[Smt-talk] Headlam on Orbifolds

Dmitri Tymoczko dmitri at Princeton.EDU
Fri Mar 13 15:10:34 PDT 2009

> I associate part-writing with counterpoint in the sense that it  
> deals with the surface elements of a composition and the threads of  
> succession. Your earlier exchange with Ildar, however, implied, on  
> his part at least, that incomplete harmonies were some pivotal  
> issue. Let us add a definition of "chord" to be an element of  
> counterpoint, which is to say, on the real surface of a  
> composition, and further say that it is a selection of elements of  
> the surface–without regard to rhythm (including simulataneity)– 
> isolated for evaluation with respect to the referent harmonies (in  
> tonal music, triads and maximally even tetrads). If we wish to give  
> a chord a name, then we are including all of its elements in the  
> concept of voice leading, while counterpoint has no such  
> obligation. Thus, I can read bars 9-11 of the Waldstein as built  
> upon GBF, three of the four notes of a G7 harmony, without their  
> representing that harmony; there need be no D. Yet bar 13 has but  
> one element of a G triad and I do read that as a full G harmony and  
> accomodate the voice leading accordingly. You and I are with Rameau  
> in this perception of voice leading, but most Schenkerians assume  
> that voice leading is what they show; sometimes it is, sometimes it  
> isn't. Züge don't need support for all their displacements.

I think I follow what you're saying.  The example is definitely useful.

What I was saying to David (not Ildar) was that you can use orbifolds  
to model surface harmonies if you want: for instance, you could model  
the relevant passage of the Waldstein as the three-voice (Ab, C, F)-> 
(G, B, F) in three-note chord space.  That's a perfectly reasonable  
thing to do.  I think you might say that we're modeling the actual  
"part writing" here.

On the other hand, you can also model voice leadings that are only  
hinted at by the surface.  For instance, in the recap of the  
Waldstein, measure 172, you find (F, F, Ab, Eb)->(Ab, F, B, D), which  
I take to be an embellishment of the common semitonal pattern (F, Ab,  
C, Eb)->(F, Ab, B, D).  This sort of semitonal play between seventh  
chords is ubiquitous in nineteenth-century music, and is the engine  
of an awful lot of chromaticism.  I think it's useful to identify it  
as lying in the background, obscured to some extent by the part writing.

What I don't quite understand is Dave's suggestion that geometry is  
useless whenever we confront incomplete chords -- as if we have to  
throw up our hands and give up the moment we see a progression like  
(F, F, Ab, Eb)->(Ab, F, B, D).  I disagree: we're still free to use  
all the standard techniques of analysis, including postulating octave  
displacements, voice exchanges, additional voices, passing tones, and  
so forth.  Part of the value of the geometrical models is that they  
give us powerful tools for understanding the ubiquitous background  
schemata that are (sometimes) only hinted at by the musical surface.


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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