[Smt-talk] Headlam on Orbifolds

[Dmitri Tymoczko]

Ildar wrote:

> I am sorry for playing a devil's advocate for a moment. Of course,  
> I appreciate your great work with geometry. It is just sometimes  
> the topoloical aspect is overshadowed in your presentations by a  
> simpler  principle of measurement by half-steps (which probably  
> come from Neo-Riemannian understanding).

No need to apologize, and thanks for the kind words!

If I understand you, the "simpler principle" you refer to is just the  
"taxicab" (or "smoothness") metric, which provides one of the  
simplest ways to measure distance on orbifolds.  From this point of  
view, (C, E, G)->(C, F, A) is has size three, since one voice moves  
by one semitone, and another moves by two semitones.  By contrast,  
(C, E, G)->(C, F, Ab) is size two, since two voices move by  
semitone.  There's no contradiction between this way of thinking  
about distance and the geometrical perspective; this is just one of  
the many possible metrics you can choose.

It would be a mistake, however, to associate this method of measuring  
distance too closely with either the Tonnetz or neo-Riemannian  
theory.  From a NR-perspective, F major is *closer* to C major than F  
minor is -- the progresson F->C is LR, whereas f->C is PLR.  (Using  
the "taxicab" metric, the opposite is the case.)  Consequently, NR- 
theory doesn't seem to explain why F minor so often appears as a  
passing chord between F major and C major -- from an NR perspective  
the progression F->f->C moves away from C major and before moving  
back toward it.  The moral is that neo-Riemannian distances (measured  
in "units" of LPR or in edge-preserving Tonnetz-flips) are *not*  
voice leading distances.  This is a complicated and subtle issue  
about which much more could be said.

As far as I understand, the first geometrical models in which all  
distances represent taxicab distance are those provided by Douthett  
and Steinbach in 1998, particularly "Cube Dance."  These models are  
all embedded naturally in the relevant orbifolds representing n-note  
chords.

> And I am ready to choose certain  metric from a number of others  
> offered by topology. For example, I will decline your proposition  
> that the E and Ab triads are the closest to C triad. In the metric  
> of tonal-functional harmony, the G triad is the closest to C triad.  
> You simply cannot fit any other triads in between in a meaningful  
> progression. That is what I meant in my rather awkward critique,   
> that the geometry of note heads on the staff presents the metric  
> which is drastically different from the one we use when playing and  
> singing most of tonal music.

Oh, OK, I should've clarified the following.  Rachel Hall and I have  
worked to describe acceptable *voice-leading* metrics.  There are  
other notions of musical distance (including tonal distance, or that  
offered by NR-theory) that can be perfectly reasonable in some  
circumstances, but that don't measure voice leading.  Acoustically,  
you might think that G4 is very close to C3, since it's the second  
overtone of the lower note.  But you're not measuring voice leading  
if you think that C3->G4 is a "small" motion.  Again, a lot more  
could be said about this.

BTW, I would say that you can fit other triads between C and G.  In C  
major, the progression C->a->d->G is perfectly reasonable.  And you  
sometimes find G->e->C, both in the baroque and in 19th-century  
music.  The question of whether tonal harmony is fifths-based or  
thirds-based is a complex one; and you can make a good case for  
thirds, rather than fifths.  Interestingly, then you're back to voice- 
leading, since third-related diatonic triads are linked by single  
(diatonic) step voice leading.

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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