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Coltrane Pitch Diagrams: Wrapping notes around a torus (medium.com)
198 points by lucasgonze on Mar 12, 2018 | hide | past | web | favorite | 68 comments

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

I have considered a spiral version.

You could see the Coltrane diagrams as viewing a spiral head-on, so that the spiral nature of the structure is hidden by occlusion.

This would solve the problem of the arbitrary nature of representing only five octaves.

Was 5 octaves really an arbitrary choice? It's the point where the circle of fifths gets back to the starting note. (Nevermind that detail that there are no positive integers m, n such that (3/2)^m == 2^n.)

That's not exactly arbitrary - there is a reason to use that range in preference to others.

Doesn't the spiral array model discard enharmonic equivalence?

I think there's a way around this if you imagine concentric shells with harmonic rotation, but I'd have to go and look up one of my old notebooks from a few years ago. I stopped writing/playing music regularly after a massive construction project in my neighborhood :-/

Oh no! Start again! Been stuck doing covers/arrangements for a few years myself...

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.

I'm totally with you, but there are 12 notes per octave (at least in the standard western scale). =)

I think he’s including the octave.

You're right I miscounted.

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.

A blog post: "Music theory for [non-music] nerds": https://eev.ee/blog/2016/09/15/music-theory-for-nerds/

Previous discussion here: https://news.ycombinator.com/item?id=12528144

The only thing that learning music theory will do for you in this case is make you realize that most of what you see in this article is nonsense. Certainly, none of it will help you make music. I say this as a songwriter and student of music theory for 20 years.

I agree with you on this as a lifelong musician and improvisor. It isn't the math of grammar we think about when conversing to convey meaning and style. We think about what we want to say and how we want to say it. We listen and build something together just like we do in conversation. There is a kind of obsession among some jazz players with spirituality though, it is true. If you are the sort of person who likes that kind of thing, you probably like prose writing like this as well. You certainly don't need to analyze all mathematical relations and artifacts to improvise masterfully. In my experience, the more time a player focuses on this type of thing, the less energy they put into actually having something to express. Or perhaps it has something to do with a fear that they will find they have nothing to say, so they avoid that challenge by making technique and mathematical-theoretical relationships the focus of their practice. The only composer I can think of who managed to be obsessed with the math of things while actually using them to create a unique language through which he expessed ideas,style, and narrative was Messaien. (in his modes studies and bird calls) I do understand why many jazz musicians reached for spirituality and really complex mathematical relationships in modes of improvisation:it's because they were not granted the legitamacy and respect they deserve (and craved, many of them, like Monk, for example) from so-called arbiters of "high culture". If what they were doing could be serious and difficult and inscrutable except with hours and hours of thinky labor, they felt it might elevate them.Of course this was totally not needed and what they (monk especially) were creating had a laguage and history of its own that was in many ways more innovative and culturally relevant than that what they sought to be included with. Of course that's easy to say now.It must have been frustrating, like for Ornette.

> most of what you see in this article is nonsense

How as a creative person are you okay with saying blank statements like that? Sure if you just want to only talk about John Coltrane's drawing out of his time and only focus on today, possibly.

As someone who has several years of Music Theory and a grandfather that retired as a NYC Jazz Musician I absolutely know many people that worked this way. Heck we had 20 years of the "Mozart Effect" (Was bogus since day one that believed passively listening to Mozart made children and lab rats smarter). BUT music is mathematical and some people have certainly taken this bogus Mozart thing and found out the math behind music. For decades the musical music was captivated with math in the 20th century. This wasn't some spiritual journey in music it was a mathematical puzzle they were trying to work out.

The best example of Music based on Math was Iannis Xenakis, who composed from 1940s into the 1990s. Now he is certainly a required taste and I would say listening to his music live MIGHT make it more palatable but he was using computers in the 1960s to help his compositions. He wasn't Thelonious Monk who just played everything in the hardest keys to prove how great he was on piano but just wrote music from the love he had for mathematics and engineering. Classical music or "European Classical Music" was more into Math and as the high brow crowd was into his stuff it certainly was inside the head of John Coltrane, Miles and Ornette Coleman.



Stravinky's Septet from 1952 can also show how this serial technique composition had huge influence on Jazz in the late 50s and early 60s. Especially when you listen to Miles Davis and Bill Evans "Sketches of Spain." You see that thousands of musicians were exploring and thinking about the "Math of Grammar."


