Friday, Jan. 26, 2007
The Geometry of Music
By MICHAEL D. LEMONICK
TIME Magazine

When you first hear them, a Gregorian chant, a Debussy prelude and a John
Coltrane improvisation might seem to have almost nothing in common--except
that they all include chord progressions and something you could plausibly
call a melody. But music theorists have long known that there's something
else that ties these disparate musical forms together. The composers of
these and virtually every other style of Western music over the past
millennium tend to draw from a tiny fraction of the set of all possible
chords. And their chord progressions tend to be efficient, changing as few
notes, by as little as possible, from one chord to the next.

Exactly how one style relates to another, however, has remained a
mystery--except over one brief stretch of musical history. That, says
Princeton University composer Dmitri Tymoczko, "is why, no matter where you
go to school, you learn almost exclusively about classical music from about
1700 to 1900. It's kind of ridiculous."

But Tymoczko may have changed all that. Borrowing some of the mathematics
that string theorists invented to plumb the secrets of the physical
universe, he has found a way to represent the universe of all possible
musical chords in graphic form. "He's not the first to try," says Yale music
theorist Richard Cohn. "But he's the first to come up with a compelling
answer."

Tymoczko's answer, which led last summer to the first paper on music theory
ever published in the journal Science, is that the cosmos of chords consists
of weird, multidimensional spaces, known as orbifolds, that turn back on
themselves with a twist, like the Möbius strips math teachers love to trot
out to prove to students that a two-dimensional figure can have only one
side. Indeed, the simplest chords, which consist of just two notes, live on
an actual Möbius strip. Three-note chords reside in spaces that look like
prisms--except that opposing faces connect to each other. And more complex
chords inhabit spaces that are as hard to visualize as the multidimensional
universes of string theory.

But if you go to Tymoczko's website http://music.princeton.edu/~dmitri) you
can see exactly what he's getting at by looking at movies he has created to
represent tunes by Chopin and, of all things, Deep Purple. In both cases, as
the music progresses, one chord after another lights up in patterns that
occupy a surprisingly small stretch of musical real estate. According to
Tymoczko, most pieces of chord-based music tend to do the same, although
they may live in a different part of the orbifold space. Indeed, any
conceivable chord lies somewhere in that space, although most of them would
sound screechingly harsh to human ears.

The discovery is useful for at least a couple of reasons, says Tymoczko.
"One is that composers have been exploring the geometrical structure of
these maps since the beginning of Western music without really knowing what
they were doing." It's as though you figured out your way around a city like
Boston, for example, without realizing that some of your routes intersect.
"If someone then showed you a map," he says, "you might say, 'Wow, I didn't
realize the Safeway was close to the disco.' We can now go back and look at
hundreds of years of this intuitive musical pathmaking and realize that
there are some very simple principles that describe the process."

That's likely to help both scholars and teachers, he argues. By showing how
compositions of various styles move through his orbifold spaces, says
Tymoczko, you can see how different styles of Western music relate to each
other and evolve. Tymoczko's maps can also be an aid to composers, says
Cohn. Most have a favorite corner in orbifold space, a set of related chord
types that they tend to explore over and over in different ways. Venturing
into a different part of space can be tough; you have to learn your way
around a whole new auditory neighborhood. You can do that intuitively by
wandering around and seeing where you get to. But with the maps, you can
plot a route that you know in advance will make some sort of sense.

That doesn't mean you can program a computer with Tymoczko's orbifold maps
and have it spit out beautiful compositions. "I don't want to sell these
maps as the royal road to composition," he warns. "They don't substitute for
the hard work of learning how to move notes around." But they can help show
when a new idea is promising and when it will probably lead to a dead end.
"They might make an O.K. composer good," says Tymoczko, "but they won't make
a good composer great."

Find this article at: 
http://www.time.com/time/magazine/article/0,9171,1582330,00.html