Rameau and Zarlino polemics

7/25/2019 Rameau and Zarlino polemics

1/17

Rameau and Zarlino: Polemics in the "Trait de l'harmonie"

Alan Gosman

Music Theory Spectrum, Vol. 22, No. 1. (Spring, 2000), pp. 44-59.

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2/17

Rameau and Zarlino: Polemics

in the

Traite de I harmonie

Alan osman

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 well-

documented debates, a struggle to si tuate 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,

i t quickly becomes apparent that Rameau is preoccupied with

Zarlino. Rameau frequently footnotes two of Zarlino's treatises,

the 157 3 version of the Istitutioni harmoniche, and the 1589 ver-

sion of the Dimostratoni harmorliche included as the second vol-

ume of De tutte l'opere. Further, Zarlino is the only person with

an entry in the Trai t i ' s Table of Terms.?

'Many authors. including Thomas Christensen and Cynthia Verba, have

documented Rameau's reactions to contemporaries who modified and threat-

ened his popular and respected theories. See Thoma s Christensen. Ranleau and

Musical Thurrght in the Enlightenn~ent New York: Camhridge University

Press. 1993). and Cynthia Verba, Music and the French Enlightennlent: Recoil-

struction qfa Dialogue 1750-1764 (New York: Oxford University Press. 1993).

'Jean-Philippe Rameau. Trait4 de I'harnlonie r4drrite u ses principes

u

turels (Paris, 17221, xxiv: reprodu ced in vol. of The Conlplete Theoretical

Writirzgs o Jearz-Philippe Ranleau (1683-17641, ed. Erwin R. Jacobi (Ameri-

can Insti tute of Mus ~col ogy, 967-72); translated hy P hilip Gossett as Treatise

on Harnlorzy

(New York: Dov er, 1971). Iv.

Other theorists, such as Descartes and Mersenne, certainly in-

fluenced 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 w ritings 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

I

that Zarlino's name is completely

ab

sent from Boo ks 111 and IV, titled Principles of Com positi on

and Principles of Accom paniment respectively. And yet

Rameau hardly neglects his precursor in the discussion of practi-

cal music. In fact, I believe that Rameau's most acute awareness

of Zarlino's shadow, an d his most powerful a ttempt to distance his

readers from his predecessor, occurs near the conclusion of Book

111. At this point one finds-with som e shock-that for his culm i-

' Zarlino, Auteur celehre en Musiq ue, qui a ecrit a p eu - p re~ ep ui s 15 an\ .

dont on ne trouve que de tr2s-foihles Copies. dans les Ouvrages qui ont

part

apr6s les siens, sur le m@m e ujet. Trait4 ile l'harmonie, Preface, second page;

Treatise on Harnlon?: xxxiv. All translations of Ra me au given here a re by

Philip Gossett .


3/17

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

book s now, one hardly expects him to cap off his composition les-

son with a strict contrapuntal form.

An explanation is suggested by the fact that the culminating

composit ions of Book I of Zarlino's Istitutioni harmoniche are

also a pair of can ons. These tw o authors' similarly placed canons,

which have been almost completely overlooked, turn out to be

summaries of their respective theories on music. Furthermore,

Rameau's composit ions can be seen as a powerful demonstration

of the inadequacies of the traditional contrapuntal explanations of-

fered by Zarlino.

close examination of Zarlino's two culminating canons, and

the methods by which Zarlino cons tructed them, will help one rec-

ognize exactly how Rameau's two canons in the Traite are a re-

sponse 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 dis-

tance. He writes, Constant practice of this close imitation has re-

sulted in such a common idiom that a fugal pattern cannot be

found that has not been used thousands of times by various com-

p o s e r ~ . ~n 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 c omm une di com porre, che a1 presente non si ri trova qua si Fuga, che non

sia stata mille migliaia di volte usata da divers1 Compositori. Gioseff o Zarlino,

Le Istitlctlon~harnloniche, reprint of the 1573 Venetia edition (Ridgew ood, N.J.:

Gregg P ress, 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 M arco and Palisca.

' . cercarem o con ogni nostro potere di fare delle Fughe. ch e siano piu

nove.

Le Istitlctioni hannonicrhe.

258:

The Art

o

Counterpoirzt,

127.