You also rightly said, "For decades the musical music was captivated with math in the 20th century. This wasn't some spiritual journey in music it was a mathematical puzzle they were trying to work out."

Yeah, you are right, for some. But actually, for some others, it was a serious spiritual journey. Like for George Crumb! (way into numerology- used it in Ancient Voices, Black Angels and other pieces- yeikes!- but some people dig this-- remember though, some people also dig a good seance or a good horoscope you know?)



Also- look into Webern's beliefs about numbers. He was really driven/controlled by this superstition. There is certainly a tradition of just this kind of spirituality that is alive in jazz today. Sun Ra was all about this spirituality without the math stuff. https://www.youtube.com/watch?v=djBKQNVj5Cc

Anyhow, computers had a lot to do with this math obsession in 20th century music composition.

And Ornette? Was he about math at all? I may not know about this. For me, having played a lot of his music, I'd say that his music is a music-of-the-people, and much more about folk music than jazz, although he's of course a great jazz musician. One of the greatest, and maybe because of this fact. He had confidence in the simplicity of his ideas. He didn't seem out to prove anything to anyone. Harmelodics has absolutely nothing to do with mathematical relationships or theory. It's all about each bringing what they think to a melody. (in very short, here) But if he was about math, I just don't know it. Please tell me, so I can look into it.

Well, it's difficult to discuss this math/music issue in a just a comment-length mini-essay and manage to get at what I'm trying to convey. I didn't write the "nonsense" comment myself, but i do hear the legit and justified frustration in it. Sketches of Spain- beautiful. Xenakis though. Man, every time I have had to perform one of his pieces, I'm thinking I should be paid double or they should just hook a digital sound making device up to his score so they don't have to pay anyone to practice that stuff! I guess some people enjoy it? He is the perfect example of what I was trying to get at. Does he have ideas? I guess so...? Do i need to practice them? Um, only if I get paid double/triple for the gig. Know what I mean? It isn't that I'm saying that musicians don't ever practice this way or compose using these devices. Tons of jazz musicians sit around with Slonimsky, mathing their way to a technique in improv that is truly justified by labor. People do it. It is a bit ridiculous. There is a pianist here in NYC who does this- she's a fantastic pianist and composer without Slonimsky and the like. She has compositional ideas and stylistic ones. I will say she is a cold player, but she's battling it out in a tough shark tank, so I get that she feels she has to join the arms race. I'm just saying that the musicians/composers I admire most focus on the human communication, style, and content aspect of improv and the aspect of music that comments upon culture and also inspires cultural change and reflection. Nuance. in As a young musician, I was forced to practice this thinky-mathy way for various professors. With age comes the perspective and mercy I hope to show my students and colleagues when I don't spend our valuable practice and rehearsal time on retrograde-inversions and the set of combinatorial relationships in a given collection, i.e. things that don't sound like much (except to a music theorist who studied serialism and 12-tone) but are justified by practice room labor. I'm definitely not trying to claim that there aren't musicians that go down this path-- there certainly are! Famous ones, even! For me though, it's is a flag waving frantically that loudly signals a lack of artistic content, usually. But not always, as I said. This is only my opinion, however. I voiced it because there aren't enough musicians/teachers/composers who voice this pushback against the macho-mathy in jazz. There is this fallacy that anyone who can't play all mathy w the Slonimsky snippets weaving in and out of their improv should not be listened to when they say, "hey! why are we practicing this stuff??? I'm not a computer! I'm an artist-human!" I'm just saying some of us don't feel bullied (not saying you are doing that-you were just raising a legit point) and we feel perfectly fine calling out this approach for its mad dash to cover up for a severe lack of artistic content and a fear that one is vulnerable to being unseated from the throne.

I'd say jazz is really suffering from a lack of artistic content right now, and I believe it's a direct result of years of this kind of prove-your-mathy-badassery-technical-arms-race among jazz musicians. This is what happens when an "island is sinking"- people start pushing others off the sinking island so there is room for themselves. Jazz (and classical) is kinda sinking. Hopefully we can heal from this and get away from analogue versions of computer music (Xenakis) and back to humanity soon. It's funny, the defense against the sinking is causing it to sink faster by being less human, less relevant. Ironic.

But I sure didn't say that people don't get all up into the math of interrelationships in music theory. They do- I would hate for a non-musician to think they had to do that though to understand music. Absolutely not!In fact, I'm just saying that isn't what music is about for me, as a composer-musician. (I want to encourage people to love music, not to feel bullied by the 'difficulty" of its theory)

also: you said, "...He wasn't Thelonious Monk who just played everything in the hardest keys to prove how great he was on piano..."

