The Geometry of Music

March 16, 2008 12:49 PM

The connection between mathematics and music is often touted in awed, mysterious tones, but it is grounded in hard-headed science. For example, mathematical principles underlie the organization of Western music into 12-note scales. And even a beginning piano student encounters geometry in the "circle of fifths" when learning the fundamentals of music theory. ...according to Dmitri Tymoczko, a composer and music theorist at Princeton University, these well-known connections reveal only a few threads of the hefty rope that binds music and math.

The Geometry of Music
See also The Geometry of Musical Chords - Dmitri Tymoczko, Science 7 July 2006: Abstract
See also Dmitri Tymoczko, Composer and Music Theoristvia

Music theorists have long found Chopin's E minor prelude puzzling. Although the chord progressions sound smooth to the ear, they don't quite follow the traditional rules of harmony. When Tymoczko looked at the piece and watched the composition's motion through his geometrical space, he saw that Chopin was moving in a systematic way among the different layers of the four-dimensional cubes. "It's almost as if he's an improviser with a set of rules and set of constraints," Tymoczko says.

Whoa... No way.

Dude, musta been high when he wrote that!

Ya think ?

Well, duh...

I knew that.

I did not know that.

Whatever....

Well... I am of no fixed opinion in regards to this matter.

Well, I must admit, however, I find that encountering the question What makes music great ? and a mention of Deep Purple on the same homepage rather more than a bit disconcerting.

posted by y2karl (30 comments total)

A friend (a conservatory classmate I hadn't seen in 19 years; we were both music theory majors) just mentioned this guy to me last week. I guess I have some reading to do, this seems rather intriguing.
posted by strangeguitars at 1:20 PM on March 16

So listening to music is actually some kind of synesthesia of multi-D spaces?
posted by MNDZ at 1:26 PM on March 16

This is quite interesting, even if I'm having a hard time wrapping my brain around some of it--thanks for this.
posted by retronic at 1:30 PM on March 16

I'm having a hard time wrapping my brain around some of it
Don't bother; it's got about as much content as the bible code.
posted by Wolfdog at 1:31 PM on March 16 [1 favorite]

See also: http://www.wnyc.org/shows/radiolab/episodes/2006/04/21

The third segment is particularly relevant and awesome.
posted by kaibutsu at 1:32 PM on March 16

I was googling around for some information about the neurological connection between doing math and doing music to link to, because I've certainly read enough references to it over the years, but most of the sites had to do with improving children's math skills by means of musical training/listening. Apparently math scores are more important than the pleasures of music.

My brother and I share a lot of the same genetic material; he is a mathematician, I am a musician.

Music and math are obviously abstract self-referential matrices...this link isn't too bad.
posted by kozad at 1:38 PM on March 16

Seconding retronic, I've been interested in this sort of thing since I read of some of the deliberate use of mathematics apparently inherent in Boards of Canada's music as well as some of the ratios between notes I've learnt about in music lectures. Will give this a good read tonight!
posted by TheWaves at 1:39 PM on March 16

Also last week, I was thinking about using a circle of minor seconds instead of a circle of fifths to visualize harmonic shapes, incorporating the idea of pitch-class sets like he does to avoid differentiating octave expression.

His circle representation of Chopin's E minor prelude does just that. It's a nice way to visualize harmony. Much more useful than doing the same thing on the circle of fifths.

I don't have any idea what's going on with the other representation inside the huge tetrahedron. Has anyone figured that one out?
posted by strangeguitars at 1:46 PM on March 16

Dmitri is an old friend of mine and I've been following this work for a long time. Coincidentally, I just brought him to UW to deliver a lecture on this stuff. Contra wolfdog, there really is a lot of beautiful content here. I wrote a bit about Dmitri and his "moduli space of chords" on my blog in order to help advertise the talk. Here's a piece (which might make a bit more sense if you've read what comes before)

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.

I'm happy to have a go at any questions about this part of DT's work, which I'm reasonably well-acquainted with.
posted by escabeche at 2:18 PM on March 16 [2 favorites]

Also last week, I was thinking about using a circle of minor seconds instead of a circle of fifths to visualize harmonic shapes, incorporating the idea of pitch-class sets like he does to avoid differentiating octave expression.
posted by strangeguitars

Eponysterical?
posted by papakwanz at 2:22 PM on March 16

His dad was a beloved logic teacher and all around great guy. I'm glad to see Dmitri is carrying on the family m.o.
posted by LobsterMitten at 2:47 PM on March 16

What of this I can understand makes sense. People have been talking about pitch class space in music theory for a while — that's the circle he talks about, where you ignore octaves, the twelve notes in the scale are all points, and transposing means rotating around the circle. And if single notes can be visualized on a circle, then it makes sense that chords can be visualized on three- or four-dimensional shapes. That's a clever step to take, but it's not necessarily all that "woo woo."

