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Coltrane Pitch Diagrams: Wrapping notes around a torus
- Maybestring 9y agoIs this the Tonnetz, or is it different?
- lucasgonze 9y agoI 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.
- stephencanon 9y agoThe Tonnetz is a torus, unwrapped onto the plane. On the first Wikipedia image (https://en.wikipedia.org/wiki/Tonnetz 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.
- lucasgonze 9y agoThat sounds promising since it does disprove my earlier line of argument. That said, there is more work to be done to show equivalence.
- lucasgonze 9y agoI'd look to see how the spiral array model can map these together.
- elihu 9y agoIt'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.
- stephencanon 9y agoIndeed--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.
- Maybestring 9y ago>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.
- MrJagil 9y agoCoincidentally, i just stumbled upon this today: https://m.youtube.com/watch?v=xUHQ2ybTejU https://m.youtube.com/watch?v=xUHQ2ybTejU Basically a musical palindrome: https://en.m.wikipedia.org/wiki/Crab_canon https://en.m.wikipedia.org/wiki/Crab_canon
- bigiain 9y agoYou might enjoy this piece too: https://www.youtube.com/watch?v=lzkOFJMI5i8 https://www.youtube.com/watch?v=lzkOFJMI5i8 (Steve Reich's Clapping Music...)
- andybak 9y agoSkip "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.
- bigiain 9y agoI saw Drumming performed in the round by Synergy Percussion in Sydney a few years back - it was _beautiful!_
- richardfontana 9y agoSpeaking 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=PLllQi5j5ulVsf7efs2X1biz977kH_MZMO 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 https://www.youtube.com/watch?v=hP6MXNkgr4A
- pault 9y agoThis looks like a perfect fit for a Max4Live patch. I've always been fascinated by the links between harmony and geometry.
- tibbon 9y agoI'd love to add this into the Ornament and Crime (http://ornament-and-cri.me/ http://ornament-and-cri.me/) for modular synth as well, which has a Tonnetz app already.
- weinzierl 9y agoThis 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.
- mtalantikite 9y agoI 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!
- lucasgonze 9y agoI 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.
- stephencanon 9y agoI believe that they're in possession of Ayesha. Stephon Alexander credits her in his 2016 book for one of these images.
- tchitra 9y agoThe 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.
- westoncb 9y agoAnyone 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.
- elihu 9y agoPerhaps Harmonic Experience by W.A. Mathieu.
- westoncb 9y agoThis seems pretty much like exactly what I was looking for—thanks!
- IAmGraydon 9y agoThe 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.
- adamsea 9y agoDamn 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?
- mrbrowning 9y agoI 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.
- anigbrowl 9y agoThis has actually been operationalized: https://en.wikipedia.org/wiki/Spiral_array_model 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 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 https://en.wikipedia.org/wiki/Head-related_transfer_function
- stephencanon 9y agoDoesn't the spiral array model discard enharmonic equivalence?
- anigbrowl 9y agoI 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 :-/
- guscost 9y agoOh no! Start again! Been stuck doing covers/arrangements for a few years myself...
- lucasgonze 9y agoI 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.
- chowells 9y agoWas 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.
- forapurpose 9y agoAre 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.
- hartem_ 9y agoI’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 :).
- ZenoArrow 9y ago> "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. http://youtu.be/DnBr070vcNE http://youtu.be/DnBr070vcNE
- UncleEntity 9y agoI 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.
- adamsea 9y agoNames 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.
- dbranes 9y agoHere'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.
- stephencanon 9y agoI'm totally with you, but there are 12 notes per octave (at least in the standard western scale). =)
- asimpletune 9y agoI think he’s including the octave.
- dbranes 9y agoYou're right I miscounted.
- lucasgonze 9y agoOne 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?
- zengid 9y agoI'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 https://music.stackexchange.com/a/33787
- beat 9y agoJust 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.
- zengid 9y agoI 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.
- danbmil99 9y agoYes, but how does it sound?