What? I don't think that's what Monk was doing at all! Monk developed a sonic language all his own. I don't think any keys were hard for him... nor are they for any professional musician. Monk thought in sound texture and color. They are just different sounds. Sometimes you need a freshness or a timbre that a special key provides. Enough jazz music is written in B-flat if you know what I mean.

I realised a while back that music theory is like modular synthesis.

You can keep collecting modules - ideas about theory - forever. But only one person in a hundred, at most, knows how to make them work together to produce something audiences are going to want to hear more than once.

Coltrane may have been one of those people. But if he was, it was because he was Coltrane.

Most people who play with these ideas aren't. And acquiring more ideas and theories won't change that for them.

exactly. well said.

> "Thelonious Monk who just played everything in the hardest keys to prove how great he was on piano..."

Can't find where I read that. I believe it was a Miles Davis biography I read years ago. I know many Jazz greats thought the same of Monk. When you listen to "Monk meets John Coltrain" you can just see the two of them just making the hardest possible way to resolve what each other was doing. It is almost like they played musical dare for the last 9 months and finally gotten to a place where we got that album.

As a saxophonist I really enjoyed playing to his work, but my keyboard friends would get so worked up. Just look at 'round midnight. The key signature and the accidentals make it almost unreadable. Plus sure we can play in any key but certain one's are easier than others and it seem like it was always the key that couldn't make it harder if one tried. It was always known if you shown any weakness in playing keys that everyone else would keep modulating to try and mess you up.

One-upping is definitely a solo strategy that has a long history, and the tradition continues today. I just don't see Monk's key choice as some kind of proof of his skill. He wrote tunes that often sat awkwardly and offered a lot of space, like "We See" or "Coming on The Hudson" or even "Blue Monk" of all pieces. I feel those as offering freedom of direction and color- offering lots of rhythmic, motivic, and harmonic interpretational options. I guess that's why I love playing Monk tunes. And also his unique accent in the way he played- accent-inflection that was so of himself and so personal. It was confident and simple. I'm personally not a fan of the one-upsman approach to soloing, but it definitely exists.

Thank I never get to geek out on Jazz anymore. I moved to a different city and never got to crack into the scene that is here.

To me Monk = craziest and most intellectually stimulating harmonies of all time. No one has ever played his music and fooled you into thinking it was by Thelonious.

Damn son, talk about dropping the hammer on ‘em.

As a longtime, trained musician with a bit of theory, I’d love to get a breakdown. Would you be willing to explain what is nonsensical about the article?

I think what the grandparent is getting at is that music is a culturally-mediated perceptual phenomenon, and so the geometric interpretations here are an artifact of the model rather than of how our minds organize pitch into some musical Gestalt. I both agree with that (I love Coltrane, but as the article accidentally implies, this is about as meaningful as numerology) and find the artifacts to be pretty cool, so I’m not so down on it as an intellectual lark as that poster.

The author of the original post seemed to miss that Coltrane was drawing a "circle of fourths" (all the notes you see that are squared in the diagram are part of this) with a +1 / -1 half step radiating from them. I'm not entirely sure what John Coltrane was trying to do here, but the circle of fourths is well established theory. From my perspective, the transformations the author did were strange to me. They made neat letter patterns but I'm not sure they made musical ones.

From my personal perspective (and all music is to some degree personal opinion), I'm not a fan of trying to orient too much "numerology" to our 12 note equal tempered system (12TET). The 12 note equal tempered system really doesn't pattern all that well to what humans typically find as "consonant": whole integer ratio harmony. EG: One can say the most pleasing ratios follow as such - 2:1 (C:C - octave), 3:2 (C:G, fifths), 4:3 (C:F, fourths), 5:4 (C:E, maj. thirds). That has a certain "math as beauty" look to it when you look at the ratios, but the letter sequence is not as pretty. So if one is trying to find "numerology sense" in music harmony, I personally I think one should use frequencies or frequency ratios instead.

My personal impression about too many 20th century music theories, is that too often theories overfocused on 12TET note relationships, and led to some music that in general is difficult to get into (atonal, 12 tone, serial, etc.). These "have their uses" (some of the theories have found great success as scoring techniques for, say, horror films), and some people do seem to genuinely enjoy it. But the theories that I see that were more successful in the public eye (and in my mind as well) are "less is more" type theories (eg modal jazz or minimalism).

