[Smt-talk] Classical Form and Recursion

dec2101 at columbia.edu dec2101 at columbia.edu
Sat Apr 4 11:32:09 PDT 2009

Dear Ildar,

In your recent post (below) you wrote:

"A product of real musical intuition is the sense of tension and
resolution. Musicians started talking about it in the 6th century
B.C. Diaphona, synphona, ho tonos, dynameis,--these are the terms of
western music theory of 25 centuries ago."

There is, however, no evidence that any of these was conceptualized in  
terms of "tension and resolution."

"Diaphonia" and "symphonia" (equivalent, more or less, to our  
"dissonance" and "consonance") were usually explained in terms of  
thelack or presence of blend and "unity" between two pitches.  
("Diaphonia" means "sounding apart"; "symphonia," "sounding with"; our  
English terms come from the literal Latin translations of these,  
"dissonantia" and "consonantia," introduced probably by Boethius.  
Adjectival forms of all four words are often found as well, e.g.,  
"diaphona" and "symphona" [though not, of course, "synphona"], and in  
Latin "dissona" and "consona.")

"Tonos," indeed, alludes to "tension," since it derives from "teino,"  
"to stretch, tighten" but the allusion is simply to the action of  
tightening a string to raise its pitch. In ancient Greek  
music-theoretical parlance, "tonos" normally denotes either (1) the  
interval of the whole tone (usually defined as the "difference"  
between a P5th and a P4th, and/or in terms of its ratio, 9:8), or (2)  
one of the "tonoi," which for most ancient writers seem to be simply  
transposition levels of the complete background scale system (the  
"greater perfect system"); hence the oblique reference to "tension,"  
i.e., pitch. (Ptolemy, though, understands the "tonoi" in a way that  
makes the allusion to pitch level irrelevant.) See also Aristoxenus's  
discussion of pitch and changes of pitch in _Elementa harmonica_, Bk.  
I, sections 10-13 (trans. Andrew Barker, _Greek Musical Writers_, vol.  
2, pp. 133-35): although he's thinking primarily of the singing voice,  
all his vocabulary concerning pitch alludes linguistically to  
"tension" ("tasis," also derived, like "tonos," from "teino"); "tasis"  
in fact is the word that actually denotes "pitch," per se, for him.

Finally, "dynamis," meaning basically "power" or "capacity/ability" to  
*do* something, is a term introduced into Harmonics by Aristoxenus  
(almost certainly borrowed from his teacher Aristotle, for whom it is  
a basic concept of his metaphysics, opposed to  
"energeia"/"entelecheia"). It's usually translated as "function" in  
English versions or discussions of Aristoxenus, but the precise sense  
that he intended has always been obscure. (But see the interpretation  
by Andrew Barker in his recent book, _The Science of Harmonics in  
Classical Greece_ [2007], pp. 183-92, which I think goes a long way  
toward answering this question.)


David E. Cohen
Associate Professor of Music
Columbia University
New York, NY 10027

Quoting Ildar Khannanov <solfeggio7 at yahoo.com>:

