Even as a recovering engineering student, this is something I do not entirely understand. I had to dust off my admittedly quite rusty music theory knowledge, and augment it with extra research before I really had a clue of what is going on with this. It’s still really cool, though, so hang in there while I do my best to explain what I can.
In short form, Dmitri Tymoczko created software that analyzes chord progressions. Instead of the familiar I-V-IV-vi four-chord song, some music uses “non-standard” progressions — the example presented is a Chopin piece. While the melody and harmony sound pleasant to the Western ear, the chord progression is difficult to explain; why it “sounds good” despite being nonstandard is even more difficult to articulate. This software uses 4-D space (it includes a time element) to map the notes, showing how close and similar each chord is to the next.
The numbers are intervals, expressed in the number of half-steps from the root (0) to the next half-step in a western twelve-tone chromatic scale. For example, the image above shows four-note chords. 0000 is a chord that, for analysis sake, is made up of four root notes. 0369, assuming the root note is C, is made up of C, E♭, G♭, and A. (Or, C, E♭, G♭, and B♭♭, to make a Cdim7). The reason for using this notation (called integer notation) is to compensate for enharmonic notes, like the A and B♭♭ just mentioned. Since the purpose is to compare how closely related each component tone is to the others, it does not make much difference if we are calling said note “A” or “B♭♭”.
I would love to try this out with a couple of my favorite songs. One of my favorites switches keys in the middle of the chorus, then switches back; I think it would be cool to visualize what is going on and why something so odd still sounds so good.
I emphatically encourage you to read this all for yourself. There are some cool videos too. Check it out!
The article: Cosmic Log – The geometry of music
The paper: The Geometry of Musical Chords
The software: ChordGeometries