[Smt-talk] Classical Form and Recursion

Dmitri Tymoczko dmitri at Princeton.EDU
Wed Mar 25 07:20:48 PDT 2009

I just wanted to add a (potentially obvious) note about recursion.

It is typically possible to interpret a sequence of symbols  
recursively.  For instance, we could take the sequence


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


(Here each symbol stands for 3 symbols in our original.)  We might  
take this to demonstrate that there is recursive structure in our  
sequence.  It is even possible that this sort of procedure could  
represent a psychologically accurate description of the way some  
particular person perceives or organizes the sequence.

The important point is that when linguists talk about recursion in  
natural languages, they are NOT talking about this sort of thing.   
They are talking about sequences such as


Whose intrinsic structure can only be described using recursive  
rules.  The "ABAB..." sequence is very different, in that we *can*  
specify its structure using very simple, non-recursive rules.  In  
this sense you might say that it is intrinsically nonrecursive,  
whereas ABCDCBA is intrinsically recursive.

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.


Dmitri Tymoczko
Associate Professor of Music
310 Woolworth Center
Princeton, NJ 08544-1007
(609) 258-4255 (ph), (609) 258-6793 (fax)

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