Rameau and Zarlino: Polemics in the Traite de l harmonie

5

Zarlino often relies on canons to demonstrate the practical ap-

plications of his contrapuntal rules, and by exploring different

canonic types, he show s his preference to seek out less comm on

varie t iesh Example 1 lists the many different canonic techniques

in the

Istitutioizi

in their order of presentation, which roughly cor-

responds 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 reproduc ed in

Examples 2 and 3.

These canons display two of Zarlino's important lessons about

composing in three or more parts. When earlier discussing three-

voice compositions in Chapter 59, Zarlino writes,

Composition may be called perfect when, in every change of chord, as-

cending or descending, there are heard all those consonances whose com-

ponents 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 dou-

hle counterpoint. He writes, Though there are many ways of writing such

counterpoints, as

I

have said, I shall demonstrate only those that seem most dif-

ticult and most elegant. This mill avoid boring the reader, who can readily infer

the other procedures for himself. The Art o Counterpoint, 205. ( Ma ancora

che molti siano li modi di comporre tali Contrapunti: come ho detto: porrb sola-

mente quelli, che mi sono paruti piu difticili piu elegant]: acc io non sia te-

dioso a i

let tor^:

da quali ciascuno ingegnoso potra comprendere, come ai

haveri da reggere in qualunque altra maniera di simili compositioni. Lr

Istitlctioni harnloniche, 297.)

- Quells com pos~ tione i puo chiamare Perfetta: nella quale in ogni muta-

tione dl chorda, tanto verso i l grave, quanto verso I'acuto, s empre si odono tutte

quelle Consonanze. che fanno varieti di suono ne i loro estremi. Et quella

6

veram ente Harmo nia perfe tta, che in essa si ode tal consonanze: ma li Suonl , o

Conson anze, 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? o Counterpoint, 186.


4/17

--

46 Music Theory Spectrum

Example

1 .

Canon types found in

L lst itut ioni hannon iche,

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)

3. Adding a two-part canon to an existing tenor (Exs. 154 and 155)

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)

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

IT

- . I '

I. '_ I -

I -

t

I

I I I

I

I

Ha$\, and consequent of the s oprano


5/17

--

-

Rameau and Zarlino Polemics in the Traite de I harmo nie 47

Example

3.

Zarlino.

I.rtit~ltiotli

Par t 111, Cha pter 66 Perpetual composition with two pairs of parts in consequence by contrary m otion

Soprano, and p ~ f the h ' ~ \ \

l r

Tenor. and consequent of the alto

-

--

-.

p e

--------~T-x---.-

-

[bpp

,

I Z

.-.

-==

~

-

--

~

.

-. .

2 _ _ I

Ba\\ . and consequent of the xiplnno

~

-=7T ~~~~==-I~r*-*~-LLPL0-~r-*~F*-~ &-----&-

ill

- .--

- L

.

-. I . I , ' I L

Zarlino's melodies are masterfully constructed to maximize the time Zarlino rnakes an adjustment to provide n Phrygian cadence,

numb er of perfect harmo nies despite the strict contrapu ntal form . the canonic frame of both exam ples simply alternate s two differ-

Almost every verticality is a chord. In each example. Zarlino is ent perfect triads.

particularly careful that the canon's frame. which define as those

The conditions of composing a double canon by contrary mo-

chords that fall at the time interval of the canon, do not spiral into tion. interesting in their own right. also demonstrate that Zarlino's

impe rfection. The time interval of the canon in each cxam ple is choice of canon type was motiva ted by the lessons of the

I.stitll-

two bars in the modern realization. Example lists the chords in tioni.

By

closely examining this presentation, one can see the con-

the odd-numbered measures of Examples 2 and 3. The list starts trapuntal mastery with which Ram eau is com peting. In a contrary-

with m. 3. because at that point enough voices have entered to cre- (notion canon, although the lrx and comes could be inversionally

ate a perfect harmony. Excep t for the end of Exam ple 2. at which symm etrical around any pitch. Zarlino decide s in both canons that


6/17


Example 4 Triadic roots at the time interval of the canon in Zarlino's

contrary motion canons

Triadic Roots in

Measure Examule 2 Examule 3

a

z

C

(

l st inv.)

D

a

z

C D

F

(I st inv.)

E etc

the note D will be the inversional center. This arrangement fits in

with the intervallic make-up of the white notes as is show n in

Example

5 .

