[Smt-talk] Headlam on Orbifolds

Dmitri Tymoczko dmitri at Princeton.EDU
Thu Mar 12 08:39:27 PDT 2009

Greg wrote:

> This discussion will go around eternally in not distinguishing  
> between voice leading and part writing.

I use "voice leading" to refer to a particular way of connecting two  
chords.  Voice leadings are described informally by phrases like "C  
major moves to F major by keeping C fixed, shifting E up by semitone,  
and G up by two semitones."  Up-to-date formal definitions can be  
found in my "Scale Theory, Serial Theory, and Voice Leading" (Music  
Analysis, 2008), as well as in Cliff, Ian, and my "Generalized Voice  
Leading Spaces" (Science, 2008).

I don't use the term "part writing" very much; but if I did, I'd use  
it to refer to something like counterpoint -- the musical art of  
constructing meaningful simultaneous melodies that articulate  
significant harmonies.  This involves voice leading, but is much more  
general -- for instance, you need to think about ranges, voice  
spacing, melodic highpoints, gap-filling, etc.


Ildar wrote:

> In music, you seem to be stuck with the signifiers, while the field  
> of signifieds is quite different in its geometry. That is what  
> exactly tone representation idea suggests. Even the calculation of  
> the size of voice leading does not work in orbifold model. At least  
> that is what I understood from the paper of Dr. Rachel Hall at the  
> last SMT meeting in Baltimore. Musical space is a topological  
> space, but its parameters cannot be calculated using your method.

I see why you might've ended up with this impression, but it's not  
quite accurate.  The whole point of the geometrical approach is to  
construct spaces in which distance represents voice-leading size.   
The amazing thing is that you can do it, but that it requires exotic  
geometry -- non-Euclidean twists and turns, singularities, etc.

To measure voice-leading distances in an orbifold, you simply need to  
choose a metric or method of measuring distance ("taxicab,"  
Euclidean, "largest distance," etc.).  (This is what you need to do  
in any other geometrical space.)  Rachel and my paper, summarized in  
Baltimore, as well as some of the supplementary sections of "Geometry  
of Musical Chords," describes a trick that allows you to *avoid*  
choosing a specific metric.  It turns out that all "reasonable" voice- 
leading metrics agree that, for instance, E and Ab are the closest  
equal tempered major triads to C major.  But the fact that you don't  
*have* to choose a metric shouldn't be taken to imply that you *can't*.

More generally, I would say that the amazing thing is that concepts  
in music and geometry are in some cases quite close.  This means that  
tools from one domain can give insight into the other.  I believe  
that contemporary mathematicians would probably say that  
"geometrical" concepts are simply more general than musical ones:  
"geometry," as they tend to think of it, is an abstract discipline  
referring to any space with the right kind of formal structure --  
which could be physical space, musical space, or various other kinds  
of abstract "configuration spaces."


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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