The moduli space of chords: Dmitri Tymoczko on “Geometry and Music”, Friday 7 Mar, 2:30pm

28Feb08

The next installment of the “Math And…” seminar is coming next week:

7 Mar 2008, 2:30 PM, Birge 145: Dmitri Tymoczko (Princeton University, music)

Title: “Geometry and Music.”

In my talk, I will explain how to translate the basic concepts of music theory into the language of contemporary geometry. I will show that musicians commonly abstract away from five types of musical transformations, the “OPTIC transformations,” to form equivalence classes of musical objects. Examples include “chord,” “chord type,” “chord progression,” “voice leading,” and “pitch class.” These equivalence classes can be represented as points in a family of singular quotient spaces, or orbifolds: for example, two-note chords live on a Mobius strip whose boundary acts like a mirror, while four-note chord-types live on a cone over the real projective plane. Understanding the structure of these spaces can help us to understand general constraints on musical style, as well as specific pieces. The talk will be accessible to non-musicians, and will exploit interactive 3D computer models that allow us to see and hear music simultaneously.

The interactive 3D computer models are made by Dmitri’s program Chord Geometries, which you can download from his website. Or you can just watch the sample movies, and see what the opening of “Smoke on the Water” looks like on a Mobius strip. Dmitri comes by his mathematical know-how familially; his younger sister is the algebraic geometer Julianna Tymoczko (with whom I wrote this paper about diameters of finite groups) and his father was the philosopher of mathematics Thomas Tymoczko.

I’ve booked a big room, and the talk is open to the public, so if the abstract sounds interesting to you, please come!

On the other hand, perhaps the abstract puts you off by means of words like “real projective plane.” Well, it’s written for mathematicians; maybe it’s worth explaining in more down-to-earth language about what a guy like Dmitri means by “the space of chords,” or, as an algebraic geometer like me might put it, the moduli space of chords. Explanation below the fold: it’s essentially a brief version of an article called “The Idea of a Moduli Space” that I wrote for Imagine magazine when I was a grad student.

Non-definition: Let S be some set of things you’re interested in. A moduli space for S is some kind of shape M with the properties that

1. The points of M are in 1-1 correspondence with the elements of S;
2. Two points of M are close together if and only if the corresponding elements of S are “close together,” with respect to whatever notion of “closeness” applies to S.

This is a non-definition because there are lots of words whose meanings I haven’t specified — “shape,” and “close together” to start with. But it’s good enough to start making examples. For instance, the moduli space of real numbers is a line — namely, the number line you learned about in grade school. And indeed, two numbers are close to each other (in the numerical sense) just when the corresponding points on the line are close to each other. Similarly, the moduli space of ordered pairs (x,y) of real numbers is a plane — this was Descartes’ great innovation, now so familiar that it seems almost obvious.

What about lines in the plane which pass through the origin? Well, such a line is determined by the angle it makes with the horizontal, which is some real number in the interval [0,pi). So you might say, “the moduli space is the interval [0,pi).” But that’s not right! Because 0 and 3.14159, for instance, are very far from each other in the interval, but the corresponding lines are almost identical. So condition 2 above is violated. (This might be clearer if I knew how to make pictures on the blog!) The moduli space needs to keep track of the fact that the two ends of this interval, rather than being far apart, are actually supposed to be close to each other; this can be accomplished by grabbing the interval at both ends and bending them together until they form a circle. And it is the circle that’s the moduli space for lines through the origin.

(Exercise for the energetic reader: what is the moduli space of all lines in the plane?)

What about unordered pairs of real numbers? Well, you might again be tempted to say “the plane” — after all, given a pair of numbers {x,y}, you just write down the point (x,y) in the Cartesian plane. But now the points (2,1) and (1,2) refer to the same unordered pair; so condition 1 is violated. We need to identify those two points, and indeed, to identify (x,y) with (y,x) for every pair of real number (x,y). You can envision this process geometrically as “folding the plane in half” along the line x=y.

This last example is the most relevant one for Dmitri’s talk. When we talk about a pair of real numbers, we have a choice — do we care what order they come in, or do we consider two pairs of real numbers the same if they differ only with respect to ordering? This choice is important — in the first case, the moduli space is the plane, in the second case, only the half plane.

Similarly, when studying music theory, one has choices about how to think of chords. When we say two chords are “the same,” do we mean they contain the exact same sequence of notes? Or are two chords the same if one is the transpose of the other by an octave? What if one is the transpose of the other by some other interval? What if chord 1 is chord 2 with the bottom note transposed an octave up, but the rest left alone? What if chord 1 is chord 2 upside down? Any one of these relations — and there are more in this vein — would lead some music theorists, in some contexts, to say the chords were “the same.” And every choice about which pairs of chords are “the same” leads to a different moduli space. Dmitri has worked out a really nice description of these moduli spaces, which apparently organizes and unifies lots of previous work in music theory — in particular, it provides a very natural and geometric notion of what it means for one chord to be “close” to another.

We’re delighted to have him here — I hope you can come.

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