[Smt-talk] Classical Form and Recursion

[Wayne Slawson]

In response to Gogins and Dmitri:

I can believe that recursion can be admitted into a geometric model;   
the question for me is whether that would be a plausible way of  
modeling recursion in classical music.  But this may be a good way  
for an old guy to get into this new stuff.  I'll think over Dmitri's  
suggestions:

> I don't see any special problem.  You can talk about voice-leading  
> distance on the chord-to-chord level, or between the start of each  
> sequential units -- as when a theorist like Caplin says that the  
> descending fifths sequence is really descending by step.
>
> More generally, Schenkerians talk about voice-leading relationships  
> at various levels of the recursive hierarchy; each level could be  
> modeled geometrically.

Regarding the first two measures of the Goldberg V. 25:  Yes, one  
could call this a sequence because the surface details from the first  
measure are repeated a whole-step down in the second.  And, yes, one  
could say that sequences are either examples of recursion---along the  
lines of my suggestion earlier---or conversely just moving a bunch of  
music up or down.  These two measures can be heard either way (I  
guess).   But the Saraband Bass is expressed as small, well-formed  
progressions _throughout the variation_.  At bars 9--11, for example,  
where the bass repeats the first three bass notes of the beginning,  
there is no sequence, but there are the lower-level progressions.   
The one on F is now a closing phrase:

			B-flat5,3/F6,3/G7,3(natural)/C7,3(natural)/F3("flat" then natural  
in the figuration)].
	In F:		  IV             I        V of V                    
V                       I

It's not possible to _prove_ that recursion is required in cases like  
these, but it seems the most plausible, and simplest explanation.    
We're in exactly the same situation as the linguists in mid-century.   
Chomsky couldn't prove that a Markov grammar was inadequate to  
explain nested relative clauses; but he could come up with hard  
cases.  (I wish I could dredge up his famous eleven-level-of- 
recursion sentence whose sentenceship was so hard to confirm that  
some thought it a counter-case to his argument.)

Recursive structures tend to be hard to understand in both language  
and music.  It's not so hard to compose them, but the burden on the  
listener/reader can be pretty heavy.  I think this makes multi-level  
recursion fairly rare.   It might account for the failure of Dmitri's  
search for harmonic recursion in Mozart's Piano Sonatas.  Mozart may  
have wanted to keep that aspect of his music simple---in contrast to  
Papa Bach.   (I think there are intimations of recursive-like  
structures in places, however:  emphasis on the melodic sixth-degree  
in K. 333 is reflected in a dominant pedal on V of VI  near the end  
of the Development.  There are other instances throughout that  
piece.)  Deep recursion's relative rarity does not mean that it can  
be ignored, and, in any case, the simplest cases, secondary  
dominants, are ubiquitous.

I agree with Dmitri that music-theoretical recognition and  
description of recursion in music would be an important and  
interesting effort.  Success would decisively generalize the  
cognitive status of features widely held to be language-specific.  A  
heady prospect, indeed!

And, yes, I've noticed the connection to the Waldstein.   Do you  
think the move to III# is a kind of reflection of the opening?

Wayne Slawson





>
> 	
> 	1. I've mainly been thinking about recursion in the harmonic  
> grammar; I leave open the question about whether there is recursion  
> in other domains.
>
> 	2. Variations structures, or places where one passages of music  
> rewrites another, may be a special circumstance.  Philip Johnson  
> Laird addresses this issue (with specific reference to Chomsky and  
> recursion) in "How Jazz Musicians Improvise" (Music Perception, 2002).
>
> About your specific example, I'm not sure I quite follow.  Are you  
> making a claim over and above the fact that this is a sequence?  If  
> the idea is that this is a sequential pattern that elaborates the  
> theme's descending bass, I agree that it forms a potential example  
> of recursion.
>
> The question about whether sequences are recursive is a complicated  
> one.
>
> 	One the one hand, someone might say: sequences aren't necessarily  
> recursive, are they?  You just take a chunk of music and repeat it,  
> transposed by some interval.  Eventually you stop.  It's not clear  
> that we need a recursive grammar to explain this.
>
> 	On the other, it's clear that sequences involve a hierarchical  
> structure, and that you can't explain them with a simple chord-to- 
> chord (first-order Markov) model.  This is a point Salzer expresses  
> quite forcefully at the start of Structural hearing.
>
> Note, BTW, that this Bach passages is one of my stepwise descending  
> VL sequences: (G, Bb, D)->(F#, A, D)->(F, Ab, C)->(E, G, C) ...   
> Basically the Waldstein with mode changes.
>
>> I take it that recursion is hard to fit into a geometric model of  
>> distance, or am I missing something?
>
>
> I don't see any special problem.  You can talk about voice-leading  
> distance on the chord-to-chord level, or between the start of each  
> sequential units -- as when a theorist like Caplin says that the  
> descending fifths sequence is really descending by step.
>
> More generally, Schenkerians talk about voice-leading relationships  
> at various levels of the recursive hierarchy; each level could be  
> modeled geometrically.
>
> DT
>
> Dmitri Tymoczko
> Associate Professor of Music
> 310 Woolworth Center
> Princeton, NJ 08544-1007
> (609) 258-4255 (ph), (609) 258-6793 (fax)
> http://music.princeton.edu/~dmitri
>
>
>
>
>

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