Mathtrek has an interesting post that explores the geometry of music.
According to Dmitri Tymoczko, a composer and music theorist at Princeton University…to grasp the true structure of music, we need to understand the geometry of hyperdimensional objects.
Tymoczko compares the structure of music to the shape of a rock face that a rock-climber is scrambling up. “If you know the conditions of the rock face, you can predict the motions of the climber,” he says. “The structure of the space makes certain choices overwhelmingly natural or convenient. There’s something similar that goes on with music. When you think about things abstractly, you can come to understand that the directions that music went aren’t completely arbitrary. Composers are exploring the possibilities that musical space presents them with.”
Tymoczko built on familiar geometrical analogs for music. For example, musical pitch is often imagined as lying on a line with low notes to the left and high notes to the right. Furthermore, as pitches go higher and higher, the notes repeat in different octaves, such that a low C, a middle C, and a high C all sound very similar. Often, the exact octave of a particular note doesn’t matter very much in music. Instead, musicians commonly visualize a “pitch class circle,” which comes from the original line by gluing together each point of the line that represents the same note in different octaves. So low C, middle C, and high C, for example, would all be glued together.
Applying the same kind of reasoning to complete pieces of music, Tymoczko created a geometric space in which he could analyze a piece of music with two notes being played simultaneously. He started with a piece of paper and made the horizontal direction represent the pitch of one note and the vertical direction represent the pitch of the other. A piece of music with two voices would correspond to dots moving around in this space.
Then he modified the space to embed musical structure within it. First, Tymoczko used the same method musicians used to create the pitch circle. He glued the left edge of the page to the right edge, turning the horizontal lines into circles and creating a cylinder from the whole page. Then he glued the bottom end of the cylinder to the top, turning the vertical lines into circles as well and creating a donut shape from the entire page.
Next, he noted that the order of the notes in a chord doesn’t much matter. That means that the point on his page that has C in the horizontal direction and E in the vertical direction is really the same as the point that has E in the horizontal direction and C in the vertical direction. So he took his space and glued all those points together. It takes a bit of effort to visualize it, but for two simultaneous notes, this turns the donut shape into a Möbius strip.
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.
Another way of visualizing Chopin’s composition is through a three-dimensional projection of a four-dimensional space, as in the video above. The chords primarily cluster in the center of the space, usually moving through small distances.
I’m not sure if I”M convinced by some of Tymoczko’s assertions – but his work offers an interesting way of visualizing relationships in music.