Rameau and Zarlino: Polemics in the "Traité de l'harmonie" Alan Gosman Music Theory Spectrum, Vol. 22, No. 1. (Spring, 2000), pp. 44-59. Stable URL: http://links.jstor.org/sici?sici=0195-6167%28200021%2922%3A1%3C44%3ARAZPIT%3E2.0.CO%3B2-G Music Theory Spectrum is currently published by University of California Press.
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Rameau and Zarlino: Polemics in the Traite de I'harmonie Alan Gosman Jean-Philippe Rameau is well known for his vociferous attacks against critics who dared to differ even slightly with him. The most celebrated of his arguments are those with Rousseau and the other philosophes.' Somewhat overlooked is a quieter struggle that occupied Rameau more than three decades before these welldocumented debates, a struggle to situate his newly emerging ideas within the legacy of Zarlino. It is in his creative handling of that legacy that we can first observe Rameau attempting to secure the acceptance of his own theories. In his first published writing, the Traite de I'harmonie of 1722, it quickly becomes apparent that Rameau is preoccupied with Zarlino. Rameau frequently footnotes two of Zarlino's treatises, the 1573 version of the Istitutioni harmoniche, and the 1589 version of the Dimostratoni harmorliche included as the second volume of De tutte l'opere. Further, Zarlino is the only person with an entry in the Traiti's Table of Terms.? 'Many authors. including Thomas Christensen and Cynthia Verba, have documented Rameau's reactions to contemporaries who modified and threatened his popular and respected theories. See Thomas Christensen. Ranleau and Musical Thurrght in the Enlightenn~ent(New York: Camhridge University Press. 1993). and Cynthia Verba, Music and the French Enlightennlent: Recoilstruction qfa Dialogue 1750-1764 (New York: Oxford University Press. 1993). 'Jean-Philippe Rameau. Trait4 de I'harnlonie r4drrite u ses principes nuturels (Paris, 17221, xxiv: reproduced in vol. I of The Conlplete Theoretical Writirzgs of Jearz-Philippe Ranleau (1683-17641, ed. Erwin R. Jacobi (American Institute of Mus~cology,1967-72); translated hy Philip Gossett as Treatise on Harnlorzy (New York: Dover, 1971). Iv.
Other theorists, such as Descartes and Mersenne, certainly influenced Rameau greatly. Rameau always returns to Zarlino's texts, however, when arguing for change. In the first footnote of the Trait&, Rameau reveals the importance that he attributes to Zarlino's writings as opposed to those of later authors: "Zarlino was a celebrated author on music who wrote approximately 150 years ago. We find only feeble restatements of his works in later writings on the same subject."' COMPOSITIONAL CANONS
The first two books of the Traite' abound in direct references to Zarlino. I will begin this investigation of Zarlino's influence on Rameau, however, by looking at Book 111. It comes as a bit of a surprise after Books I and I1 that Zarlino's name is completely absent from Books 111 and IV, titled "Principles of Composition" and "Principles of Accompaniment" respectively. And yet Rameau hardly neglects his precursor in the discussion of practical music. In fact, I believe that Rameau's most acute awareness of Zarlino's shadow, and his most powerful attempt to distance his readers from his predecessor, occurs near the conclusion of Book 111. At this point one finds-with some shock-that for his culmi'"Zarlino, Auteur celehre en Musique, qui a ecrit a peu-pre~depuis 150 an\. & dont on ne trouve que de tr2s-foihles Copies. dans les Ouvrages qui ont part apr6s les siens, sur le m@mesujet." Trait4 ile l'harmonie, Preface, second page; Treatise on Harnlon?: xxxiv. All translations of Rameau given here are by Philip Gossett.
Rameau and Zarlino: Polemics in the Traite de l'harmonie 45
nating compositional tour de force, his last word in compositional technique, Rameau presents two strict canons. The shock occurs because, having followed Rameau's harmonic agenda for three books now, one hardly expects him to cap off his composition lesson with a strict contrapuntal form. An explanation is suggested by the fact that the culminating compositions of Book I11 of Zarlino's Istitutioni harmoniche are also a pair of canons. These two authors' similarly placed canons, which have been almost completely overlooked, turn out to be summaries of their respective theories on music. Furthermore, Rameau's compositions can be seen as a powerful demonstration of the inadequacies of the traditional contrapuntal explanations offered by Zarlino. A close examination of Zarlino's two culminating canons, and the methods by which Zarlino constructed them, will help one recognize exactly how Rameau's two canons in the Traite' are a response to his precursor. In the Istitutioni, Zarlino makes it clear that advanced musicians show their talent and knowledge by taking on and solving challenging problems of canonic writing. When Zarlino is just beginning to describe canons, he says that he is most impressed by canons which are not garden-variety, two-voice compositions whose voices are separated by a short distance. He writes, "Constant practice of this close imitation has resulted in such a common idiom that a fugal pattern cannot be found that has not been used thousands of times by various comp o s e r ~ . "In~ demonstrating advanced canonic techniques, Zarlino intends to teach his readers to "apply all our ingenuity to write fugues that are fresher."'
'". . . ma il troppo continuare cot a1 vicinita fece, che si casch in un certo mod0 commune di comporre, che a1 presente non si ritrova qua si Fuga, che non sia stata mille migliaia di volte usata da divers1 Compositori." Gioseffo Zarlino, Le Istitlctlon~harnloniche, reprint of the 1573 Venetia edition (Ridgewood, N.J.: Gregg Press, 1966). 258; Gioseffo Zarlino. The Art of Counterpuirzt, trans. Guy A. Marco and Claude V. Palisca (New Haven: Yale University Press, 1968), 127. All translations of Zarlino given here are by Marco and Palisca. '". . . & cercaremo con ogni nostro potere di fare delle Fughe. che siano piu nove." Le Istitlctioni hannonicrhe. 258: The Art of Counterpoirzt, 127.
