[Smt-talk] Headlam on Orbifolds

Ildar Khannanov solfeggio7 at yahoo.com
Fri Mar 13 12:38:50 PDT 2009

Dear Dmitri,
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). 
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.

--- On Thu, 3/12/09, Dmitri Tymoczko <dmitri at Princeton.EDU> wrote:

From: Dmitri Tymoczko <dmitri at Princeton.EDU>
Subject: Re: [Smt-talk] Headlam on Orbifolds
To: "smt-talk Talk" <smt-talk at societymusictheory.org>
Date: Thursday, March 12, 2009, 10:39 AM

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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