> Dear Olli,
> sorry to interfere in your exchange with Dmitri.
> What makes you think that "closure" and its synonim "prolongation"   
> are the norms of music, and that they come directly out of musical   
> intuition?
> Or, maybe it is something which was common in   
> music history? Kirnberger writes about the progression "comming to a  
>  close," but that means that progression arrives at the cadence.   
> Cadence is an important device, but is it the goal and essence of   
> music?  Of course, there is a sense of closure at the end of any   
> progression, but this is not  a product of  specifically musical   
> intuition. Rather, it comes from the intuition of a lawyer. The case  
>  is closed. The relatives of a victim receive the sense of closure   
> after the death penalty has been administered. Whatever happened in   
> the middle should be forgotten.
> A product of real musical intuition is the sense of tension and   
> resolution. Musicians started talking about it in the 6th century   
> B.C. Diaphona, synphona, ho tonos, dynameis,--these are the terms of  
>  western music theory of 25 centuries ago. However, in your   
> "prolongation" the role of the chord which creates tension and   
> requires resolution is reduced to almost nothing. It looses its   
> harmonic function, becomes a "contrapuntal chord" or Nebenakkord.    
> The most important agency is being reduced, the most important   
> event--overlooked. By the way, you cannot not notice it while   
> listening to it, but it is possible to "reduce" it in visual analysis.
> Let me through my 2 cents into the analysis of the Three Blid Mice   
> motive (3^  2^  1^). A very common example, which is used to   
> demonstrate the validty of "prolongation," is the voice exchange   
> progression. And you would say that it has a "passing 6/4 chord in   
> the middle." What is the function of this middle chord:  "Passing."   
>  How about passing Dominant 6/4? Or the fact that it is the Dominant  
>  is unimportant?
> But then your students will have a surprise for you. They will write  
>  a ii5/3 in the middle. They do this  very often.  They are not that  
>  stupid: they are just following the recommendations concerning   
> adjacency, "voice-leading" and contrapuntal, passing function of the  
>  middle chord. Indeed, why not to harmonize all the notes in a  
> melody  with parallel triads: for 3^ 2^ 1^ to use iii5/3  ii5/3 and   
> I5/3? Tell me that this progression does not create ultimate   
> parsimony, ultimate voice-leading economy and ultimate adjacency,    
> true Nebenakkorden!
> Why, then should we bother with  root motion on the fifth, all this   
> basso fondamentale influence, a French disease (according to Oswald   
> Jonas's introduction to Harmony)?
> That is why we discuss the heterogeneous character of music in   
> general and harmony in particular. And I cannot agree more with   
> Nicolas when he mentioned basso fondamentale as another example of   
> parsimony, or economy and laconicity of musical expression.
> I do not see the musical intuitive basis for "reduction" of the   
> middle element.
> As in  ABA= A. That is exactly what Dmitri has said, which tells me   
> that he is a post-Schenkerian.
> Resolution of the Dominant is not only and never only   
> "concatenational." Plase, read Riemann's analyses and notice the   
> discussion of large-scale dominants. The Dominant function is   
> capable of stretching its resolution power over a great nunber of   
> measures.
> As for the phrase Americans care only about Americans, it is an   
> excellent example of recursion. It does it on all levels, from   
> syntactic to rhetoric. And how naive is to try to separate them, or   
> to reduce one to another!
> Best wishes,
> Ildar Khannanov
> Peabody Conservatory
> solfeggio7 at yahoo.com
> --- On Wed, 4/1/09, Olli Väisälä <ovaisala at siba.fi> wrote:
> From: Olli Väisälä <ovaisala at siba.fi>
> Subject: Re: [Smt-talk] Classical Form and Recursion
> To: "Dmitri Tymoczko" <dmitri at Princeton.EDU>
> Cc: "smt-talk Talk" <smt-talk at societymusictheory.org>
> Date: Wednesday, April 1, 2009, 3:57 AM
> First, on music versus language:
> There were really a pair of issues.  One is grouping -- getting from  
>  ABAB... to (ABA) ...  But the other is reduction -- getting from   
> (ABA) to A.  The point of the "Americans care only about Americans"   
> example was that this latter process is also problematic: the mere   
> presence of ABA (as in "Americans care ...") does not automatically   
> license or motivate a reduction to A ("Americans").
> Dmitri, your analogy between music and language fails in an   
> illuminative way.  Beginning and ending the sentence with the same   
> word plays no role for syntactic closure in language. In your   
> example sentence, the subject happens to be the same as the object,   
> but this coincidence has no significance for syntax (only for   
> semantics and rhetoric). In tonal music, by contrast, there is a   
> norm that closed harmonic progressions begin and end with I (I hope   
> you will agree that there is such a norm). If a phrase starts on I   
> and proceeds to other harmonies, we are expecting a convincing   
> return to I until this happens. (If our expectations are not   
> fulfilled and the phrase does not return to I, we do not hear it as   
> closed phrase, but await continuation.) This demonstrates that the   
> referential status of a single element (tonic chord in this case)   
> may have significance for musical syntax in a way that differs   
> fundamentally from that of a single word for linguistic
>  syntax. The perception of the syntax in a tonal progression may be   
> governed by an element in that progression in a sense for which   
> there is no linguistic counterpart. (Closed tonic-to-tonic   
> progressions are by no means the only way to acheive such governing   
> status, but they are a prime example.)
> Owing to this property, music has, in my view, much stronger   
> potential for extensive recursive (prolongational) structuring than   
> has language. Hence, when I received the first mail in this thread,   