Zarlino often relies on canons to demonstrate the practical applications of his contrapuntal rules, and by exploring different canonic types, he shows his preference to seek out less common varietiesh Example 1 lists the many different canonic techniques in the Istitutioizi in their order of presentation, which roughly corresponds to their compositional difficulty. The final three canon types are all in four voices, and in the last two, each of the voices takes part in the canon. The final two compositions of Zarlino's counterpoint text appear in Chapter 66, and they are meant to be the most ingenious of his canon types. Both are pieces with two pairs of parts in canon by contrary motion. They are reproduced in Examples 2 and 3. These canons display two of Zarlino's important lessons about composing in three or more parts. When earlier discussing threevoice compositions in Chapter 59, Zarlino writes, Composition may be called perfect when, in every change of chord, ascending or descending, there are heard all those consonances whose components give a variety of sound. Where such consonances are heard, the harmony is truly perfect. Now these consonances that offer diversity to the ear are thg fifth and third or their compound^.^
hA similar tendency to explore can be found in Zarlino's discussion of douhle counterpoint. He writes, "Though there are many ways of writing such counterpoints, as I have said, I shall demonstrate only those that seem most difticult and most elegant. This mill avoid boring the reader, who can readily infer the other procedures for himself." The Art of Counterpoint, 205. ("Ma ancora che molti siano li modi di comporre tali Contrapunti: come ho detto: porrb solamente quelli, che mi sono paruti piu difticili & piu elegant]: accio non sia tedioso a i let tor^: da i quali ciascuno ingegnoso potra comprendere, come ai haveri da reggere in qualunque altra maniera di simili compositioni." Lr Istitlctioni harnloniche, 297.) -"Quells compos~tionesi puo chiamare Perfetta: nella quale in ogni mutatione dl chorda, tanto verso il grave, quanto verso I'acuto, sempre si odono tutte quelle Consonanze. che fanno varieti di suono ne i loro estremi. Et quella 6 veramente Harmonia perfetta, che in essa si ode tal consonanze: ma li Suonl, o Consonanze, che possono fare diversita a1 sentimento sono due, la Quinta & la Terza. over le Replicate dell'una & dell'altra." Le Istitlctioni harmoniche, 287: The Al? of Counterpoint, 186.
Music Theory Spectrum
46
Example 1. Canon types found in L'lstitutioni hannoniche, Part I11 1. Canon at the distance of three to five minims (Exs. 88/89 at 8ve, 94/95 at 3rd, 98/99 at 5th) 2. Canon in contrary motion (Exs. 90/92a, 91/92b, 96/97)
4. Adding three parts, two of which are in canon, to an existing tenor (Ex. 161) 5. Composition with two pairs of parts in canon by contrary motion (Ex. 162) 6. Perpetual composition with two pairs of parts in canon by contrary motion (Ex. 163)
3. Adding a two-part canon to an existing tenor (Exs. 154 and 155)
Example 2. Zarlino, Ijtitutioni, Part 111, Chapter 66, Composition with two pairs of parts in consequence by contrary motion
plus minor third a b m c
[p; &
-
I
Ha$\, and consequent of the soprano
1
I UU
II.'_
I
I
I
I
1
-
IT ,
-.
II -
I
CI
L'
-
',
I
2'
-it
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I
-1
Rameau and Zarlino: Polemics in the Traite de I'harmonie 47
Example 3. Zarlino. I.rtit~ltiotli,Part 111, Chapter 66, Perpetual composition with two pairs of parts in consequence by contrary motion
Soprano, and p ~ d of e the h ' ~ \ \
-
'
L l r
Tenor. and consequent of the alto
. [b pp 1 -
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-
-
-
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-
-
*
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--
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~
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-
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.
Ba\\. and consequent of the xiplnno
y q + f -=7T$~~~~==-I~r*-*~-LLPL0-~r-*~F*-~ I
ill'
R
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-
Zarlino's melodies are masterfully constructed to maximize the number of perfect harmonies despite the strict contrapuntal form. Almost every verticality is a 2 chord. In each example. Zarlino is particularly careful that the canon's frame. which I define as those chords that fall at the time interval of the canon, do not spiral into imperfection. The time interval of the canon in each cxample is two bars in the modern realization. Example 3 lists the chords in the odd-numbered measures of Examples 2 and 3. The list starts with m. 3. because at that point enough voices have entered to create a perfect harmony. Except for the end of Example 2. at which
+
.
L
I -I ,
~
'
!