The chromatic notes Bb and F# are also included be-

cause Zarlino uses them in both pieces. Exam ple

6

show s that the

alternating chords (cf . Ex. are also, as one would expect, those

that are symmetrical around D . The first column sh ows the inver-

s ional mappings between notes of the A and C tr iads . These

chords alternate throughout most of the f irst canon . The second

column shows the mappings between the G and

D

tr iads. These

chords alternate in the second can on.

For Zarlino. however, the mere presence of perfect ha rmonies

does not assure variety. He stresses throughout his text that there

are dis t inct ive types of per fect harmonies . For example. in

Chapter 3 of Part 111. Zarlino writes:

Th e varie ty of the harm ony in auch s i tuat ions does not consis t so le ly in

t h e v a r i e ty o f t h e co n s o n an ces t h a t a re fo u n d b e t w een t w o v o i ce s . b u t a l so

in the varie ty of the harmonies-which [varie ty] i s determ ined by the po-

si t ion of the note that makes a t h ird o r t en th a b o v e t h e l o w es t v o i ce o f t h e

co mp o s i t i o n . E i t h e r t h e se [ i n t e rv a l s ] a re mi n o r , an d t h e h a rmo n y t h a t

a r i se s i s d e t e rmi n ed b y o r co r re sp o n d s t o t h e a r i t h me t i ca l p ro p o r ti o n o r

d iv is ion . or they are major, and such a harmony i s determin ed by or corre-

sponds to the har mon ic mean . On th is varie ty depe nds a11 the d iversi t )

an d p e r fec t i o n o f h a rmo n i e s . . F o r a s I h av e sa i d e l sew h e re , w h en t h e

ma j o r t h i rd i s b e l o w , t h e h a rmo n y i s ch ee r fu l , an d w h en i t i s p l aced

ab o v e . t h e h a rmo n y i s s ad .8

Zarlino did not just happen to choose to write a canon in contrary

motion as a contrapuntal ch allenge. 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 Exa mple 2, the dux of the canon be-

tween 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 pat-

tern continues until In. 13. For almost the entire piece, the inner-

voice canonic frame consists of the major third from C to E.

Zarlino has the choice of whether to place a mino r third abo ve 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 mino r third abov e 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 sou nds 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

ux

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 iar ie ti anco delle Harmonie. la quale con\iste nella positlone della

chorda. che h la T er n, over la Dccima wpr a In partc gra \e della cantilenu.

Onde. overo che cono minor1 I'Harmonia che nasce. C ordinata.

i

a \ -

\imiplia alla proportlonalith, o mediations Arithmetica: overo one ~ n ag p o r - i

t a le Harmonla ord inata , over s i as \ i~ n~ gl ialla mediocrith Harmonica:

da

questa varieth dipende tutta la div er\ iti la perfettione dells Harm onie per-

cioche (como hh detto altro\e) cluando \ pone la T e r n mappiore nella parte

grave. I'Harmonia

\ i fh

allegra: quan do ci pone nell'acuto si S ~ I e\ta. LC

I ~ t i r u r ~ o ~ i ~cir-r~ic~~~ic.I~c~10-1

:

The 4r t

of

Cour~rc~ipoi~it .

9-70.

Becaus e of the utrict cond ~ti ons or con\tructing triad\ in a double controrq

motion canor:. it is clear that Zarlino

i i

adding a single note or

C;

to a pitct,

pairing that

i \

set C and

E).

Bq this method. mm.

3.

7. and

I are

C rnajor tri-

ads. What is inte reh ng is that the chord in

m .

7 ic

a C

major triad In fir\[ in\t.r-

sion. Ba\ed on his theoretical text. Zarlino would not have recopnired

a

relation


7/17

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 B C D E F G

Distance (step s): 112 1 1 112

Example

6.

Inversional mapping of triads in Zarlino's contrary mo-

tion canons

Chord Root$ C -A D-G

F

-E

G-A A -G C -E

E t , C

F t , B b A -G

C-E D-D F 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-numb ered b ars 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 an d 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 writ ing lead to an early instance of inversional thinking.

Pursuing this subject could easily generate another essay.

Rarneau and Zarlino Polemics in the


9

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

on e fifth (D-A) is divid ed 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 com-

pose. More importantly, however, it directs the reader's attention

to the structures recomme nded in Zarlino's text-perfect chords

that demonstrate the diversity of harmony. In Example 3, the

ux

of this composition is only seven notes, but the piece gains con-

siderable length because it is a perpetual canon. This is striking,

because i t 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 pro-

vides his own commentary on issues raised in Zarlino's canons

about which type of chord to privilege, and how to obtain diver-

sity 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 in-

troduced w ith enorm ous fanfare, they m ay well be the first pieces

specifically designed for harmonic an alysis.