> I was surprised to see that someone had claimed just the opposite.   
> Of course, the existence of this recursive potential does not mean   
> that composers have actually utilized it. For studying this   
> question, we need empirical research of their music, and I have   
> tried to present some ideas how this issue may be approached.
> Next, let us return to this example:
> (3^) ? V (2^) ? I (3^) quarter rest / V (2^) ? I (1^) ? V (2^) q.r.   
> / I (1^) ? V (7^) ? I (1^) q.r.
> As an additional feature, let us suppose that the bass line is   
> C2?G2?C2, G2?C3?G2, C2?G2?C2, thus further weakening the I in m. 2   
> and reinforcing the perceptual analogy between bars 1 and 2.
> A crucial difference between a prolongational and concatenational   
> perception of this progression is as follows. Under prolongational   
> perception (= I (3^) ? V (2^) ? I (1^), the I in m. 3 offers closure  
>  for the entire progression; under concatenational perception, it   
> only offers closure for the I?V?I succession starting from bar 2,   
> beat 2. Frankly speaking, I find the latter alternative utterly   
> unintuitive. (I am not sure whether you agree, Dmitri, but sometimes  
>  I almost cannot avoid the impression that, whereas you suspect that  
>  I or other analysts may claim to hear something that we do not   
> actually hear, you might be claiming not to hear something that you   
> actually hear.)
> If we accept the prolongational interpretation, this example   
> illustrates that I is not the only harmony that can be prolonged. If  
>  we hear tonal closure only in bar 3, the I in bar 2 prolongs the   
> surrounding V. The V at the downbeat of bar 2 creates the   
> expectation of I, but there are stong perceptual reasons why the   
> immediately following I fails to fulfill these expecations in a   
> convincing way. Not only is it rhythmically and registrally weak and  
>  surrounded by stronger dominants, but the similarity between mm. 1   
> and 2 guides the listener to perceive this I in a way analogous to   
> the V in m. 1.
> For testing whether a listener actually perceives tonal closure in   
> m. 3, one might consider the following experiment, though it has a   
> deficiency. Listen to the progression (1) as written above and (2)   
> as a truncated version, breaking of after bar 2, beat 2. If one   
> finds (1) embodying more convincing closure than (2), this speaks to  
>  prolongational perception. The deficiency in this experiment is  
> that  (2) does not include all the information that supports  
> perceiving  bar 2, beat 2 as subordinate to the surrounding  
> dominant, since part  of this information comes retrospectively  
> through the return of V  (2^) at beat 3. Nevertheless, even without  
> this retrospective  information, I find (2) less satisfactory than  
> (1) in terms of  closure.
> (The case is different if we break off after bar 3, beat 1. The last  
>  V (7^) and I (1^) are actually superfluous for the sense of  
> closure.  In fact, one might say that the sense of closure is  
> enhanced if the  goal status of the last I (1^) is marked by the  
> cessation of the  sequential model.)
> In order to overcome the "I hear this ? I hear that ? No, you only   
> claim so" type of discussion, I have tried to focus on the   
> compositional evidence that there may be for prolongational   
> structuring. I suggested that if a composer had written the above   
> progression, there would be a certain amount of such evidence. The   
> prolongational model would explain the emergence of several   
> compositional features, including the feature that the composer has   
> stopped the top-voice sequence on 1^?if we suppose that the   
> progression occurs in circumstances that support its perception as a  
>  closed entity. (A crucial feature in the explanatory power of the   
> Schenkerian approach to sequences concerns the participation of the   
> framing points in the larger context; in this case, however, we have  
>  not identified a larger context.)
> I did not claim that the evidence "proves" the prolongation   
> hypothesis. There might be alternative explanations, but at the very  
>  least the facts are well concordant with that hypothesis. For   
> strengthening the case for the hypothesis, we would have to allow   
> for the larger context and for the composer's general practices, but  
>  this, of course, is impossible for this artificial example.
> Instead, I presented some observations of the descriptive and   
> predictive power of the prolongation hypothesis for Bach's music. I   
> discussed how a passage in G Major Invention involves several   
> features of design, register, emphasis, and meter that can be   
> elegantly explained on the basis of the hypothesis that Bach had in   
> mind a prolongational pattern II (4^) ? V7 (4^) ? I (3^). (I do not   
> mean he was consciously aware of that pattern; one does not have to   
> be aware of syntactic or quasi-syntactic rules for following them.)   
> I also related this 4^?3^ pattern to the piece as a whole and to   
> Bach's general practices (referring to "the predictive power of the   
> Urlinie"). My point was that there are objectively identifiable   
> compositional features in Bach's music that can be explained on the   
> basis of the hypothesis that prolongational (=Schenkerian) patterns   
> affected his composition and for which it is not easy to see what   
> would be equally satisfactory
>  theories. While this cannot "prove" the hypothesis, it justifies   
> and motivates it in a way that is largely comparable to any   
> scientific hypothesis.
> (Incidentally, I do not think that my approach to empirical evidence  
>  repeats arguments overly familiar from previous Schenkerian   
> literature, although the significance of register and design has   
> certainly been focused on by authors such as Oster and Rothgeb. For   
> example, I am not aware of precursors for my systematic study of the  
>  predictive power of the Urlinie for the corpus of 15 Inventions.)
> Olli Väisälä 
> Sibelius Academy
> ovaisala at siba.fi
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