'
I
&-----&. -1
L
-L
time Zarlino rnakes an adjustment to provide n Phrygian cadence, the canonic frame of both examples simply alternates two different perfect triads. The conditions of composing a double canon by contrary motion. interesting in their own right. also demonstrate that Zarlino's choice of canon type was motivated by the lessons of the I.stitlltioni. By closely examining this presentation, one can see the contrapuntal mastery with which Rameau is competing. In a contrary(notion canon, although the dlrx and comes could be inversionally symmetrical around any pitch. Zarlino decides in both canons that
48
Music Theory Spectrum
Example 4. Triadic roots at the time interval of the canon in Zarlino's contrary motion canons Triadic Roots in Measure
Examule 2 a C ( l st inv.) a C F (I st inv.) E
Examule 3
z4 D
z4 D a o
etc
the note D will be the inversional center. This arrangement fits in with the intervallic make-up of the "white notes" as is shown in Example 5. The chromatic notes Bb and F# are also included because Zarlino uses them in both pieces. Example 6 shows that the alternating chords (cf. Ex. 3 ) are also, as one would expect, those that are symmetrical around D. The first column shows the inversional mappings between notes of the A and C triads. These chords alternate throughout most of the first canon. The second column shows the mappings between the G and D triads. These chords alternate in the second canon. For Zarlino. however, the mere presence of perfect harmonies does not assure variety. He stresses throughout his text that there are distinctive types of perfect harmonies. For example. in Chapter 3 1 of Part 111. Zarlino writes: The variety of the harmony in auch situations does not consist solely in the variety of the consonances that are found between two voices. but also in the variety of the harmonies-which [variety] is determined by the position of the note that makes a third o r tenth above the lowest voice of the composition. Either these [intervals] are minor, and the harmony that arises is determined by o r corresponds to the arithmetical proportion or division. o r they are major, and such a harmony is determined by or corresponds to the harmonic mean. O n this variety depends a11 the diversit) and perfection of harmonies. . . . For as I have said elsewhere, when the
major third is below, the harmony is cheerful, and when it is placed above. the harmony is sad.8
Zarlino did not just happen to choose to write a canon in contrary motion as a contrapuntal challenge. Rather, he utilized the special features of that form to highlight the inversional relationship of major and minor chords. This can be observed by looking at the role of each pair of canonic voices separately. In Example 2, the dux of the canon between the alto and tenor begins with E. This maps to a C in the tenor in nl. 3. which is set by another E in the alto. This again maps to C in the tenor in bar 5 and again is set by an E. The pattern continues until In. 13. For almost the entire piece, the innervoice canonic frame consists of the major third from C to E. Zarlino has the choice of whether to place a minor third above or below this fixed major third. His choice will result in a major or minor triad respectively. By taking advantage of the inversional pair G-A, which provides the notes a minor third above and below the fixed major third C-E. Zarlino is able to alternate between C major and A minor chords. In m. 3. the dux of the outer-voice canon sounds a G. thus combining with the inner voices to form C major. The G maps to the bass A in ni. 5, thus sounding A minor. The G is found again in the dux of m. 7, again forming C major." ""Conciosia che la varieth dell'Harmonia in simili accompagnamenti non consi\te solarnente nella varieth delle Consonan7e. che i i troia tra due parti: rn'i nella iarieti anco delle Harmonie. la quale con\iste nella positlone della chorda. che fh la T e r n , over la Dccima wpra In partc gra\e della cantilenu. Onde. overo che cono minor1 & I'Harmonia che nasce. C ordinata. 0 \i a\\imiplia alla proportlonalith, o mediations Arithmetica: overo bone ~nagpor-i& tale Harmonla 5 ordinata, over si a s \ i ~ n ~ g l alla i a mediocrith Harmonica: & da questa varieth dipende tutta la diver\iti & la perfettione dells Harmonie . . . percioche (como hh detto altro\e) cluando \ I pone la T e r n mappiore nella parte grave. I'Harmonia \ i fh allegra: & quando ci pone nell'acuto si S ~ me\ta." I LC, I ~ t i r u r ~ ohcir-r~ic~~~ic.I~c~, ~i~ ? 10-1 I : The 4rt of Cour~rc~ipoi~it. 69-70. "Because of the utrict cond~tionsfor con\tructing triad\ in a double controrq motion canor:. it is clear that Zarlino i i adding a single note ( A or C;) to a pitct, pairing that i \ set ( C and E). Bq this method. mm. 3. 7. and I I are C rnajor triads. What is interehng is that the chord in m. 7 ic a C major triad In fir\[ in\t.rsion. Ba\ed on his theoretical text. Zarlino would not have recopnired a relation
Rarneau and Zarlino: Polemics in the Traite de I'harmonie 49
Example 5 . Inversional mapping of notes in Zarlino's contrary motion canons D-D c-E B W F Bb t-,F# A W G Notes: A Distance (steps):
C
B I
112
D 1
E 1
F 112
G 1
Example 6. Inversional mapping of triads in Zarlino's contrary motion canons Chord Root$ C -A G-A E t , C C-E
D-G A -G F# t , B b D-D
F C A F
-E -E -G t , B
The alternating pattern continues until the last bar of the piece when a G # is substituted for the Gh as part of a Phrygian cadence. The canonic frame in Example 3 alternates between major and minor chords using a slightly different method. Instead of adding up thirds as in Example 2, Example 3 divides a perfect fifth. The perfect fifth is found in the soprano-bass canon. Two inversional pairs are found in the odd-numbered bars starting in m. 3: the D-D pair first mapped between mm. 1 and 3, and the A-G pair first mapped between mm. 3 and 5. Thus m. 3's outer-voice twelfth between a C chord in root position and a C chord in first inversion. Rather, one would be a chord with a third and a fifth, and the other would be a chord with a third and a sixth. In the practical context of this composition, however, Zarlino is pressed to recognize the equivalence of the two sets of notes. Therefore, the demands of canonic writing lead to an early instance of inversional thinking. Pursuing this subject could easily generate another essay.