Rameau 's four-part canon at the fifth is reproduced in Example

7. This piece is what have termed a stacked canon, because the

tenor and alto, each of w hich is an imitative voice, also each plays

the role of

ux

for another part enterin g later.'"hus Ram eau

'Osee Alan Gosman, "Stacked Canon and Renaissance Composi t ional

Procedure," Journal of usic

heor

4112 (1997): 289-3 17.


8/17


Example

7 .

Kameau Trait6 de l han?iorzie, Book 3 Chapter 44 Canon at the Fifth

Ah

I

Loin

A

I I L

4

de ri

rc.

Plcu rolls Pleu ran\.

8

Dux repeated up

a n


9/17

Ram eau and Zarlino Polemics in

the

Traite de I harmo nie 51

xample 7 cont inued]

Dux repeated up another major th~r d

CI: a l i DIP Gd ed- AI4 Dd- b% Eli AdQ

f 1

bl

Dux is an auemen ted \ekenth hove the uartl ng nltch


10/17

5

usic Theory Spectrum

revives Zarlino's practice of displaying canonic acumen by stray-

ing from a straightforward m odel.I1 But Ram eau's four-voice

canon is not merely a dem onstration of compositional technique.

He a lso intends to remind the reader of what he considers a funda-

mental 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 com pose som e four-

part 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 re-

lates closely to the T rrrite"~eachings. 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 propul-

sion, particularly when used on the do minan t. In Book 11. Chap ter

2. Ra~neau rites of the seventh chord.

T h e d ibe r s i t) tha t thi s c hord b r ings

b

in t roduc ing a c e r ta in t a r tne s s

w h i c h s i ~ n u l t a n e o u s l y n h a n c e s t h e s w e e t n e ss o f t h e p e r fe c t c h o rd , ~ n a h e s

us de s i r e i t s p r e s e nc e , no t r e jec t i t . W e m us t thu s p la c e i t a m o ng the fun -

da m e n ta l c hords , s inc e i t in no wa y d e s t roys the s ourc e wh ic h s ubs i s t s in

the lowe s t s oun d o f the pe r fe c t c hord . '

"Ranieau crroneou\ly procla im\ h~m self o hc th e t i n t t o h a ~ conipo\ed a

four-part canon of thlh type

(T rru ri te or7 Hcrr.~jro~i\:70 .

" "L es e r r eu r s q u ~ r trou\ent dans Ir\ Rcglrs de Zarlin proLienncnt cn pnr-

t le dr ce qu ' i l n ' en\isagroit que deux Sons la fois." Trr~it is lr / ' i ~r i r~? ~or j ir ,

xx i \ : Tretrr ite or Hrirrriorr); I \ .

l 'l n Z a r l~ no ' \ ou r - \o ic e c anons hy in ~e r s io n . n ly two \e \ rn t h c hord \

occur. The\e are in Example 1 on the last quarter note of mm.

8

and 13.

In

each

case, the \e\enth has a clear pabsing function.

""La d~\e rs i tC ue c r t accord y cause en y introduisant unr crrtalne anier-

tunic. qui relc\c en nitnir tcnips la doucrur de I 'accord parfait. doit nous Ir

f111e souhalter, hicn l o ~ n e le rejrtter, ne po u ~ an t ous dl\p enscr pour lors de

la mettre au nomhrr dea accords fondamentaux. puisqu'il ne dCtru~tpolnt le

pr in c~ pe ui subaistr toC?jouu dans le Son gr a\e de I 'Accord parfait." Trfiit P rlu

l ' hc~r n~or l i r , on H ar m or n ,

61

2 , T r ~ a t i ~ r

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 im possible, one

is aware of their approach as the imitative voices are added. In m.

2 the parts sugge st a C7 chord (V 3IIV) 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 sev-

enth 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 ~n ea u arefully planned the harmonic pro-

gression first, and allowed it to guide his melody. He thereby pro-

vides compositional evidence for on e of his principal assertions,

that harmony is prior to melody.

