Math in… Voice Leading
A standard chord progression in jazz is the 2-5-1 (ii-V-I) progression. Here are some voicings in the key of C:
Image: Scot Ranney (learnjazzpiano.com)
If you look at the treble staff (the top set of five lines), you’ll see that when transitioning from one chord (cluster of notes) to another, the notes barely move. Moving from Dm7 to G7, we see the top note slide down from C to B while the bottom two notes remain F and A. Moving from G7 to C△, we see the top not remains B while the bottom two slide from F and A to E and G. The notes on the bass staff are simply plunking out the roots of the chords: D, G, and C.
There are many other ways to voice these chords — “voice” meaning choose the notes that make it up but leave its sound the same in some essential way. Strictly speaking, the notes in each of Dm7, G7, and C△ are DFAC, GBDF, and CEGB, respectively so this composer made a choice to go with A instead of D in their G7 chord. There are all sorts of music theoretic explanations for why you might make that substitution, but for our purposes I’ll just note that it makes the note transitions much more subtle!
To create an extremely subtle chord progression, you could try changing one note at a time and only bump it up or down by a half-step or whole-step, the basic increments in a major or minor scale. In 1739, the mathematician Leonhard Euler described a network of how to move from chord to chord in this way now called a Tonnetz (German for “tone net”).
I vibe-coded an app below that allows you to experiment with major and minor triads (special 3-note chords). Clicking a colored circle in a triangle will play the triad. You can click the corners of the triangle to play the individual notes. Enabling “Smooth voice-leading” will make it so that clicking a disk and then its neighbor will choose the version of that chord that differs in one note by a half or whole step.
What do you notice if you start at a C major and end at a difference C major? Try getting from C major to C major by clicking through a path of adjacent triangles. Is the result always the same?
Clifton Callender, Ian Quinn, and Dmitri Tymoczko took this idea and generalized it to other sorts of triads by creating equivalence classes of triads. The idea is to take the chromatic scale extending infinitely in both directions and assign each note a number:
A C major triad (CEG) could then be denoted 0.4.7. Another triad that involves C, E, and G is 4-7-12, but we could just choose the representative for each note in the range 0-11, which would pull it back to 0.4.7. A G major triad (GBD) would then be 7.11.14, which reduces to 7.11.2 and reorders to 2.7.11. (If you’re familiar with modular arithmetic, this is wrapping around is arithmetic modulo 12.) In this scheme, we could give any triad an equivalent 3-number label.
They go further with their equivalency and say two triads of the same type will be equivalent (e.g., major triads are all equivalent, minor triads are all equivalent, etc). Since a chord’s type essentially comes from the spacing relationship of its notes, the way they handle this is translating chords by adding the same thing to all numbers, wrapping around so the numbers stay in the range 0-11. This means that our G major triad is equivalent to each of the 12 chords
2.7.11, 3.8.12 = 0.3.8, 1.4.9, 2.5.10, 3.6.11, 4.7.12 = 0.4.7, 1.5.8, 2.6.9,3.7.10, 4.8.11, 5.9.12 = 0.5.9, 1.6.10Each of these is indeed the code for a major triad — for example, 0.4.7 is our C major triad. Similarly, any minor triad is equivalent to 0.3.7, our code for a C minor triad. If we take every possible triad triple X.Y.Z (where X, Y, Z are allowed to have the same value), it turns out there are only 31 different types. Since notes can be reordered so that any spacing larger than 4 is at the back, we can choose a representative number for each starting with 0 and with no digits larger than 8, getting rid of the need for punctuation. Here are representatives for the 31 classes:
000, 001, 002, 003, 004, 005, 006, 011, 012, 013, 014, 015, 016, 022, 023, 024, 025, 026, 027, 033, 034, 035, 036, 037,044, 045, 046, 047, 048,055, 056You can then construct a graph with edges based on whether you can move from one triad type to another by increasing or decreasing the pitch of a single note by a half step.
I vibe-coded an app below so you can play with that graph, as well. It starts in “Smooth voice-leading” mode, which means pressing a neighboring node will choose a chord that differs by a half step in one note from the last chord played. It turns out this choice is well-defined except when moving between three triads: the major triad, the minor triad, and the augmented triad. (Why might that be?) In those cases, you’ll have the ability to choose from the two or three candidate chords. You can preview their sounds by right clicking.
An interesting property of this network is that, even in parts of the graph where you’re not given a choice, you can find triangles where if you travel around them starting and ending at the same node, the chord can end up higher pitched than it started! See if you can find one. If you travel around such a triangle multiple times, the shift in pitch will become more apparent.
One of the authors of the paper describing this model, Dmitri Tymoczko, is interested not only in the math of music, but also the music of math! At the International Congress of Mathematicians, Dmitri and some musician friends entertained us with a fantastic part-lecture, part-performance on the music of math. The American Mathematical Society recorded it and recently uploaded it to YouTube. Check it out here:
In what other ways is music mathematical?
