[Smt-talk] Headlam on Orbifolds

[Dmitri Tymoczko]

Dave,

I separated out your comments about orbifolds since they aren't  
relevant to sequences, and since they continue an old discussion  
between you, Brown, and myself.

> The orbifold approach will work for complete chords, but rather  
> than the permutational equivalence that orbifolds are based on,  
> something like 000 040 004 047 (CCC CEC CCE CEG) equivalence more  
> closely matches tonal practice, to avoid constant careening from  
> the middle to the edges and back to pick up incomplete chords (see  
> Headlam and Brown reply to Tymoczko in Science).  We also have the  
> Chopin situation of combined triads and seventh chords that require  
> coordination between orbifolds of different cardinalities.

I have several things to say about this.

First, the point of using orbifolds is to faithfully model voice  
leading distances.  In this context, incomplete chords (or chords  
with doublings) are very different from permutations and octave  
transpositions.  Octave displacement and permutation preserve voice  
leading distance (at least when applied uniformly): hence, for any  
three-note chord (C, E, G) in any octave and any order, there is an E  
major chord two semitones away from it (using the taxicab metric).   
This is not true of an incomplete C major chord.  For instance, there  
is no E major chord, either complete or incomplete, that is *two*  
semitones from (C, E) or (C, C, C).  A tripled unison and a complete  
major triad simply have different voice leading potentialities.
	Consequently, if you try to glue together (C, C, E) and (C, E, G)  
you destroy your space's connection to voice leading, and this is a  
large sacrifice indeed.  This is why -- as discussed in my reply to  
you guys -- I think one typically wants to use orbifolds to represent  
background or conceptual structures (often involving complete triads)  
which may be only incompletely represented by the musical surface.  I  
really don't see any difficulties here -- this is not very different  
from what we do when we engage in Roman numeral analysis ... we say  
an incomplete C-E "represents" a conceptual C-major triad.

Second, if you want to faithfully model the musical surface, then  
it's not a problem that your representation of musical objects moves  
back and forth from the center of an orbifold to the edges. Why  
should it be?

Third, voice leadings between chords of different cardinalities are  
discussed in Cliff, Ian, and my "Generalized Voice Leading  
Spaces" (under the rubric of "C-equivalence").  While there are some  
technical issues, there's no fundamental problem: you can say, for  
example, that the C7 chord is closer to E major than Eb7 is, in  
(infinite-dimensional) OPC space.  (An efficient algorithm for  
calculating minimal voice leadings in that infinite-dimensional space  
is provided in "Geometry of Musical Chords.")  The real problem is  
that these spaces are difficult to visualize.

Fourth, there's a small problem of ambiguity -- CE could represent A  
minor in the right context, so you don't simply want to glue CEG and  
CE together.  Again, I think this suggests the strategy of using your  
musical intuition to translate CE into a complete triad, and then  
model the result geometrically.  Since this strategy produces  
analytical insights that I find useful, I'm quite happy with it.

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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