Whoa... No way.

Dude, musta been high when he wrote that!

As for Chopin "improvising in four-dimensional space," I wonder if that makes him sound more mysterious than he was. Look: when you're graphing something, you need one axis for each variable. Here, there are four variables, because that prelude has four notes sounding at any given moment. When I walk around my apartment turning the volume on my iPod up and down, that also creates a pattern you'd need a 4-D plot to represent — latitude vs. longitude vs. altitude vs. volume — but I assure you I can do it stone sober.

This is still a really cool way to think about music, and a really cool post. I'm not trying to hate on it at all. I just think the "four-dimensional" business is making it seem more far-out than it needs to be. (Then again, I do music but I don't do higher math. There may be far-out mystical insights in there than I'm missing as a result. Any mathematicians out there want to clue me in?)
posted by nebulawindphone at 3:35 PM on March 16 [1 favorite]

The mathematics of (western 20th century) music:

- There is a base note n whose frequency F is arbitrary.

- Note n+m has frequency F * 2^(m/12)

- Call the notes n=(0,...,11) the names (C, C#, D, D#, E, F, F#, G, G#, A, A#, B)

- Two notes whose ratio of frequencies is 2^n where n is an integer have the same name, and the higher note is said to be n 'octaves' above the lower.

- Two notes with the same name in different octaves sound very similar, for psycho-acoustic reasons.

- Notes whose frequencies are integer multiples of each other tend to sound nice together, for psycho-acoustic reasons.

Most of music is intuitive/psychological and mathematics doesn't get you very far.
posted by snoktruix at 4:27 PM on March 16

(Sorry, say the base note is n=0)
posted by snoktruix at 4:29 PM on March 16

Mathematics "underlies" all tonal systems, in a sense making the point trivial, and there is nothing special about the western 12-note tonal system except the way it actually has required a distortion of "pure" mathematical rations in order to achieve equal temperament.

This is still mysticism, disguised as science. Mathematics describes nature. Music is part of nature.
posted by fourcheesemac at 5:12 PM on March 16

ratios, not rations
posted by fourcheesemac at 5:14 PM on March 16

I'm very curious to read his actual paper---some of what the article on the science blog makes it sound eerily close to some material on which I wrote my Master's thesis...
posted by vernondalhart at 5:44 PM on March 16

I'm happy to have a go at any questions about this part of DT's work, which I'm reasonably well-acquainted with.

Does he think Deep Purple made great music ?
posted by y2karl at 5:50 PM on March 16

I dunno about Deep Purple but we certainly was a Steve Forbert fan in 1979 :)
posted by jdfan at 6:01 PM on March 16

err, he (DT).
posted by jdfan at 6:02 PM on March 16

Nice post; thanks. Regardless of whether you want to call this type of study "hard science," or argue the relevance, it's still fun and interesting to think about (for geeks, at least).
posted by p3t3 at 6:13 PM on March 16

This is still mysticism, disguised as science. Mathematics describes nature. Music is part of nature.

I don't see why it's mysticism rather than the science it appears to be. If your point is just against the pervasive sense of "the unreasonable effectiveness of mathematics", then I suppose I agree. But what thesis, exactly, do you take yourself to be refuting? And, moreover, where in this is the mystical, where the scientific?
posted by voltairemodern at 6:25 PM on March 16 [1 favorite]

Hmmm. I'm a musician with a degree in mathematics and I'm having a hard time seeing that there is really anything there.

For example, the first video with the beads on the wire moving; that seemed meaningless to me. Not patternless! I've found from years of fiddling with this sort of thing that if you take ANY sort of visualization of a famous work of music, you see interesting patterns. But does this actually reveal anything about the underlying music? Sometimes yes; in this case, I can't see it.
posted by lupus_yonderboy at 6:25 PM on March 16

If mathematics describes nature and music is part of nature, why is it mysticism to try and describe music with mathematics?

