This has actually been operationalized: https://en.wikipedia.org/wiki/Spiral_array_model
It's conjectural on my part, but if you apply D-weighting to the resulting pitch cylinders you can conceptualize musical space as an aesthetically pleasing spherical shape: https://en.wikipedia.org/wiki/A-weighting
If you further consider the nature of stereo hearing, head-related transfer functions, and sympathetic resonance, then you could go a bit nuts in the visualization department: https://en.wikipedia.org/wiki/Head-related_transfer_function
Here's a generalization that yields an infinite family of similar observations.
Step 1. Take any "torus knot with n marked points": which for our purposes will mean homeomorphism classes of embeddings from a circle with n marked points, to the torus.
Step 2. Draw it on paper.
In the case of the Coltrane drawing, n = (5 octaves * 13 notes per octive) = 65. The author exhibited 3 non-homeomrphic embeddings of this circle into the torus in the three images below the protractor picture. In particular these are embeddings generated by iterated Dehn twists on the unknot. You can classify them by winding number.
The image right below the aforementioned ones also shows an embedding of a circle with marked points into the torus if we choose to identify the 2 'c's. This time there's only 13 marked points. This suggests the following. To a circle with marked points, one can associate a canonical family, labelled by integers, of circles with marked points as follows: take any covering map of the circle, and declare the union of the fibers over the marked points to be the new marked points. In the case of the music scale analogy, we take the circle with 13 marked points, and this in particular gives a family of circles with 13 * k marked points for any positive integer k, and the musical interpretation is that k is how many octave one chooses to have. In the images mentioned above k = 3 and k = 1 are exhibited.
There's a simple further generalization of all this: we can replace the torus with any topological space. For example, doing this on a higher genus surface or a non-orientable surface would both yield probably interesting-looking diagrams.
This is as much about Coltrane as it is about Lateef, which makes this story even more fascinating. It's a shame that it ends so abruptly.
Lateef was successful with many things, but I enjoy his performances from the 60s best. When Lateef joined Cannonball Adderley's quintet at that time, Cannonball said that it wasn't really justified to call it a sextet now, because Lateef played so many instruments so well, that he should really count as more than one member.
Anyone know a nice source for learning some basic music theory? To be more specific, something that would teach whatever would be required to understand the musical concepts in this article (for example).
The ideal would be something directed towards readers with some mathematical background (but not a PhD), conversational, and focused on concepts and maybe historical development rather than just stating facts without context.
Coincidentally, i just stumbled upon this today: https://m.youtube.com/watch?v=xUHQ2ybTejU
Basically a musical palindrome: https://en.m.wikipedia.org/wiki/Crab_canon
I'm excited to see others on the journey of connecting music theory and math.
I feel like taking equal-temperament for granted obscures a lot of the simplest connections. Just intonation has a treasure trove of interesting implications (and challenges), and the math is elementary school stuff .
The diagrams remind me of performing a Dehn twist. Perhaps what you're doing at each grouping corresponds to a Dehn twist on the Tonnetz? Of course, since the Tonnetz is a particular subdivision of the torus, only certain groupings that satisfy some divisibility relations will correspond to Dehn twists.
Is this the Tonnetz, or is it different?
Are there musicians of today, with popularity similar to Coltrane's in 1960, that study and explore the theory of music at this level? If not, why not? Was jazz, and bebop in particular, unique in this regard?
It's a serious question; I don't know. I can't think of any current musicians but that's not conclusive. Also, I'm not sure exactly how popular John Coltrane was in 1960; certainly jazz was orders of magnitude more popular than it is now.
This looks like a perfect fit for a Max4Live patch. I've always been fascinated by the links between harmony and geometry.
I just recently started playing out of Yusef Lateef’s book and was wondering about the Coltrane diagrams at the front of the book myself. Did Yusef ever respond? I was planning on cleaning them up myself, glad to see someone has already given a lot of thought to it!
One of the most interesting things about this is reconstructing Coltrane's thinking. What steps did he follow to get here?
Here's a genius whose profession is music, not math. What was his intellectual path during his private sessions with the protractor?
Yes, but how does it sound?