Coltrane Pitch Diagrams: Wrapping notes around a torus

https://medium.com/@lucas_gonze/coltrane-pitch-diagrams-e25b7d9f5093

by lucasgonze • 8 years ago

198 68 8 years ago

Is this the Tonnetz, or is it different?

lucasgonze6• 8 years ago

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.

stephencanon5• 8 years ago

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.

MrJagil• 8 years ago

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

bigiain2• 8 years ago

You might enjoy this piece too:

https://www.youtube.com/watch?v=lzkOFJMI5i8

(Steve Reich's Clapping Music...)

andybak• 8 years ago

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.

richardfontana• 8 years ago

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

pault• 8 years ago

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

tibbon• 8 years ago

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.

weinzierl• 8 years ago

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.

mtalantikite• 8 years ago

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!

lucasgonze6• 8 years ago

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.

stephencanon5• 8 years ago

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

tchitra• 8 years ago

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.

westoncb3• 8 years ago

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.

elihu2• 8 years ago

Perhaps Harmonic Experience by W.A. Mathieu.

westoncb3• 8 years ago

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

IAmGraydon• 8 years ago

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.

adamsea4• 8 years ago

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?

westoncb3• 8 years ago

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.

brooklyn_ashey8• 8 years ago

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.

snaily• 8 years ago

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

wyclif2• 8 years ago

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:

https://youtu.be/mNmEq-cP9Oo

https://youtu.be/9lSifKZ80H4

beat2• 8 years ago

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.

mdahlstrand• 8 years ago

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

anigbrowl2• 8 years ago

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

stephencanon5• 8 years ago

Doesn't the spiral array model discard enharmonic equivalence?

anigbrowl2• 8 years ago

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 :-/

lucasgonze6• 8 years ago

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.

chowells• 8 years ago

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.

forapurpose• 8 years 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? 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_• 8 years ago

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 :).

ZenoArrow• 8 years 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

UncleEntity• 8 years ago

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.

adamsea4• 8 years ago

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.

brooklyn_ashey8• 8 years ago

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.

Demiurge• 8 years ago

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

https://www.youtube.com/watch?v=zFRUCd4gQe4

brooklyn_ashey8• 8 years ago

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.

dbranes2• 8 years ago

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.

stephencanon5• 8 years ago

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

asimpletune• 8 years ago

I think he’s including the octave.

dbranes2• 8 years ago

You're right I miscounted.

lucasgonze6• 8 years ago

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?

zengid2• 8 years ago

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

beat2• 8 years ago

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.

zengid2• 8 years ago

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.

danbmil99• 8 years ago

Yes, but how does it sound?

Coltrane Pitch Diagrams: Wrapping notes around a torus

Related Stories