Thank you both for the thoughtful answers :). I think what you both said makes a lot of sense.

The one thing I'll throw in is that graphic notation (which I don't know a lot about) is an area where something like the OP can also have relevance - a way of visually expressing musical ideas (including, perhaps, some ideological content meant to inform attitude of the musician). It's also a blurry place between "functional" music notation and art.

I hadn't looked closely at the actual notation in the article - sounds like it may have lacked some rigor.

My main interest is just that it seems like an interesting system—and it's cool that it can be put into correspondence with sounds.

This is only going to be useful in a partial way, and more likely to be helpful if you play guitar, but Richard Lloyd (formerly of the band Television) explains the "circle of fourths and fifths" here:



I took a lesson from Richard Lloyd once. In that format, he goes much deeper into the Circle of Fifths. He finds as much spiritual meaning in it as Coltrane did in the symmetries found in the drawings of the original article.

You are lucky. He's an idiosyncratic and eccentric, but unique and brilliant guitar teacher. Also one of the most influential guitarists ever. He has issues too, but I think his unique approach overcomes his mental health history and he's stabilized a lot after getting the right medication. I've heard he now conducts lessons over Skype and I should check to see if he's taking new students.

A bit late to the game, but I've really enjoyed reading Musimatchics by Gareth Loy, it explains music theory from a mathematical and historical perspective: https://mitpress.mit.edu/books/musimathics

Perhaps Harmonic Experience by W.A. Mathieu.

This seems pretty much like exactly what I was looking for—thanks!

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

Speaking of Coltrane and musical palindromes, he played a tune that was partially a palindromic twelve-tone melody, called Miles' Mode: https://www.youtube.com/watch?v=J6BkV37OQK0&index=3&list=PLl... It actually may have been written by Eric Dolphy (called by him Red Planet): https://www.youtube.com/watch?v=hP6MXNkgr4A

You might enjoy this piece too:


(Steve Reich's Clapping Music...)

Skip "Clapping". It's fun but you wouldn't listen to it for anything other than intellectual interest. Other Reich though - Drumming is a wonderful midpoint between formalistic process-driven music but it still can give you goosebumps. He gets less process driven as time goes on - which I think is a good thing in many ways. For me probably the high point of his work is something like Music for 18 Musicians. Simultaneously a work of symmetry and structure - but it sounds like art. It sounds like an artist.

I saw Drumming performed in the round by Synergy Percussion in Sydney a few years back - it was _beautiful!_

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 [0].

[0] https://music.stackexchange.com/a/33787

Just intonation leads to giving up on key changes, though. And it gives up the flexibility of the piano and fretted instruments - if you want to play well in a different key, you have to re-tune the piano!

The dominance of equal temperament can probably be traced directly to the rise of the piano in importance.

I don't know about you but I don't own a piano. I do have several computers though, and they all have a soundcard. Dynamic tuning is a problem that can be solved by software!

I'm fine with equal temperament dominating, but it shouldn't be taken for granted as the only system for selecting an intonation. When I went to school for music the only time we talked about intonation was in music history. I'd like to change that.

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?

I haven't thought about it in depth, but my guess is that it is not, because a lattice is not isomorphic with a cylinder or torus.

Figuring out the relationship in detail would be a good project.

The Tonnetz is a torus, unwrapped onto the plane. On the first Wikipedia image (https://en.wikipedia.org/wiki/Tonnetz) you can see the circles of major- and minor-thirds on the 60˚ lines, whole-tone and semi-tone scales along 30˚ and vertical lines, and the circle of fifths on the horizontals.

Different presentations skew it in different ways, to emphasize different features, but it's very hard to make a diagram like this that isn't topologically equivalent to the Tonnetz.

If you find this sort of stuff interesting, you might also like to check out the writings of Fred Lerdahl and Dmitri Tymoczko, who come at these scale and chord topologies from a more academic perspective.

>it's very hard to make a diagram like this that isn't topologically equivalent to the Tonnetz.

And more generally, it's just hard to have a good idea that Euler hasn't had first.

It's equivalent to a torus in equal temperament, because the pattern repeats in a predictable way when the notes alias each other.

In a just intonation 3-5 lattice, it extends infinitely without repetition.

Indeed--you can still map it onto the torus but the lattice doesn't line up neatly; you get irrational spirals that cover the surface.

But, since this is in the context of Trane and Lateef, I think ET is a reasonable assumption.