The natural fit between the structure that R a~ ne au reates (a

four-voice canon) and his penchant for seventh chords suggests to

his audience that the contrapuntal devices of the past have actu-

ally 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 be-

fore 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 ~ Yare exactly how long i t take s fot- the

even-numbered m easures (the canonic frame) to present a dom i-

nant seventh chord on each of the chromatic scale's twelve notes.

In addition to displayin g all possible dominant seventh chord^

Rameau takes very seriously how they are connected. Relying on

his explan ation in Boo k 11. progressio ns of f und a~ ne nta l bass

notes follow the findings based on the div ided string. In particular.

they dem onstrate the desire d 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


11/17

Example 8. Chords at each time interval in Rameau's Canon at the

Fifth

m. 8

A;

m. 16

C : m. 24 E J

=F;

m. 10

E?;

m. 18

G : m. 26 B j= C ? ;

m.12 B 7

m. 20

D ' m. 28

F x7 = G 7

m. 14

F ;

m. 22

A ?

m.

30

C x; =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 inv ersions 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 funda-

menta l bass progression of mm.-7-8 is transpose d by a fifth: F/i

2

B?, and Ei . Rameau must have been proud that his canon so spec-

tacularly 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~lxs 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 conta ine d in the dux at m. 8-the fifth-re-

mains 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 de-

grees 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

Rameau and Zarlino Polemics in the Traite de I harmonie 5

m. 12, and the bass comple tes the cycle of four notes by sounding

the third of the chord in m.

14

Becau se 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 inver-

sions of a seventh chord within each pass through the dux. 'Wnce

again, the canon in Example 7 vividly portrays a melody's depen-

dence on harmonic principles.

Like Zarlino, Rameau utilizes a second canon, reproduced in

Example 9, to convey his theoretical ideas in practice. In his first

canon, the fou r entrances related by fifth resulted in the d ~ ~ seing

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 cy-

cles 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 tw elve pitches. Because the canon has

only three voices, however. Rameau has to work harder to show

the comm on 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

G 7,

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 lis-

tener for a chain of dominant to tonic relationships that will con-

tinue until the performers have the good sense to stop.

It is obvious that Rameau's decision to conclude the T~a i te

with two canonic compositions was not motivated merely by a de-

sire to engage a topic that had not been mentioned until that point.

His decision to end with tw o canons seem s inspired by the similar

conclusion of Zarlino's Part 111, and more importantly by Ra-

meau's belief that Zarlino failed to express important theoretical

' < T h eem ph as ~s n seventh chords makes the l in t note of the plece Teen1

l ~ k e he only place where a t r lad is implied. Inter es t~n glj ,when the~ ~ n p l e

openlng C returns twenty-four bars later (as a

B ) .

it is the tifth of a seventh

chord.


12/17

5 usic Theory Spectrum

Example

9.

Rameau, Traitt de l harmonie, Book 3 , Chapter 44 Canon at the Fourth

vec du vin, en do r- mons nous. en dor

mans

I IOU\ ,

en dor mons

nous.

G f ~ _ _

c

Ci F:

Dux

repeated

up

a

m nor

t h ~ r d

Bbll:--: ~ ~ ;

Dux repeated up another

m nor

t h ~ r d


13/17

Rameau and Zarlino Polemics in the Traite

e

I harmonie

Example

9

[conrinued]

,~

7

Dux

repeated up a nother rnlnor third

ek

-

7

Dux

15 a dinlinishsd ninth abobe the starting pitch

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 funda-

mental bass movement. Rameau's soi-disant lesson in "counter-

point" is subverted to become a lesson in harmonic resources and

harmonic movem ent. By using cano ns to demonstrate his theories,

he diminishes the distinctions between counterpoint a nd harmony.

His ingenious pieces create the illusion that the older tradition of

counterpoin t has long demanded Rameau 's harmonic explana-

tions. What at f irst glance appears to be an homage to Zarlino

turns out to be a persuasive appea l for change.

etc.

A d

Rameau's canons may thus be seen as part of an effort to dis-

play continuity between Zarlino's theoretical/practical tradition

and Rameau 's revolu tionary 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") ac company the intro-

duction 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


14/17

56

Music Theory Spectrum

topics on which Rameau manipulates Zar l ino' s legacy are the

source of intervals, motion from the major third, and the funda-

mental bass. In the case of each, Rameau's persuasive method

employs an artif icial agreement between Z arlino and himself .

