[Smt-talk] Classical Form and Recursion

[Olli Väisälä]

Dear list,

I would like to make some points that I find crucial for the  
discussion about recursion in prolongational (Schenkerian) structures  
and about pertinent empirical evidence.

First of all, I do not think this issue can be adequately treated  
merely on the basis of symbol sequences, as suggested by Dmitri  
Tymoczko's post:

>
> It is typically possible to interpret a sequence of symbols  
> recursively.  For instance, we could take the sequence
>
> ABABABABAB...
>
> And group it as follows:
>
> (ABA)(BAB)(ABA)(BAB) ...
>
> Then, by asserting that (ABA) "stands for" (or "represents" or  
> "prolongs") A, while (BAB) "represents" B, we could come up with a  
> "higher level" version of the same pattern
>
> ABABABAB ...

Musical events are not adequately determined in terms of such  
sequences, since each occurrence of "A" and "B" may relate  
differently with aspects such as meter, design, register, emphasis,  
etc. , and these aspects are crucial for both the expression (by the  
composer) and perception (by the listener) of prolongational  
relationships. Consider a very simple example in 4/4 time:

bar 1: I–V–I–quarter rest
bar 2: V–I–V–quarter rest
bar 3: I–V–I–quarter rest

If this occurs in a composition, we may speak of empirical evidence  
for the hypothesis that the recursive interpretation, (IVI)(VIV)(IVI)  
= IVI, models a pattern relevant for composition. The model has  
considerable explanatory power for the way in which the composer has  
related the chords with  features such as meter and grouping.

Naturally, meter and grouping in this example also enhance the  
probability that the recursive model pertains to the listener's  
perception in this example. What I would stress, however, is the way  
in which features in the score may testify or fail to testify to the  
compositional significance of prolongational models. Therefore, I  
would like to comment on Dmitri's following suggestion:

>
> The big complication is that it is possible to perceive  
> intrinsically nonrecursive sequences in a recursive fashion.  I  
> think music theorists could probably stand to pay more attention to  
> this possibility.  For instance, I think you can make a good case  
> that the grammar of elementary tonal harmony is, intrinsically,  
> (largely) non-recursive.  Theorists such as Schenker, Lerdahl,  
> etc., propose that we nevertheless *perceive* music recursively --  
> imposing a recursive organization on an intrinsically not-recursive  
> structure, roughly as we did with the ABAB sequence above.  This is  
> potentially true, but it is quite hard to test empirically.  I  
> think discussions of the issue of recursion could stand to pay more  
> attention to the distinction between intrinsic structure and our  
> psychological organization of a structure.
>

As my example clarifies in a very simple way, a sequence may be  
"intrinsically nonrecursive," but perceiving it recursively may be  
supported by features that are intrinsic to the music. Whether the  
"recursive potential" of a sequence becomes realized does not depend  
only or primarily on the listener's psychology, but on the composer's  
acts: on the ways in which he/she supports (or fails to support)  
certain elements and relationships through parameters such as meter,  
design, register etc. While it may be hard to test empirically  
whether people perceive music recursively (and different people have,  
of course, different perceptions), it is possible to approach  
empirically the question whether composers did (do) so, by assessing  
the extent to which recursive (prolongational) models are supported  
by such parameters in their music.

Of course, the question is much more complicated in real music than  
in the above example, but for those interested I might mention that  
that I have attempted to approach this issue in my recent (and  
ongoing) work on Bach.

Olli Väisälä
Sibelius Academy
ovaisala at siba.fi

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