[Smt-talk] Classical Form and Recursion

[Dmitri Tymoczko]

> 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 realize that I was being a bit coy here.  In fact, I do think that  
there's a way in which tonal music is recursive, because efficient  
voice leading occurs between scales as well as chords.

So, I think that a modulation from C major to G major involves an  
efficient voice leading whereby F moves to F#.  This is the same  
basic thing that happens when you have, say, a voice leading from the  
C major to G major triads, such as (C, E, G)->(B, D, F).  Here, two  
notes of the C major triad move down by step to G major.  So you have  
the same basic process (efficient voice leading between structurally  
similar objects) occurring on two different temporal levels (that of  
the chord, and that of the scale).  I've talked about voice leading  
between scales in "Scale Networks and Debussy"; in my book, I claim  
that the same thing happens in traditional modulation.

Scales can be thought of as large "chords" inhabiting a seven- 
dimensional geometrical space.  In "Scale Networks and Debussy" I  
show how to depict the relevant portion of the space that contains  
familiar scales.  It actually turns out that the geometry  
representing seven-note scales in chromatic space is very, very  
similar to the geometry representing three-note diatonic triads.  So  
it turns out the local geometry of chords-in-a-scale is almost  
identical to the geometry of scales.

All of this, I think, is musically pretty plausible, as well as being  
theoretically quite elegant.  But of course, it's a longer story than  
I can really explain in a short email like this.

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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