from D to A maps to m. 5's outer-voice twelfth from G to D. This fifth is mediated by one of the notes from the inversional pair F#-Bb in the inner voices. Since each of these notes is a major third from D, and D is found in both of the fifths, it is clear that one fifth (D-A) is divided with the major third on the bottom, forming D major, and the other fifth (G-D) is divided with the major third on top, forming G minor. The simple alternation of two chords at the canonic frame of Examples 2 and 3 makes such a complicated canon easier to compose. More importantly, however, it directs the reader's attention to the structures recommended in Zarlino's text-perfect chords that demonstrate the diversity of harmony. In Example 3, the duxof this composition is only seven notes, but the piece gains considerable length because it is a perpetual canon. This is striking, because it is Zarlino's only perpetual canon. In some sense, Zarlino's final compositional statement of Part 111 is meant to linger in the reader's head forever. As has been suggested, it seems more than coincidental that Rameau also includes, as his final compositions in the Trait&, a pair of four-voice perpetual canons. In these pieces Rameau provides his own commentary on issues raised in Zarlino's canons about which type of chord to privilege, and how to obtain diversity in harmony. Rameau's canons constitute a carefully designed response to Zarlino, with the intention of revolutionizing musical thought and, in many ways, turning music theory away from its contrapuntal explanations. Although Rameau's canons are not introduced with enormous fanfare, they may well be the first pieces specifically designed for harmonic analysis. Rameau's four-part canon at the fifth is reproduced in Example 7. This piece is what I have termed a stacked canon, because the tenor and alto, each of which is an imitative voice, also each plays the role of dux for another part entering later.'"hus Rameau 'Osee Alan Gosman, "Stacked Canon and Renaissance Compositional Procedure," Journal of Music Theor) 4112 (1997): 289-3 17.
50
Music Theory Spectrum
Example 7. Kameau, Trait6 de l'han?iorzie, Book 3, Chapter 44, Canon at the Fifth
.---AI
A
I 4
%
I L ,
\
Ah'
8
Dux repeated up a n
Loin de
ri
rc.
Plcu
rolls.
Pleu
- ran\.
Rameau and Zarlino: Polemics in the Traite de I'harmonie 51
Example 7 [continued]
~
I
Dux repeated up another major t h ~ r d
CI:
I~
ali
DIP
Gd!
Dux is an auemented \ekenth hove the uartlng nltch!
ed-
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b%
Eli
AdQ
f.1
bl
52
Music Theory Spectrum
revives Zarlino's practice of displaying canonic acumen by straying from a straightforward model.I1 But Rameau's four-voice canon is not merely a demonstration of compositional technique. He also intends to remind the reader of what he considers a fundamental mistake by Zarlino, a mistake that the "Zarlino" entry in the Truircs Table of Terms specifically criticizes: "The errors found in his rules arise partly because he envisaged only two sounds at a time."'? While Zarlino did in fact compose some fourpart canons. the fourth voice in these pieces is almost always a doubling. Zarlino exhibits no great desire to build up chords with four different notes.13 Rameau ventures into the realm of four-voice canonic writing not only because Zarlino did, but also because the exploration relates closely to the Trrrite"~teachings. Significantly. the four-voice texture aswres the possibility of a complete seventh chord once all of the voices have entered. This is consistent with Rameau's preference for the seventh chord as a means of harmonic propulsion, particularly when used on the dominant. In Book 11. Chapter 2. Ra~neauwrites of the seventh chord. T h e dibersit) that this chord brings . . . b! introducing a certain tartness which si~nultaneouslyenhances the sweetness of the perfect chord, ~ n a h e s us desire its presence, not reject it. We must thus place it among the fundamental chords, since it in no way destroys the source which subsists in the lowest sound of the perfect chord.!' "Ranieau crroneou\ly proclaim\ h~mselfto hc the tint to h a ~ cconipo\ed a four-part canon of thlh type (Trrurite or7 Hcrr.~jro~i\:370). ""Les erreurs q u ~sr trou\ent dans Ir\ Rcglrs de Zarlin proLienncnt cn pnrtle dr ce qu'il n'en\isagroit que deux Sons i la fois." Trr~itisrlr /'i~rir~?~orjir, xxi\: Tretrrite or! Hrirrriorr); I\. l'ln Zarl~no'\four-\oice canons hy in~ersion.only two \e\rnth chord\ occur. The\e are in Example 1 , on the last quarter note of mm. 8 and 13. I n each case, the \e\enth has a clear pabsing function. ""La d~\ersitCque crt accord y cause en y introduisant unr crrtalne aniertunic. qui relc\c en nitnir tcnips la doucrur de I'accord parfait. doit nous Ir f'111e .
souhalter, hicn l o ~ nde le rejrtter, ne p o u ~ a n tnous dl\penscr pour lors de la mettre au nomhrr dea accords fondamentaux. puisqu'il ne dCtru~tpolnt le princ~pequi subaistr toC?jouu dans le Son gra\e de I'Accord parfait." TrfiitP rlu 5 2 , T r ~ a t i on ~ r Harmorn, 61 l'hc~rn~orlir,
In the canon of Example 7, once all four voices have entered in m. 6, every vertical sonority (senzufine) is a complete seventh chord. Before m. 6, where complete seventh chords are impossible, one is aware of their approach as the imitative voices are added. In m. 2 the parts suggest a C7 chord (V3IIV) as the notes Bb and G are embellished by an eighth-note upper-neighbor C in the bass. In In. 4 the piece inches closer to stating a full seventh chord with G, D. and F sounding as half notes or longer. Finally. in m. 6. the seventh chord (D7) is complete. While it may look as if the harmony is a byproduct of melody, the extraordinary features of this canon make it clear that Ra~neaucarefully planned the harmonic progression first, and allowed it to guide his melody. He thereby provides compositional evidence for one of his principal assertions, that harmony is prior to melody. The natural fit between the structure that Ra~neaucreates ( a four-voice canon) and his penchant for seventh chords suggests to his audience that the contrapuntal devices of the past have actually been in service of more basic harmonic principles, with the phenomenon having gone unnoticed before Rameau. Although the canon goes on perpetually, because it transposes each eight bars up a major third. three cycles through the melody are required before the dll.r which begins on C (now technically B# ) is repeated in In. 24. This brings up interesting tuning issues. although these will be avoided at the present time. Example 8 shows that these three passes through the ~ L L . Yare exactly how long i t takes fot- the even-numbered measures (the canonic frame) to present a dominant seventh chord on each of the chromatic scale's twelve notes. In addition to displaying all possible dominant seventh chord^ Rameau takes very seriously how they are connected. Relying on his explanation in Book 11. progressions of funda~nental bass notes follow the findings based on the divided string. In particular. they demonstrate the desired progression by fifths. In this respect, the canon in Example 7 succeeds masterfully as a pedagogical tool. Beginning an analysis at the repeat of the melody in the bass voice at m. 8, where all four voices are present, it is easy to see that harmonically the piece moves in two-measure fragments (the
Rameau and Zarlino: Polemics in the Traite de I'harmonie 53
Example 8. Chords at each time interval in Rameau's Canon at the Fifth m. 8 m. 10 m.12 m. 14
A; E?; B7 F#;
m. 16 m. 18 m. 20 m. 22
C#: G#: D#' A#?