T H E S O U R C E O F I N T E R V L S

Rameau believed that his four-voice canon at the f if th was the

first piece in which a single part generated so many other parts by

strict imitati on. 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 Ram eau's dis-

covery that a single source (the bass-line dux) can generate com-

posi tional mater ia l beyond the presumed two- or three-voice

cano nic boundary.17 This can on is not the TraitP's first exam ple of

Rameau revealing the generative potential of a single source. The

theories in Books

I

and I of the Trait6 revolve aroun d the devel-

opment of a single source which Rameau calls the fundanzerztal

sound.

Rameau uses the venerable monochord to demonstrate that the

fundamen tal sound is the source of intervals. He writes:

Each part o f the d iv ided s t r ings ari ses from the f i rs t s t r ing , s ince these

part s are conta ined in that f i rs t , un ique s t r ing . Thus, the sounds which

t h e se d i v i d ed s t r i n g s p ro d u ce a r e g en e ra t ed b y t h e f i r st so u n d , w h i c h i s

co n seq u en t l y t h e i r so u rce an d t h e i r fu n d amen t a l. l X

l 'Nous ne croyons pas que I'on puisse en employer ici plus de quarte,

puisque m& me l n'en a point encore parCi de la forte h quarte Parties. Trait4 de

I harrnonie,

360:

Treat ise on Harrn on~ ,

70.

See my Stacked Canon and Renaissance Com posi t~on al rocedure for

example s of stacked canons by Ockeghem, Mouton, and Willaert . Rameau was

apparently unaware of these pieces

ln Chaque partie de ces cordes provient de la prem iere, 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 so und generates the intervals,

Rameau extends the reach of his source to include other elements

of his the ory. First, the in tervals g enera ted fro111 the un divid ed

string are used as the building blocks for Rameau's two funda-

mental types of chords: the consonant tr iad 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 im portan ce he attributes to the firzdamerztcil sou nd.

He writes: A principle on which everything is founded cannot be

established too firmly; to lose sight of it for a moment is to de-

stroy it. Iy Rameau develops his music theory in a way similar to

the compo sition of a canon. By reiterating the principles of

a

sin-

gle 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 val-

ues the gene rative pow ers of two different sources-the octav e

and the unison. In the Istiturioni, Zarlino writes:

Th e u n i so n i s a b eg i n n i n g , b ecau se f ro m eq u a l i t y s t em s i n eq ua l i ty . Th e d i -

apas on [octav e] is a beginning , beca use fro m i t s duple ra t io , the f irs t un-

equal propor t ion , s tem the o the r proport ions of inequali ty . )

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 conso-

nance as it does not fulfill the necessary condition for one, i.e., a

difference in the sou nds with regard to low an d high. ?l

' Cependant I'on ne peut trop hien etablir un principe, sur lequel tout est

fondC. c'est le ditruir e. que de le perdre un moment de vCie. Trait4 de I har -

rnonle,

49:

Treatise on Harrnor~ y

59.

?u CioP I'Unison o per la Eq ualita, dalla qua le hi3 principio la Ineq ualita;

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'es t pas une Conson ance. parce qu'il ne s'y

trouve pas la condition necessaire pour en faire une: spavoir la difference des

Son s h 1'Cgard du grave de l'aigu. Trait6 de I harrnonie, : Treatise on H ar-

mon?: 8.


15/17

Since there is no pitch difference in the two sounds of a un ison,

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 d iapas on , eve n though cons t i tu t ed o f two s ounds o f d i f f e r ent lo -

ca t ion , s eems to the s ens e to be bu t one s ound , becaus e the two ar e s o

much a l ike . Th i s r esu l t s f rom the c losenes s o f the number to uni ty, and

thes e a r e the tw o t e rms of i ts r a t io , which i s dup le . Th i s r a tio con ta ins two

beginn ings : un i ty , which i s t he beg inn ing of the number s and i s t ha t

amon g them which i s i ndiv i s ib le ; and the number 2, which i s t he beg in-

n ing of the con junc t ion o f un i ti es and i s t he s mal l es t number tha t 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 arith-

metic 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 re-

assesses Zarlino's two sources in light of the fundamental sound.

