[Smt-talk] Classical Form and Recursion

[Dmitri Tymoczko]

> I think Dimitri ...

Actually "Dmitri" with no "i" between the "D" and "m."  You wouldn't  
believe the trouble this causes with telephone-operator types.

> rightly cites secondary dominants as evidence of "hierarchical"  
> recursion, but there's more (as they say in those TV ads).  I think  
> it's very hard indeed to analyze Goldberg Var. 25 without a kind of  
> recursion that "rewrites" a single unit from the underlying bass of  
> the Saraband---a well-formed harmonic progression itself, of  
> course---as an entire well-formed progression on the level of that  
> unit.  For example, the A-flat in the second bar is a pretty  
> convincing suggestion that we're now starting a one-bar "Opening  
> Phrase (I to V)" here in F-minor (i.e., the F that is the second  
> unit of the bass in its minor version).  A similar spot in bar 10-11 
> (also on an F from the bass) is even more elaborate, with,  
> incidently, many secondary dominants.   We seem here to have at  
> least two levels of recursion.

Two general points:
	
	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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