Skip to content

Dmitri Tymoczko: A Geometry of Music

I have always been a sucker for music theory and analysis. The combination of the fairly strict rules underpinning the way in which music works with the creative freedom expressed on top of them is really appealing to me. It is probably true of any art, but music is the one I know best and it feels especially true there. This new book promised “a revolutionary approach to music theory”, which set off my bullshit detector a bit, but everything else about it (blurbs, published by Oxford University Press) checked out, so I gave it a shot.

And it was great. There is a lot of exciting new stuff here, but Tymoczko doesn’t claim to have replaced the entire field of music theory, just to have discovered an additional way of looking at music that provides interesting insights, and he totally succeeds there. A quick overview, focusing on the stuff that was interesting to me:

n-part counterpoint can be visualized as the movement of a point through an n-dimensional space. Pretty obvious stuff in retrospect, especially if you have a math background, but it lets him do some neat analytical things, especially when he gets to 4+—note chords. Also, that n-dimensional space repeats in a very interesting way (the 2-dimensional case is a Möbius strip; the higher-dimensional ones are even weirder).

There’s a continuum all the way from local 2-part counterpoint to long-range modulation. Basically, Tymoczko is looking at music as much as he can through the concept of efficient voice-leading (transitioning from one set of pitches to another with each individual part moving as little as possible). In the small, this is about melodic counterpoint. In the middle, the same principles can be applied to harmonic motion (and he shows how a lot of chromatic music from Schubert on is best analyzed from this viewpoint). And in the large scale, you can treat scales (and therefore tonalities) as being akin to 7-note (or so) chords and do exactly the same sort of analysis. So for example, your standard modulation from C major to G major can be thought of as following a voice leading from C-D-E-F-G-A-B to C-D-E-F♯-G-A-B. Obviously there are qualitative differences as you move along this spectrum, but the fact that you can be using similar tools at each scale is really neat.

20th century tonality is a natural evolution of classical tonality, not a clean break. The standard history of music is that tonality slowly got stretched and stretched, as harmony got further and further out, until it reached a breaking point in the early 20th century, where it pretty much split into complete atonality on the one hand, and on the other a “tonality with non-functional harmony” that was qualitatively different from the tonality that came before in that the chords in it, although they were still consonant, had lost much of the semantic meaning that they had had through the 19th century. Tymoczko argues pretty strongly that rather than there being a real break between old and modern conceptions of tonality, the transition is actually relatively smooth, in that early 20th century composers were solving perfectly natural problems that had arisen in perfectly natural ways. These problems, as above, tend to be ones of voice-leading and the relationship between chords and scales. He also draws a compelling line from 19th century harmony through 20th century harmony through jazz harmony to 21st century harmony. Clearly everyone can hear jazzy chords in Debussy, for example, and you can think of it as being kind of a coincidence, but he shows that impressionist composers and jazz musicians were faced with similar musical problems, and solved them in similar ways.

There’s a ton more in here, and pretty much all of it was thought-provoking at the very least and genuinely conception-altering at the best. As far as background needed: although it doesn’t have much in the way of music-theoretical prerequisites (because it is approaching a lot of ideas from a different direction), it probably wouldn’t be that interesting to anyone who wasn’t already interested enough in music theory to have learned the more standard approach (if that makes sense). There’s a bit of math terminology but I don’t think it’s that scary. Highly recommended.