: 'Et 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: per-

cioche sono molto simili: cib aviene per la viciniti del Binario alla Unith. che

rono contenuti ne gli estremi della sua ihrma, che la Dupla: Onde tal forma

contiene due principii: la Uniti. che principio de i Numeri. uella che tra

loro non si pub dividere: il Binario, che il principio della congiuntio ne delle

Unita; il min imo numero, che dividere si possa. Lr Ist i t~if iorzi artt ioti ic/z(~.

174; Th e Art of Courzterpoirzt. 7-8.

Rameau doe s not acknowledge Zarlino's statement that the unison is the

true source. Zarli no writes. Moreover. sinc e inequality originates in equality,

the diapason originates in the unison. Zarlino, The Al of C o~mt r rpo i n t ,7.

( Che de lln Equalith h i principio In Ineq unlit i: cosi bibogna dire. che

dall'Unisono habhin principio In Diapason. Le Istitutiorzi harmorriche, 174. )

Rameau and Zarlino: Polemics in the


7

This approach is similar to Rameau's reassessment of contrapun-

tal 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 sen-

tences from Zarlino's lengthy explanation for inclusion in the

Trait :

The oc tave is the mother , the source, and the or igin of al l intervals . By the

divis ion of i ts two ter ms al l other harm oniou s 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 equa l i h

of the two pitches in a unison with the

u z ~

of his correc t sing le

source-the fund amen tal sound. In the follow ing passage. Ra-

nieau sets up Zarlino's ultimate fall by first endorsing Zarlino's

opinion that the unison

is

a source.

After having s tated that the uni t , whic h is the sourc e of numbers , rep-

resents the sonorous body f rom which the proof of the relat ionship be-

tween s ounds i s der ived . and tha t t he un i s on i s t he s ource o f cons onances ,

Zar l i no forgets al l this in his demonstrat ions and rules . Far f rom fol lowing

the p r inc ip le he has just announced . the fu r ther he goes the more h e d raws

a w a y

fro m 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, : Twu tise on Hurmoriy 10.

De \orte que pour faire valoir le sentiment de Zarl~ n. 'on ne peut di\-

penser d' y ajofiter, que pour l o r le Son fondamental se sert de son Octave

comme d'u n second terme.

Trait; clr I'harrrronic. 8 : Trecrtisr orr Hurtrrorzj: 10

'h Zarlin nprCs avoir remnrquC que I'unitC qui eit le principe des notn-

bres. nous represen te 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'htrr-

rrzonie. 18; Treatise on Harmon?; 22.


16/17

58

Music Theory Spectrum

In the middle of this passage, Rameau seems to be accepting the

unison as source. The unison may be o ne step closer to Ram eau's

fundamental sound than the octave is, but it certainly is not equiv-

alent to the fundamental sound.?' The beginning of this passage

confuses the situation more by implying that Zarlino recognized

a

third source-the fundam ental sound. expresse d by the unit.

Rameau's indication that Zarlino actually considered the single

string as a \ource is rather hopeful. Zarlino's method for generat-

ing intervals begins by positing a string, but the material of the

string is used only in servi ce of the unison a nd 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

rifir111

past

theories that hahe been betrayed. rather than dispel Zarlino's theo-

ries. 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 F R OM

THF

M 4 l O l i

THIR

On the topic of motion from the major third, Rameau c onti n~le s

to tnanip~llate arlino'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 fro111 the 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

root of

a

dominant cho rd acco rd ing to R a m e a ~ ~ ,roceeds with the

upper part moving by semitone. then it is inehitable that the bass

descends bq f if th . R at n ea ~ ~hus reassures his readers that Zarlino

' -Rumeau'\ indication that Zarlino al\o co ns~ der ed he single string as a

source 5 rather hopeful. Zarlino's method for generating inter\al\ begins h?

positing a str ing, but the material of the string i\ used on11 in \er vicr of the

unison and oc ta \ r .

z E " Q ~ ' i lalloit monter dc la Tiercr mdjeure de la Sixte mq eur e h

I'Octave." Trciitc;

r

I ' h u r n ~ o ~ ~ i r .

6:

Ti-c.cltite oil Hor rtzor i).

64-65.

understood the typical progression of the fundamental bass-

movement 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 (m ajor third) and the

major hexachord (major sixth) desire to expand into a fifth and an

oc tave [ r e~pec t ih e ly ] . " ' ~

here is no mention whatsoever here (or

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 intro-

duce\ 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 descen d bq fifth. Exa mp le

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

bass moving down by a f if th, R a m e a ~ ~eplaces Zarlino's original

~o ice -le ad ing rescriptions (that a third expands to a fif th) with

one option of the two exceptions Zarlino allow s. Apparently this

is enough of a connection for Ratneau to avoid acknowledging

any break on his part fro m the earlier tradition.

