[Smt-talk] Classical Form and Recursion

[Olli Väisälä]

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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