Granted, the final quote from Tymoczko in Rehmeyer's article is at best, misleading. To say that Chopin was thinking in four dimensions tends to imply that he was thinking in four spatial dimensions, which is kind of silly. But if you take the looser interpretation of "dimension" to mean "degree of freedom", then it's not such a ridiculous thing to say. Nebulawindphone's example has three spatial dimensions and one dimension of the amplitude of sound. Imagining at a piece of paper decorated in various colors with varying shades is also thinking in four dimensions: two spatial, one for "color" and one for "lightness". (Note: due to peculiarities of physiology and psychology, the dimensions of color that humans naturally think of (hue, lightness, saturation, brightness, etc.) don't always correspond to the most obvious physical characteristics (wavelength, amplitude, etc.) Humans are actually not great at imagining three spatial dimensions because we tend to visualize them, and pure static imagery is better suited for two-dimensions. We're also not very good at keeping separate three different color dimensions, although we can distinguish three of them (e.g., hue-saturation-brightness or red-green-blue). Before I get further side-tracked, I'll just say that it's hardly revolutionary to think in four dimensions, but it does require some talent.

I'd like to share my interpretation of some of the more interesting parts of the article to hopefully help those who are a bit confused understand what's going on, and because I love topological visualizations.

The idea of mapping one note to a dimension is hardly new. To keep the visualization useful, when we visualize multiple notes, traditionally this is by imagining multiple points on the same one-dimensional scale. Mathematically, this is not any different than a single point in a multidimensional space. What this buys us is not just some nifty visualizations because those start to lose usefulness past two dimensions (note how hard it is to read the second video without camera movement, context, focus, or bioptic cues to help us see that third dimension). The main benefit here is that it enables us to use the powerful tools of geometry and topology to talk about what's going on. One of the simplest topological tools we have is the "identifying" of points in a space that Tymoczko talks about. The simplest example of this is the identifying of the "same" notes in different octaves, turning a single open-ended line into a closed loop (still one-dimensional, just loopy). The motivation for precisely how to make that identification is psychological and cultural, but it's a good way to capture in mathematics what we hear when we listen to notes (provided the identification we make matches the one from his model, which, for a lot of modern Western music, is generally the case). Doing this for one note is hardly ground-breaking, but he goes further, identifying the points on the two dimensions of a pair of notes, yielding a torus (the surface of a donut, just like the space in which the game Asteroids is played). For a set of three notes (like a major or minor chord), we still have only a three-dimensional manifold, but you'd need a fourth dimension to imagine this if you're stuck thinking about curving things back on themselves as in the donut case (instead, imagine a 3-D Asteroids game, where going through any of the 6 "walls" leads to the same point on the opposite wall; this is the space that these triples live in). Going one step further, Tymoczko notes that if the notes are all being played on the same instrument simultaneously, then we certainly would view the chord C-E-G as the "same" as G-E-C, but those represent different points in the space. But identifying points is easy in topology, provided you don't care how easy it is to imagine things. Even in the two-dimensional case, you don't really end up with a Mobius strip (but you do get something close to that).

Why do something like that, other than for the sheer joy of it, which is enough for me? Well, in this space, some chords that seem very different if you think in terms of their names are actually quite close. The four dots on-a-line video shows this fairly well (four points on a one-dimensional manifold are much easier to see than one point in a four-dimensional manifold), though the conceit of moving the points to the new locations instead of just having them jump there instantly hides the fact that a couple of the chord changes are not as small (i.e. covering a large distance in this space).

Chopin probably did not think consciously along any of these lines when he sat down to compose music and certainly did not think consciously along these lines when he was improvising, but that doesn't mean that we can't use mathematics to describe explicitly what many music lovers can understand implicitly.

I could go on for hours, but as usual, I've already written too much.
posted by ErWenn at 6:26 PM on March 16 [2 favorites]

All I know is that when I was in college the music department was the largest consumer of mainframe computer (and that was all there was then) resources of any department on campus, even more so than the computer science, math or any engineering department. Their composition programs made some lovely musci.
posted by caddis at 11:11 PM on March 16

What? I waited all day and still nothing about Donald in Mathmagic Land. I'd swear there was another music and math post recently and I waited to see something there. I must have watched this a dozen times when I was a kid. Great stuff.
posted by ericales at 2:51 AM on March 17

I'd go on and on about it, as an anthropologist of music and a musician and a former musicologist, but it comes down to this for me: so what?
posted by fourcheesemac at 12:46 PM on March 17

Which, y2Karl, is not meant as a slam at the post. You are one of my all time faves and you post great music stuff.

I have just dealt with too many music theorists who have trouble hiding their erections around equations they believe trump any other explanation of what "music" even is. And Tymockzo is one of them.
posted by fourcheesemac at 12:47 PM on March 17

I see it as a very coherent and useful way of visualising chordal harmony.
posted by unSane at 2:51 PM on March 17

I read some articles about this connection on Modern classical a while ago. Interesting stuff.
posted by mysticalfairy at 6:39 PM on March 28