch der Harmonie. 166-67, 180. 191, 217

Kollmann, August Friedrich. 180 Krehbiel, James V,, lOOn, 124n, 139n, 159n Lampe, Johann, 179 Landini, Francesco, 37, 39, 40 Langle, Honore Frangols-Marie, TraltS d’harmonie et de modula­ tion, 152-54, 197 L’Escurel, Jehannoi, 34, 39 Levens, Abrege des regies de l'harmonie. 133-37, 139-51, 151, 166, 168, 169, 195, 196, 208 Lirou, Chevalier de, Explication du systeme de l'harmonie, 151-52, 197, 200n Lowinsky, Eduard, 15 Harchettus of Padua, lucldarlum in arte muslcae planae, 30-32 Marpurg, Freidrlch Wilhelm, 159, 161, 163, l67n, 174-75n, 180, 191, 201 Hattheson, Johann, 149, 207 Die exemplarlsche OrganlstenProbe. 84-85•~204 KLelne General-BassSchule. 85-86. 155n. l 6l Hercadier, Jean-Baptiste, Nouveau systeme de musicLue thSorique et pratique, 147-51 Mitchell, John W., 72n, l4ln, 204n Modification of harmony, 48-49 Modulation, 15, 16, 63, 65, 82, 84 152, 163, 170, 180, 183n Momigny, Jerome-Joseph de, Cours complet d*harmonie, 194-97, 209 Monteverdi, Claudio, 13, 15, 59, 6l, 63, 64, 71, 208, 219, 221-25 Muris, Johannes de, De Discantu, 39-42 Neidt, Friedrich, Muslkalische Handleltung. 81-82, 83 Palestrina, Pierluigi da, 50, 55,

58, 60, 70, 220-21 Partimenti, 79, 84 Pepusch, John C«, l46n, 179

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242 Progression Arithmetical, 134-35» 139? 141, 156, 157, 160, 169, 170, 195, 196, 204, 208 Geometric, 130n, 138, 139, 14-5, 160 Harmonic, 134-38, 156, 169, 209 Triple, 145, 146, 152n. 204 Prolongation (suspension), 45, 83, 94, 99, 130, 133, 143, 154, 157, 158, 162, 164, 165, 177-78, 188, 189-91, 196, 211, 212-13 Bameau, Je&n-Fhilippe, 35, 82, 112, 113, 120, 125, 128, 132, 134, 137, 140, 141, 144, 148, 150, 152, I55n, 157, 158, 159, 162, 163,. .17*, 179, 185, 191, 198, 200n« 204, 211, 217 Traite de l'harmonie, 85, 89-94, 97, 101, l42n, 15* Nouveau systems de muslque theorique, 95, 15* Reese, Gustave. Music in the Middle Ages, 25, 34n,~^7n Music in the Renaissance, 42n, ------------------

Reicha, Anton, Gouts de composi­ tion musicale, 191-9*, 220 Res facta, 50-51 Retardations (suspensions), 35, 37, 41, 47-49, 60, 94, 165-66, 176 RouBsier, Abbe, Traite des accords, 141-45, 151, 197 Rubeli, Alfred, 122n, 123n, 126n Rudolph, Johann Joseph, 184 Rule of the octave, 78, 79, 80, 83n, 89 Sabbatini, Luigi Antonio, la vera idea, 173-78 Schicht, Johann Gottfried, Principes d'harmonie, 202, 209 Schneider, Friedrich, KLementarbuch der Harmonie und Tonsetzkunst, 202, 203, 217 Schrade, Leo, Polyphonic Music of the Fourteenth Century, 38n, 45* Schrffter, Christopher, GSttlieb, Deutllche Anweisung zum Generalbasse, l8r 163-66, 190, 191, 217

Senario, 92n„ 132m Sequence, 13, 23-6-17 Seventh ,;hord on second degree, 12, 96, 163, 164, 167, 189? 190, 199-, 215 Shixlaw, Matthew, The Theory of Harmony,- 1, 13n, 125n, 152n, 158n Sonorous body, 95# 100-01, 112, 122, 123, 128, 132n, 146, 151# 156, 198, 199? 204 Sorge, Georg Andreas, Vorgemach de musicalischen Composition, 7..S, 155-59# l6l, 163# 165, 190, 191, 195, 197, 217 Strunk, Oliver, Source Readings in Music History, 25n, 55a, Tin, 74n, 125n, 152n, 158n Substitution, 88, 143, 154, 164, 165# 183, 188, 190, 211-12, 213 Supposition, 92, 93# 96 Suspensions. See Retardation and Prolongation Tartlnl, Giuseppe, Trattato di musica, 121-33, 172, 174n, 175n Third sound, 124-25 Tinctorls, Johannes, 38, 46-48, 51n, 55n Tonality, 95, 99, 101, 132. 151# 152, 159# 161, 162, 165, 170, 175# 180, 187, 205-17 Omnitonique, 14, 16-1,4 119# 184, 215 Pluritonique, 16, 119 Transitonique, 15, 64, 119, 182, 183 Onitonique (Plainchant), 14-15, 31, 44, 45, 50, 54, 58, 60, 61, 62, 63, 64, 66, 69 Triade harmonique, 81 Trias deficiens, 156 Trias minus perfects, 156 Triple progression. See Progres­ sion Vallotti, Francesco Antonio, Della Sclenza teorlca a pratlea della modema musica, T68, 172-75, 17*, 176 VerzSgerung, I65 Viadana, Lodovico, 72-78

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Vogler, Abbe, Tonwlssenschaft und Tonsetzkunst, 167-71» 203» 217 Wangemnee, Robert, In, 2n, 3n, 5a, 43n Weber, Gottfried, Versuch einer geordaelen Theorle der Tonsetzkunst, 203-04,217 Wilkins, Nigel. 32n, 35* Zarlino, Gloseffo, Le Istltutloni harmonlche, 56-57f 65a, 89, 90» 92n

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VITA

Namei

Mazy Irene Arlin

Bom:

26 June 133?* Lycn2*

York

Education: Diploma, Dtica Free Academy, 1957 Bachelor of Science, Ithaca College, 1961 Master of Music, Indiana University, 1965 Current Position1 .Chairman, Music Theory, School of Music, Ithaca College Professional Affiliations: Pi Kappa Lambda Sigma Alpha Iota American String Teachers Association Music Educators National Conference

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