
Coltrane Pitch Diagrams: Wrapping notes around a torus - lucasgonze
https://medium.com/@lucas_gonze/coltrane-pitch-diagrams-e25b7d9f5093
======
anigbrowl
This 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)

~~~
lucasgonze
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
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.

------
dbranes
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.

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

~~~
asimpletune
I think he’s including the octave.

------
weinzierl
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.

------
westoncb
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.

~~~
IAmGraydon
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.

~~~
brooklyn_ashey
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.

~~~
baldfat
> 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.

[https://en.wikipedia.org/wiki/Iannis_Xenakis](https://en.wikipedia.org/wiki/Iannis_Xenakis)

[https://www.youtube.com/watch?v=n2O8bMlEijg](https://www.youtube.com/watch?v=n2O8bMlEijg)

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."

[https://www.youtube.com/watch?v=ijSmRoPzMaY](https://www.youtube.com/watch?v=ijSmRoPzMaY)

~~~
brooklyn_ashey
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.

~~~
TheOtherHobbes
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.

~~~
brooklyn_ashey
exactly. well said.

------
MrJagil
Coincidentally, 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
You 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
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.

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

------
zengid
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](https://music.stackexchange.com/a/33787)

~~~
beat
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.

~~~
zengid
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.

------
tchitra
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.

------
Maybestring
Is this the Tonnetz, or is it different?

~~~
lucasgonze
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.

~~~
stephencanon
The 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.

~~~
elihu
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.

~~~
stephencanon
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.

------
forapurpose
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.

~~~
adamsea
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_ashey
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.

~~~
adamsea
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."

~~~
brooklyn_ashey
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.

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

~~~
tibbon
I'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.

------
mtalantikite
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!

~~~
lucasgonze
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.

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

------
lucasgonze
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?

------
danbmil99
Yes, but how does it sound?

