[Smt-talk] Classical Form and Recursion

[Dmitri Tymoczko]

Over the last few years, I've been working on the problem of trying  
to figure out whether the rules of functional harmony are recursive,  
and if so, to what extent.  This is one reason I enlisted a group of  
theorists to produce a corpus of Roman Numeral analyses of the Mozart  
sonatas -- that is, to ask whether we need recursive rules to  
describe the chord progressions we find in the music.  So far, the  
data seems to suggest that the answer is "no."  Simple rules like "ii  
goes to V but not vice versa" account for the data pretty well.  One  
important exception is the practice of secondary dominants, which  
permits V/x->x to be embedded inside I->...->V->I in a different  
key.  This seems genuinely recursive.

Some early work on these ideas is summarized here:

http://music.princeton.edu/~dmitri/tonaltheories.pdf

Published (in French, in Nicholas Meeus's excellent translation) in  
Musurgia.  More is coming in my book.

To make further progress on this topic, one needs -- as Richard  
Hermann suggests -- to define terms more precisely.  In particular,  
one needs to distinguish

	1. Psychological theories, which claim we need recursion to account  
for our *perception* of music, from grammatical theories, which claim  
we need recursion to account for the formal structures in pieces,  
whether perceived or no.

	2. Hierarchical theories, which suggest (incontrovertibly) that  
different sorts of rules govern different levels of musical  
structure, from recursive theories, which claim that tonal structures  
are built up by embedding musical units within other units of the  
same type.  (For instance, ii/ii->V/ii->ii embedded within I->ii->V->I.)

It's certainly true that many theorists -- including Schenker, Sadai,  
Lerdahl, and Jackendoff -- have claimed tonal harmony is recursive.   
Figuring out whether this is in fact true is, to my mind, a very deep  
and important problem.  It speaks to fundamental issues about the  
human mind: about the extent to which recursion is basic to human  
cognition, and about the extent to which music and language are similar.

Interestingly, this is also a hot topic in linguistics.  I've  
recently been giving a talk, to philosophers and linguists, about why  
I think linguistics is a bad model for music theory.  Whenever I try  
to contrast language with music, claiming that the former is more  
obviously recursive than the latter, I encounter dissident linguists  
who say, in essence, "a lot of what you say about music is true about  
language too."  This issue is, as they say, above my pay grade, but  
it's interesting that some of the same issues arise in the two  
disciplines.

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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