Introducing the Chord Torus
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Wait, don’t run! This musical honeycomb has fascinating properties.
I developed this thing as a tuning exercise for my old a cappella group with Mike Smith, Jon Napolitano, and Johannes Pulst-Korenberg. I doubt we’re the first to see it, but I can’t couldn’t1 find any relevant papers, so.
Here’s an intervallic compass to help you orient yourself:
Pick any vertex on a hexagon, and the three neighboring tones form a valid major or minor triad. Because I’ve spelled every note with sharps to make the torus’s cyclic nature clear, these chords aren’t music-theory approved: C minor has E♭, not D♯. But they’re enharmonic.
You can move from any chord to any neighboring chord by moving just one note—a whole step if moving vertically; a half step if diagonally:
Moving like this, we’d wander a whiteboarded honeycomb as an exercise. Here’s what the above progression sounds like:
But now you’re feeling betrayed, bereft. Where’s the torus here? I’m no 3D painter, so you’ll need a little imagination to assemble it.
We’ll use the figure at right.
Start by folding each green number onto its mate, forming a tube made of rings moving in major thirds.
Then do the same with the orange numbers, twisting the tube so that D connects to A and so on. You’ll form a donut that spirals around the circle of fifths as you trace its rim.
So? On this torus, distance corresponds directly to dissonance.
Take C. Here are the closest notes, in order of increasing distance:
- F, G, G♯, E, A and D♯, making fourths, fifths, thirds, and sixths;
- D, A♯, B and C♯, making seconds and sevenths;
- and F♯, making a tritone.
You can break these groups down further if you realize that folding the torus distorted its distances, but even this naïve ordering is remarkable.
I should print giant glossy posters of this thing for music classrooms. I don’t know any other way to get an amateur choir to sing an improvised chord progression.
Update: Peter Schankman pointed me at the AXiS MIDI controllers, which use just this layout. They even have interactive demos.
AXiS also notes that standard chords have regular shapes, no matter the key.
They do seem to be missing dissonance increasing with distance, but that may be of interest only to math nerds like me.
