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:
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
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.
That's not exactly arbitrary - there is a reason to use that range in preference to others.
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.
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.
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.
Previous discussion here: https://news.ycombinator.com/item?id=12528144
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."
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.
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.
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.
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.
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.
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.
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?
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).
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.
Basically a musical palindrome:
(Steve Reich's Clapping Music...)
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 dominance of equal temperament can probably be traced directly to the rise of the piano in importance.
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.
Figuring out the relationship in detail would be a good project.
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.
And more generally, it's just hard to have a good idea that Euler hasn't had first.
In a just intonation 3-5 lattice, it extends infinitely without repetition.
But, since this is in the context of Trane and Lateef, I think ET is a reasonable assumption.
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.
Wadada Leo Smith,
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.
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
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.
Jacob Collier springs to mind, though he's less well known than John Coltrane.
Don't know if it's true since I never got into Phish so YMMV.
The Coltrane originals were probably part of his estate, and I would guess have been donated to the university where he taught.
Here's a genius whose profession is music, not math. What was his intellectual path during his private sessions with the protractor?