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Back in the late 90s I came across this drawing in Lateef’s book and studied it closely. I found that it actually describes not a flat disk but a torus (aka don
by lucasgonze 10y ago
Back in the late 90s I came across this drawing in Lateef’s book and studied it closely. I found that it actually describes not a flat disk but a torus (aka donut) with a string winding around it. There are versions of the torus for all the symmetrical intervals (semitone, whole tone, minor third, major third). What changes from one interval to the next is the angle of the string.
I diagrammed each of the intervals and shared my work with Lateef. We had a warm conversation.
He mentioned to me that “Coltrane was always drawing things like this.” This particular drawing was something Coltrane did between set breaks at a gig they did together. Coltrane gave it to Lateef at that gig.
- zhemao 10y agoOh man, I'm so jealous that you got the chance to talk with Yusef Lateef. That must have been awesome to pick his brain about music. He's one of my favorite musicians (alongside Coltrane himself of course). Eastern Sounds is still one of my favorite albums. He was one of those musicians who had a unique sound but was always trying new things.
- lucasgonze 10y agoI'm so happy I got to have that meeting. He's a hero of mine too. He did a particular live solo that left a deep impression. The whole thing was integrated. There was a simple little motif that he built on, playing it over and over again, elaborating on bits, remembering the variations, repeating them, adding little elaborations to the elaborations. Eventually, it became like a labyrinth with the original motif in the center.
- jesuslop 10y agoHolly cow, the torus is topologically the product of two circles S1*S1 so he must have been thinking of something cycling in two dimensions, not just one as the circle of fifhts or chord-scale theory.
- lucasgonze 10y agoThe actual shape in this diagram contains five octaves. That's 60 semitones. The 60 semitones correspond to minutes on a clock, of course. The correspondence between music theory and time is not accidental. Both came out of Babylonian numerology, which had a thing with the integers that pop out of trigonometry - 12, 60, 360. Thanks for pointing out the "two circles" way of factoring a torus. I hadn't thought of that.
- qiuwob 10y agocould you elaborate on the toroidal structure for people like me who don't know music theory please ? what are the two fundamental cycles that make ZxZ ? I see semitones going around the main circle - but how are they connected back to themselves the other way ?
- lucasgonze 10y agoHere is a cross-section drawing I did of the toroidal structure. This is for minor thirds instead of whole tones, meaning that there are three semitones per winding instead of two. https://goo.gl/photos/U6gKHvRTUtM3p49z7 https://goo.gl/photos/U6gKHvRTUtM3p49z7
- theoh 10y agoI'm not a music theory guy, but Dmitri Tymoczko's work comes to mind here. That uses moebius bands rather than torii: https://www.simonsfoundation.org/multimedia/mathematical-impressions-making-music-with-a-mobius-strip/ https://www.simonsfoundation.org/multimedia/mathematical-imp...
- jesuslop 10y agoOh my, a scribbled diagram of Coltrane as an application of Tymoczko. That would be a nice motivation to study him. Sweet video, I think in the end they show the moebius strip obtained by glueing opposite points, opposite meaning the two points pinned by a needle perpendicular to the donut.