That sounds promising since it does disprove my earlier line of argument. That said, there is more work to be done to show equivalence.

I'd look to see how the spiral array model can map these together.

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.

Names which spring to mind (though bear in mind popularity is quite a relative term when it comes to jazz). And that I’m not super-up on the current jazz scene:

Wadada Leo Smith, Vijay Iyer, Branford Marsalis, Dave Douglas, Kamasi Washington

They all have varying degrees of popularity and avant-gardness. Also, it’s not really fair to compare any of them (or any other contemporary players) to Coltrane - that was a special time for jazz. Plus music has moved on — new genres, instruments, it’s new times.

Candidates for the new John Coltrane, Id argue, would be hip hop producers:

J Dilla, Madlib, Flying Lotus (who is actually related to Alice Coltrane), DJ Premier, Pete Rock, RZA, Q-Tip, are some of the most influential producers who have strong connections to jazz. By which I mean knowledge of, musical influence from, and also using samples of jazz (though arguably that’s the least relevant part) in their work.

You could make an argument for rappers, too (Rakim, Nas, etc), as hip hop is the inheritor of the jazz tradition, but it is trickier comparing rapping - oral poetry at heart - to instrumental music.

One reason, imho, jazz had that golden age was that it was during a time in NYC when there was a critical concentration of talent and opportunities for jazz musicians to make a living playing gigs there.

Totally agree with you. Excellent answer. Vijay most of all with the theory/math. I sat through a lecture he gave on ragas in jazz once, and it was essentially a math class about ragas.

That must have been awesome! I was lucky enough to see Vijay & Wadada in Chicago last year. Indian music is crazy/complex/awesome - I learned a tiny bit about it when learning about Olivier Messiaen's "Quartet for the End of Time."

It was so boring. North and South Indian music is wonderful-I love it. But hearing Vijay break that down was mind-numbing. I like some of his music. Wadada can be great too! But I really wish we could stop justifying jazz by the labor it takes to produce/understand it. I need a break from jazz versions of Radiohead and Hendrix tunes as well. This trend is heralding the content crisis I'm afraid.

Yes, there are a plethora of extremely talented musicians, of every genre who take music theory extremely seriously in order to reach top competitive level.

If you look at any live performance of a pop star, most likely their band will include drummer, guitarists, keyboard player, sound engineers at the top of their game. They might even be playing jazz on the side :)

Check out this pretty much random video of two great young guitarist getting pretty deep in applying theory


Most jazz musicians today under the age of 45, went to music school. Brad Mehldau went to the New School. (someone mentioned Brad) The jazz faculty at the New School, when he went there at least, was ramming theory down every student (most music school does this as one of the main requirements of graduation). The piano/comp/theory teachers then (Gary Dial and Armen Donelian?) were all about the theory, especially for pianists, who kind of have to have the structure nailed down in their heads in order to accompany others at least.

But nearly every musician you will hear on a recording or at a concert these days went to music school. Music school has to employ all of its academic faculty, so as a result, they have to require several years of varying kinds of theory, counterpoint, jazz theory, etc. I could go into why the music schools feel that they have to overload every musician with all kinds of required theory classes, but you'd likely be bored. Let's just say it makes everyone feel comfortable with how legit pro they are.

So to answer the question, no, jazz is not unique. Bebop is not unique in this way. All musicians who go to a serious top-10 conservatory must waste a boatload of time taking all sorts of theory classes. Some programs are better than others. The conservatory I went to put musicians through a brutally boring and in-depth multi-year course of theory and analysis.

> "Are there musicians of today, with popularity similar to Coltrane's in 1960, that study and explore the theory of music at this level?"

Jacob Collier springs to mind, though he's less well known than John Coltrane.


I heard (back in the '90s) Trey Anastasio was heavy into the music theory.

Don't know if it's true since I never got into Phish so YMMV.

I’d argue that Mehldau today is not less popular. I wouldn’t be surprised if it turned out that he did explore the theory of music at that same level, but I don’t have any evidence :).

This looks like a perfect fit for a Max4Live patch. I've always been fascinated by the links between harmony and geometry.

I'd love to add this into the Ornament and Crime (http://ornament-and-cri.me/) for modular synth as well, which has a Tonnetz app already.

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!

I never heard from him.

The Coltrane originals were probably part of his estate, and I would guess have been donated to the university where he taught.

I believe that they're in possession of Ayesha. Stephon Alexander credits her in his 2016 book for one of these images.

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?

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