m. 24 m. 26 m. 28 m. 30
E#J = F ; B # j =C?; Fx7 = G 7 Cx; =D;
time interval of the canon) related by ascending fifth (the imitative interval of the canon). The ascending-fifth sequence can be seen in the major-minor seventh chord arrivals in mm. 8, 10, 12, and 14. which proceed through different inversions of A7, E7, B7, and F#7. Each of the fragments within these two-bar divisions consists of its own descending-fifth sequence. In m. 7 there is a B i going to an E7 which resolves to an A$ in m. 8. In mm. 9-10 the fundamental bass progression of mm.-7-8 is transposed by a fifth: F/i B?, and Ei. Rameau must have been proud that his canon so spectacularly displayed his preferred method of fundamental bass movement by fifths, both ascending and descending. The manner in which Rameau builds up these chords can be gleaned from the shape of the dux. From a harmonic point of view. the main task of the d ~ l xis to introduce each degree of the seventh chord, the root, third, fifth, and seventh. Once introduced, the degree remains constant when it reappears in each cornex. This is shown by the line in Example 7 connecting mm. 8, 10, 12, and 14: the chord degree contained in the dux at m. 8-the fifth-remains the fifth of the chord when it is imitated in the tenor, alto, and soprano voices. Since the initial appearance of a chord's fifth in the dux assures that the fifth will be found in the upper voices two, four, and six bars later, the dux can introduce other chord degrees at these points. In m. 10, the bass melody introduces the seventh of the chord. with the result that this chord degree is locked into the chords at the succeeding three even-numbered measures. The root of the seventh chord is found in the bass of
2,
m. 12, and the bass completes the cycle of four notes by sounding the third of the chord in m. 14. Because Rameau designs each pass through the dux to be eight bars long before its transposition, there is exactly the required time to introduce systematically all four of a seventh chord's degrees. In contrast to Zarlino, who presented the diversity of harmony by alternating major and minor chords, Rameau demonstrates diversity by displaying each of the inversions of a seventh chord within each pass through the dux.'Wnce again, the canon in Example 7 vividly portrays a melody's dependence on harmonic principles. Like Zarlino, Rameau utilizes a second canon, reproduced in Example 9, to convey his theoretical ideas in practice. In his first ~ s canon, the four entrances related by fifth resulted in the d ~ being repeated up a major third every eight bars. In the second one. the du.u is repeated up a minor third every six bars. Therefore four cycles must take place before the dux returns to its starting pitch E. now spelled as an Fb (m. 24). Since the time interval of the canon is again two bars, the canonic frame of this piece also presents a chord built on each of the twelve pitches. Because the canon has only three voices, however. Rameau has to work harder to show the common occurrence of seventh chords. For example, from the middle of m. 6 to the middle of m. 7, a G minor chord ends up as a G7, having moved through a passing chord on the downbeat of m. 7. In m. 8, the G7 resolves to a C minor triad, preparing the listener for a chain of dominant to tonic relationships that will continue until the performers have the good sense to stop. It is obvious that Rameau's decision to conclude the T~aite' with two canonic compositions was not motivated merely by a desire to engage a topic that had not been mentioned until that point. His decision to end with two canons seems inspired by the similar conclusion of Zarlino's Part 111, and more importantly by Rameau's belief that Zarlino failed to express important theoretical '
54
Music Theory Spectrum
Example 9. Rameau, Traitt de l'harmonie, Book 3 , Chapter 44, Canon at the Fourth
A
nous.
-
vec du
Gf
_
vin,
en
- - ~ _ _ --
-
dor- mons
~- - - -
-
-
Dux repeated up a minor t h ~ r d
Bbll:--: 1 Dux repeated up another minor t h ~ r d ~~
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-
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en - dor
mans
c
IIOU\, en
Ci
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-
dor
-
mons
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Rameau and Zarlino: Polemics in the Traite de I'harmonie 55
Example 9 [conrinued]
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--
--
3
- -
7
t
Dux repeated up another rnlnor third
etc.
I 8
-
ek
-
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ideas in hic compositions. As Rameau must have seen it, Zarlino had failed to construct seventh chords, had not demonstrated a knowledge of inversion, and had little sense of appropriate fundamental bass movement. Rameau's soi-disant lesson in "counterpoint" is subverted to become a lesson in harmonic resources and harmonic movement. By using canons to demonstrate his theories, he diminishes the distinctions between counterpoint and harmony. His ingenious pieces create the illusion that the older tradition of counterpoint has long demanded Rameau's harmonic explanations. What at first glance appears to be an homage to Zarlino turns out to be a persuasive appeal for change.
Rameau's canons may thus be seen as part of an effort to display continuity between Zarlino's theoretical/practical tradition and Rameau's revolutionary ideas. Rameau recognizes that Zarlino offers the TruitP's theories a degree of authority when the two authors appear to concur. In the first two books of the TruitP, Rameau's surprising endorsements of Zarlino's ideas (or what Rameau misinterprets as "Zarlino's ideas") accompany the introduction of several, far-reaching theoretical concepts. While Rameau adheres to many of Zarlino's methods and charitably accepts certain of Zarlino's theories as correct, Rameau usually leaves room for himself to be more correct than Zarlino. Three
56
Music Theory Spectrum
topics on which Rameau manipulates Zarlino's legacy are the source of intervals, motion from the major third, and the fundamental bass. In the case of each, Rameau's persuasive method employs an artificial agreement between Zarlino and himself.
