[Smt-talk] Classical Form and Recursion

[Ildar Khannanov]

Dear David,
 
thank you very much for your suggestion to read the books. I hope that you understand that the book you have suggested is an interpretation of events of the past and not the required literature for an UG student. I happen to be able to read Aristoxenus in the  original language. There were other translations and intepretations into other languages which I find very interesting and helpful. I would suggest that you read them too.
 
As for the terms, could you elaborate what do you mean by "blending," and how it is different from my interpretation of synphona and diaphona? Was there any relationship between these two groups? If yes, was it related to the concept of harmonia? What is harmonia? I keep hearing recently that harmony is a "vertical structure." Then, I assume that counterpoint is defined as a "linear structure." Both are wrong! Harmony, a universal signifier of music, cannot be explained by visual metaphors of verticality and horizonatlity. Harmony is the relationship of  the opposites. It can be the relationship between darkness and light, but also between the wasp and the orchid, or even the car and the traffic light. In music it is diaphona and symphona. 
 
Latin translations and later English translations present the problem and it is not only the problem of musical terminology. All terms, including Being and Nature, were translated into Latin from Greek with great loss of meaning. There is much written on this topic  by Heidegger. For example, the Greek preposition dia- does not mean the same thing as Latin diss-. The fact that we are having a dialogue on the list is a clear proof of it. 
 
Today, we can only reconstruct the meanings of Greek terms and the intentions of Greek authors by deconstructing our prevalent views and formulations. We've changed and in need of developing a different organ to see the past events. When I read in Aristoxenus that all intervals smaller than the diatessaron are diaphona, I ask myself, what does this mean? And what can be gained by reaching the diatessaron after passing through the smaller intervals? How this can be translated into a more modern language? Is not this a clear case of tension and resolution? Of polemon ending with synousia? When the voice passes through ho tonos, we do not control two "pitches" simultaneaously. We are taken off guard by the power of melodic motion. We are moving through (dia) something which does not have the beginning and the end.. In this case, even "the path up and the path down are the same." Only when we reach the diatessaron, the skies are cleared and we start seeing
 the horizon. This is the junction, connection (syn-) of the begining and the end of our travel, and the resting point "Ruhepunkt." The quantity of motion turned into a new quality. 
  In general, we should see the forest behind the trees: harmony in Greek understanding(s) means the opposition of Chaos and Cosmos, the reconciliation of the irreconcilables. Music since Greeks has been always operating with this definition of harmony. It has been realized in different ways, but it was present at all times. Not to see this evidence is strange, to say the least!
 
The idea that tonoi are the "transposition levels of complete background scale system" seems too revisionist to me. For imagining that, a Greek theorist must have had an idea of pitch collection, pitch class, transposition levels, background structure, etc. Try to translate these terms back into ancient Greek language! And by the way, try to translate the word "pitch" into any modern language. I have tried to translate it in into Russian, my native language, and I am at loss. There is such a degree of polysemy and such a rich folkloric etymology that it is very difficult to operate with this word on a reflective  terminological level.  Systhema teleion (magnon ou pyknon) is the greatest achievement of Greek theory. It is a result of a purely theoretical effort to see the invisible (Platonic idea). It is obvious that early in Greek music history only shorter melodic patterns were used and they were limited to small number of notes (mostly four).
 The similar layers of folk music elsewhere have similar oligotonic character. IN Greek theory, the two genuine melodic patterns were either connected (synemmenon) or placed next to each other (diezeugmenon). The functional complexity of this system does not allow to treat it as a "collection" or a "set." Music is open for higher levels of abstraction (since it has been introduced in the West together with theory and philosophy), but harmony seems to remain an ultimate irreducible prerequisite. Collection and set are exactly the mathermatical tools which are intended for disregarding functional differentiation of the elements. Sets are internally a-systematic. In mathematics itsels, the category of set is used selectively. (See the reaction of Poincare to Georg Cantor's idea of transfinite sets). Schenker himself would take my side: for him a group of notes could not be a "set" of "collection" and harmonic prerequisite would never be negleted.
 
 Dynameis  (FUNCTION!) creates a problem. Should we, then, lable it as "obscure"?
 
In general, I suggest  reconsidering our approach to history of music theory. It is not something that we can "criticize" at present. It is like a 25-centuries-old cathedral. I suggest walking inside in soft shoes  and enjoying the beauty of harmony.
 
 
Best,
 
 
 
Ildar Khannanov
Peabody Conservatory
solfeggio7 at yahoo.com
 
 
 
 
 
 
 


--- On Sat, 4/4/09, dec2101 at columbia.edu <dec2101 at columbia.edu> wrote:


From: dec2101 at columbia.edu <dec2101 at columbia.edu>
Subject: Re: [Smt-talk] Classical Form and Recursion
To: smt-talk at societymusictheory.org
Date: Saturday, April 4, 2009, 1:32 PM


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.)

Best,

-David
---------------
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
> 
> -----Inline Attachment Follows-----
> 
> 
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