[Smt-talk] Sequences

[Dmitri Tymoczko]

> what do you mean by "harmonic terms" with respect to the orbifold  
> approach (factorized voice-leading) you are working on?

Sorry, I suppose that was a bit too brisk.

In general, I think a lot of nineteenth-century chromatic practice  
can be understood contrapuntally: composers exploit efficient voice  
leading to move between tonally distant chords.  So, you see Schubert  
exploiting the efficient voice leading among major-third related  
triads to move from C major to E major, for example.  Or Wagner  
exploits the efficient voice-leading among minor-third related  
seventh chords to move from D half-diminished to B7.  Looking broadly  
at nineteenth-century practice, I think you can make a case that many  
of these unusual tonal moves are motivated by voice leading  
relationships.

One piece of evidence for this is that there's a marked asymmetry  
between triadic and seventh chord routines: triads are more likely to  
move chromatically by major third and seventh chords are more likely  
to move by minor third or tritone.  This reflects the underlying  
voice-leading relations: C major is "closer" (in voice leading terms)  
to E major than Eb major, but C7 is closer to Eb7 than E7.  You can  
actually see this asymmetry in the statistics -- just count up all  
the chromatic progressions in Mozart or Schubert.

In the case of our sequences, I think it's hard to make the case that  
these minor-third progressions are motivated contrapuntally,  
precisely because composers seem to avoid the most efficient  
(stepwise, descending) voice leading in favor of ascending parallel  
motion.  This makes it look like they're choosing the harmonic  
destination, rather than it being an effect of voice-leading  
relationships.  On the other hand, with "Hey Joe" perhaps it's the  
descending voice leading that produces the unusual (ascending fifth)  
root motion.

I'd never thought about trying to express this in terms of  
orbifolds.  I suppose you could though: the contrapuntal approach,  
roughly, involves thoughts like "I'm here at chord X; what chords are  
nearby?"  The harmonic approach involves thoughts like: "I'm here at  
X, and I want to go to Y, and distance isn't really a factor right now."

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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