T H E SOURCE OF INTERVALS
Rameau believed that his four-voice canon at the fifth was the first piece in which a single part generated so many other parts by strict imitation. He writes: "We do not think that more than four parts can be used here, for hitherto no such piece has appeared even in four parts."16 The four-part canon marks Rameau's "discovery" that a single source (the bass-line dux) can generate compositional material beyond the presumed two- or three-voice canonic boundary.17 This canon is not the TraitP's first example of Rameau revealing the generative potential of a single source. The theories in Books I and I1 of the Trait6 revolve around the development of a single source which Rameau calls the fundanzerztal sound. Rameau uses the venerable monochord to demonstrate that the fundamental sound is the source of intervals. He writes: Each part of the divided strings arises from the first string, since these parts are contained in that first, unique string. Thus, the sounds which these divided strings produce are generated by the first sound, which is consequently their source and their fundamental.lX l"'Nous ne croyons pas que I'on puisse en employer ici plus de quarte, puisque m&meil n'en a point encore parCi de la forte h quarte Parties." Trait4 de I'harrnonie, 360: Treatise on Harrnon~,370. "See my "Stacked Canon and Renaissance Composit~onalProcedure" for examples of stacked canons by Ockeghem, Mouton, and Willaert. Rameau was apparently unaware of these pieces ln"Chaque partie de ces cordes provient de la premiere, puisque ces parties sont contenues dans cette corde premiere & unlque; donc les Sons que doivent rendre ces cordes divisies. sont engendrez du premier Son. qui en est par consequent le principe & le fondement." Trait6 de I'harrnonie, 5; Treatise on Harmony, 7-8.
After showing that the fundamental sound generates the intervals, Rameau extends the reach of his source to include other elements of his theory. First, the intervals generated fro111 the undivided string are used as the building blocks for Rameau's two fundamental types of chords: the consonant triad and the dissonant se\ enth chord. Then, the favored movement of the fundamental bass is shown to derive from these same intervals. Rameau is very aware of the importance he attributes to the firzdamerztcil sound. He writes: "A principle on which everything is founded cannot be established too firmly; to lose sight of it for a moment is to destroy it."Iy Rameau develops his music theory in a way similar to the composition of a canon. By reiterating the principles of a single source, he creates a unified theoretical whole. Rameau's singular focus on the fundamental sound conflicts with Zarlino's explanation of the origin of intervals. Zarlino values the generative powers of two different sources-the octave and the unison. In the Istiturioni, Zarlino writes: T h e unison is a beginning, because from equality stems inequality. The diapason [octave] is a beginning, because from its duple ratio, the first unequal proportion, stem the other proportions of inequality.")
This dichotomy hinges on the belief that the unison is not the first consonance because it is neither interval nor consonance. Even Rameau concurs with Zarlino on the distinction between unison and consonance and writes: "The unison is not called a consonance as it does not fulfill the necessary condition for one, i.e., a difference in the sounds with regard to low and high."?l '""Cependant I'on ne peut trop hien etablir un principe, sur lequel tout est fondC. & c'est le ditruire. que de le perdre un moment de vCie." Trait4 de I'harrnonle, 49: Treatise on Harrnor~y59. ?u"CioP I'Unisono per la Equalita, dalla quale hi3 principio la Inequalita; & la Diapason. che & prima d'ogn'altra Consonanza. per la Dupla. dalla quale ha principio le altre proportioni della inequalith." Le Istitrrtioni harmoniche, 174: The Art of Counterpoint, 7. ""D'ou I'on dit que I'Unisson n'est pas une Consonance. parce qu'il ne s'y trouve pas la condition necessaire pour en faire une: spavoir la difference des Sons h 1'Cgard du grave & de l'aigu." Trait6 de I'harrnonie, 6 : Treatise on Harmon?: 8.
Rameau and Zarlino: Polemics in the Traite de I'harmonie 57
Since there is no pitch difference in the two sounds of a unison, it is numerically represented by the ratio of 1:l or equality. Zarlino explains that the octave, whose ratio is 1:2, is the first ratio of inequality. It follows that the octave is the closest of any of the intervals to the perfection of the unison. Zarlino writes that the octave is nearly perceived as a unison, and then he explains how numbers reflect this perception: Thus the diapason, even though constituted of two sounds of different location, seems to the sense to be but one sound, because the two are s o much alike. This results from the closeness of the number 2 to unity, and these are the t w o terms of its ratio, which is duple. This ratio contains t w o beginnings: unity, which is the beginning o f the numbers and is that among them which is indivisible; and the number 2, which is the beginning of the conjunction of unities and is the smallest number that can be divided."