The reader's first reaction might be that Rameau is misreniem-

bering the I.~rirutioni,or at least being careless with his presenta-

tion. However, sehen chapters later in the

TmitP.

Ranieau's ac-

"'Latcl- in the 7i.clitP. Rameau \\.rite\ that Zarlino

sa ? \

of the ha>\.

I t \

rial-

UI-al rogl-ession in perfect cadences is to dc\cend

a

fifth." T,rciri$e 111 Hiii~it ioii \ ,

159. ("Sa PI-ogres\ionn,iturelle da ni Ic\ C aden ces pill-faite\. est de de\ceridre dc

Quinte." Tt.cliti t r I'lzciririoiric. 1 16. )

'""I1 Ditono lo Her achor do maggiore desiderano di fr11-\1 i i~ gg iol -I.C

nendo I'uno nlla Quinta I'altro alla Ott a\a. " c I.~rifrrti(~rrr

1x2:

l ~ ~ r t i i o t l i ~ ~ i r ~ ' .

Tlic rt

of

Co~lt~f(,r.poi~lr,3.

Za1.11no.Tlrr Art

of

Colrtifei-poi~ir.80.

"lhid.


17/17

Rameau and Zarlino: Polemics in the


59

Example 10 Zarlino's tw o 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 conscio us of his initial selectivity. Appare ntly,

Ra ~n ea u elieves 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 M E N T L B S S

In his discussion of fundamental bass, Rameau again empha-

sizes a connection to Zarlino's theoretical tradition, perhaps be-

cause 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 s ource is repre-

sented by the part called the buss in I I I US ~C. to which the epithet

, f~~ ndut net~ ru l Rameau merely seems to be relabelings added.

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 con-

taining the fundamental sound is always the lowest and deepest,

we call it the bass. Here is what Zarlino says on the subject.

35

Trecifi.se or1 Harri~orl\:

103.

Le principe y est pour lors represent6 danr la Partie de Musique qu'on

appelle Basse. laquelle on ajot ite l'ipit hete de Fondamentale.

Trtirti

dc

/'heir.-

rrlorlie.

Preface:

Trecirire or1 Hcirrr~orz?; xxv .

On appelle

Bcisse,

la partie ob regne ce Son fondamental, parce qu'il est

toi?jours le plus gra \e. le plus bas: \oi cy com men t Zarlin r'explique \ur ce

sujet:

TrtritP dr l'harri1or1ir.

:

Trc,ati.\r or1 Harr~ror~~.

9 .

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 writ-

ten. In Book I1 of the TruitP, Ra ~n ea u urns to on e of Zarlino's

examples as ev idence for the fundamental bass . Ra ~n ea uustifies

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 over-

throw Z arlin o's t heo ries fro111 within . An interesting psycho logical

battle occu rs within the TrczitP's pages, as has been observed in the

compositional canons, and the theories about the origin of inter-

vals, motion from the major third, and fundamental bass. Rameau

allows himself a certain freedom in reporting Zarlino's ideas, and

Ram eau's motive s color the Zarlino that emerge s 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 theo-

retical landscape.

A B S T R A C T

R ameau ' s methods to s ecure the fu tu r e o f h i s own theor i es invo lve a car e -

ful t reatment of Zar l ino 's theor iej . Rameau. in the rriiri

d r I ' Irurmoir ir ,

wres t l e j wi th Zar l ino 's i deas and us es them for h i s own end . The cu lmi -

nat ing composi t ions of each author 's t reat ise. which are all canons . are

analyzed. In addi t ion, a var ie ty o f theore t i ca l i s we s a r e exam ined . inc lud-

ing the source of intervals . motion f ro m the major third. and the funda-

mental hass .

'h Ob je vous prie de remarquer qu'il ne peut r'empescher de reconnoitre

une

ha.$e

dans I'Harmon ie, qu'il temoigne la souhaiter lorsqu'il ne l'entand

point.

Triri tP dr I 'ha rn~ ot~ ie .

8 :

Trrcirise orr Hcirnlorry

9 2 .

Documents

Rameau and Zarlino polemics