The octave, with its 1:2 ratio, is the first interval of inequality, and therefore is the first interval that can be divided by either the arithmetic or harmonic mean. In the TraitP, Rameau addresses both of Zarlino's choices for source.?' Logically, we could expect Rameau to conlplain that neither of Zarlino's sources are the fundamental sound. Instead of taking this confrontational tack, however, Rameau gently reassesses Zarlino's two sources in light of the fundamental sound. :"'Et P in tal maniera semplice la Diapason, che sehen 6 contenuta da due suoni diversi per il sito: dirb cosi: paiono nondimeno al senso uno solo: percioche sono molto simili: & cib aviene per la viciniti del Binario alla Unith. che rono contenuti ne gli estremi della sua ihrma, che P la Dupla: Onde tal forma q uella che tra contiene due principii: la Uniti. che e principio de i Numeri. & .i loro non si pub dividere: & il Binario, che P il principio della congiuntione delle Unita; & .i il minimo numero, che dividere si possa." Lr Istit~ifiorziharttiotiic/z(~. 174; The Art of Courzterpoirzt. 7-8. "Rameau does not acknowledge Zarlino's statement that the unison is the true source. Zarlino writes. "Moreover. since inequality originates in equality, the diapason originates in the unison." Zarlino, The Al? of Co~mtrrpoint,7. ("Che delln Equalith h i principio In Inequnliti: cosi bibogna dire. che dall'Unisono habhin principio In Diapason." Le Istitutiorzi harmorriche, 174.)
This approach is similar to Rameau's reassessment of contrapuntal canons in light of his own harmonic theories. Let us consider how Rameau approaches Zarlino's statement that the octave is the cause. Rameau selects a pair of short sentences from Zarlino's lengthy explanation for inclusion in the Trait&: The octave is the mother, the source, and the origin of all intervals. By the division of its t w o terms all other harmonious chords are generated.'i
Despite needing to correct his predecessor, Rameau is reluctant here to criticize. Instead he offers a rephrasing that somehow manages to endorse Zarlino's opinion while simultaneously changing it. Rameau writes: "To validate Zarlino's opinion . . . we must add the following: the fundamental sound uses its octave as a second term."?' Regarding the unison as source, Rameau conflates the equalih of the two pitches in a unison with the u r z i ~of his "correct" single source-the fundamental sound. In the following passage. Ranieau sets up Zarlino's ultimate fall by first endorsing Zarlino's opinion that the unison is a source. After having stated that . . . the unit, which is the source of numbers, represents the sonorous body from which the proof of the relationship between sounds is derived. and that the unison is the source o f consonances, Zarlino forgets all this in his demonstrations and rules. Far from following the principle he has just announced. the further h e goes the more h e draws away from it.'" '4"L'Octave est la mere. la source & I'origine de tous les intervales. c'est par la division de ces deux termes que s'engendrent tous les accords Harmonieux." TrtritP de l'hurttlor~ie,8: Twutise on Hurmoriy 10. ""De \orte que pour faire valoir le sentiment de Zarl~n.I'on ne peut di\penser d'y ajofiter, que pour l o r le Son fondamental se sert de son Octave comme d'un second terme." Trait; clr I'harrrronic. 8: Trecrtisr orr Hurtrrorzj: 10. 'h"Zarlin nprCs avoir remnrquC que . . . I'unitC qui eit le principe des notnbres. nous represente le corps Sonore, dont on tire la preuve du I-apport de\ Sons. & que I'Unisson est le principe des Consonances; Zarlin. dis-je, oublie tout cela dnns ses DCmonstrations & dans ses Regles: loin d'y suivre le principe qu'il vient de declarer. plus il penitre. plus il s'en Cloigne." Truite tie I'htrrrrzonie. 18; Treatise on Harmon?; 22.
58
Music Theory Spectrum
In the middle of this passage, Rameau seems to be accepting the unison as source. The unison may be one step closer to Rameau's fundamental sound than the octave is, but it certainly is not equivalent to the fundamental sound.?' The beginning of this passage confuses the situation more by implying that Zarlino recognized a third source-the fundamental sound. expressed by the unit. Rameau's indication that Zarlino actually considered the single string as a \ource is rather hopeful. Zarlino's method for generating intervals begins by positing a string, but the material of the string is used only in service of the unison and octave. By aligning Zarlino's theories with his own, Rameau screens his own radical moves. Rameau's complaint is not that Zarlino was wrong. but that Zarlino strayed from an initial recognition of the truth concerning origins. Rameau's purpose is to rifir111past theories that hahe been betrayed. rather than dispel Zarlino's theories. Rameau does not seem bothered by the fact that he has to stretch Zarlino's beliefs to engender agreement.
\ O I ( E - L F 4 D I U G FROM T H F M 4 l O l i THIRD
On the topic of motion from the major third, Rameau contin~les to tnanip~llateZarlino's ideas in order to stress that the ideas in the Trciitc; are compatible with, rather than divergent from, the ideas in the Isriturioni. Ratneau first quotes Zarlino as saying, "We should ascend fro111the major third and the tnajor sixth to the ~ c t a h e . " ' ~ Given this statement. if the major third, whose lower note is the ~, with the root of a dominant chord according to R a m e a ~ proceeds upper part moving by semitone. then it is inehitable that the bass descends bq fifth. R a t n e a ~thus ~ reassures his readers that Zarlino '-Rumeau'\ indication that Zarlino al\o cons~deredthe single string as a source 15 rather hopeful. Zarlino's method for generating inter\al\ begins h? positing a string, but the material of the string i\ used on11 in \ervicr of the unison and octa\r. z E " Q ~ ' i falloit l monter dc la Tiercr mdjeure & de la Sixte mqeure h I'Octave." Trciitc; iir I ' h u r n ~ o ~ ~56: i r .Ti-c.cltite oil Horrtzori). 64-65.
understood the typical progression of the fundamental bassmovement by fifth."' Rameau has adroitly introduced his ideas so that his readers, and perhaps also he himself, need not feel that they are dismissing premises of previous music theory. Philip Gossett, howeher, has noticed that Rarneau practiced a deception in claiming to reproduce the original passage from Zarlino. Zarlino actual!\ wrote, "The ditone (major third) and the major hexachord (major sixth) desire to expand into a fifth and an There is no mention whatsoever here (or octave [re~pectihely]."'~ in the surrounding text) of tnajor thirds going to octaves, although that is the very detail that Ratneau stresses as similar between the two theories. Rameau's misreading relies on an exception that Zarlino introduce\ altnost thirty chapters after giving the typical voice leading for the third." Ehen in this later chapter. it is clear that Zarlino does not require that the bass rrlrt,ci>,.s descend bq fifth. Example 10. taken from Le I,trirutioni, shows that Zarlino also accepti a progression in which the lower voice descends by semitone.'! Bq narsowing in on the major third moving to the octave with the Zarlino's original bass moving down by a fifth, R a m e a ~replaces ~ ~oice-leadingprescriptions (that a third expands to a fifth) with one option of the two exceptions Zarlino allows. Apparently this is enough of a connection for Ratneau to avoid acknowledging any break on his part from the earlier tradition. The reader's first reaction might be that Rameau is misreniembering the I.~rirutioni,o r at least being careless with his presentation. However, sehen chapters later in the TmitP. Ranieau's ac"'Latcl- in the 7i.clitP. Rameau \\.rite\ that Zarlino sa?\ of the ha>\. "It\ rialUI-alprogl-ession in perfect cadences is to dc\cend a fifth." T,rciri$e ,111 Hiii~itioii\, 159. ("Sa PI-ogres\ionn,iturelle dani Ic\ Cadences pill-faite\. est de de\ceridre dc Quinte." Tt.cliti tlr I'lzciririoiric. 116.) '""I1 Ditono & lo Herachordo maggiore desiderano di fr11-\1~ii~ggiol-I. LCnendo I'uno nlla Quinta & I'altro alla Otta\a." Lc I.~rifrrti(~rrr il~~rtiiotli~~ir~'. 1x2: Tlic Art of Co~lt~f(,r.poi~lr, 23.
Za1.11no.Tlrr Art of Colrtifei-poi~ir.80.
"lhid.
Rameau and Zarlino: Polemics in the Traite de I'harmonie 59
Example 10. Zarlino's two acceptable progressions from a major third to an octave
knowledgement of what Zarlino actually said regarding the major third, namely that the major third ordinarily expands to the tifth, suggests that he is conscious of his initial selectivity." Apparently, Ra~neaubelieves his system has so impressed his readers by now that they will not recognize or be disturbed by his inconsistent reporting.
T H E F U N D A M E N T A L BASS
In his discussion of fundamental bass, Rameau again emphasizes a connection to Zarlino's theoretical tradition, perhaps because of the radical implications of the new theory. In a prefatory description of the fundamental bass, Ra~neaumakes his concept seem as innocent as possible. He writes, "The source is represented by the part called the buss in IIIUS~C. to which the epithet ,f~~ndutnet~rul is added."" Rameau merely seems to be relabeling the traditional term "bass" with the fancy term "fundamental bass," making the fundamental seem familiar. Rameau appeals to Zarlino to stress the familiarity of the concept: "As the part containing the fundamental sound is always the lowest and deepest, we call it the bass. Here is what Zarlino says on the subject. . . ."35
Rameau implies that Zarlino discussed the term fundamental bass in his writings. Of course, Rameau's term is a label that Zarlino would not have recognized as synonymous with "bass." Zarlino never conceived of the bass as anything but the lowest note written. In Book I1 of the TruitP, Ra~neauturns to one of Zarlino's examples as evidence for the fundamental bass. Ra~neaujustifies adding a fundamental bass to Zarlino's examples by saying, "Notice that he cannot avoid recognizing a bass in harmony, and that he desires it even when it is absent."'(1 Rather than position himself outside of Zarlino's theories and argue for change, Rameau consistently displays a desire to overthrow Zarlino's theories fro111 within. An interesting psychological battle occurs within the TrczitP's pages, as has been observed in the compositional canons, and the theories about the origin of intervals, motion from the major third, and fundamental bass. Rameau allows himself a certain freedom in reporting Zarlino's ideas, and Rameau's motives color the Zarlino that emerges in the TruitP. An important part of the TruirP's story is how Rameau embraces Zarlino's presence wh,le at the same time introducing a new theoretical landscape. ABSTRACT Rameau's methods to secure the future of his o w n theories involve a careful treatment of Zarlino's theoriej. Rameau. in the Trriiri d r I'Irurmoirir, wrestlej with Zarlino's ideas and uses them for his o w n end. The culminating compositions of each author's treatise. which are all canons. are analyzed. In addition, a variety of theoretical i s w e s are examined. including the source of intervals. motion from the major third. and the fundamental hass.
"Trecifi.se or1 Harri~orl\: 103.
""Le principe y est pour lors represent6 danr la Partie de Musique qu'on appelle Basse. i laquelle on ajot"ite l'ipithete de Fondamentale." Trtirti dc /'heir.rrlorlie. Preface: Trecirire or1 Hcirrr~orz?;xxxv. ""On appelle Bcisse, la partie ob regne ce Son fondamental, parce qu'il est toi?jours le plus gra\e. & le plus bas: & \oicy comment Zarlin r'explique \ur ce sujet:" TrtritP dr l'harri1or1ir.19:Trc,ati.\r or1 H a r r ~ r o r 59. ~~.
'h"Ob je vous prie de remarquer qu'il ne peut r'empescher de reconnoitre une ha.$e dans I'Harmonie, & qu'il temoigne la souhaiter lorsqu'il ne l'entand point." TriritP dr I'harn~ot~ie. 78: Trrcirise orr Hcirnlorry 92.
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[Footnotes] 10
Stacked Canon and Renaissance Compositional Procedure Alan Gosman Journal of Music Theory, Vol. 41, No. 2. (Autumn, 1997), pp. 289-317. Stable URL: http://links.jstor.org/sici?sici=0022-2909%28199723%2941%3A2%3C289%3ASCARCP%3E2.0.CO%3B2-6
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