Hacker News new | past | comments | ask | show | jobs | submit login
John Coltrane Draws a Picture Illustrating the Mathematics of Music (openculture.com)
208 points by qazwse_ on Apr 13, 2017 | hide | past | web | favorite | 64 comments

Back in the late 90s I came across this drawing in Lateef’s book and studied it closely. I found that it actually describes not a flat disk but a torus (aka donut) with a string winding around it. There are versions of the torus for all the symmetrical intervals (semitone, whole tone, minor third, major third). What changes from one interval to the next is the angle of the string.

I diagrammed each of the intervals and shared my work with Lateef. We had a warm conversation.

He mentioned to me that “Coltrane was always drawing things like this.” This particular drawing was something Coltrane did between set breaks at a gig they did together. Coltrane gave it to Lateef at that gig.

Oh man, I'm so jealous that you got the chance to talk with Yusef Lateef. That must have been awesome to pick his brain about music. He's one of my favorite musicians (alongside Coltrane himself of course). Eastern Sounds is still one of my favorite albums. He was one of those musicians who had a unique sound but was always trying new things.

I'm so happy I got to have that meeting. He's a hero of mine too.

He did a particular live solo that left a deep impression. The whole thing was integrated. There was a simple little motif that he built on, playing it over and over again, elaborating on bits, remembering the variations, repeating them, adding little elaborations to the elaborations. Eventually, it became like a labyrinth with the original motif in the center.

Holly cow, the torus is topologically the product of two circles S1*S1 so he must have been thinking of something cycling in two dimensions, not just one as the circle of fifhts or chord-scale theory.

I'm not a music theory guy, but Dmitri Tymoczko's work comes to mind here. That uses moebius bands rather than torii:


Oh my, a scribbled diagram of Coltrane as an application of Tymoczko. That would be a nice motivation to study him. Sweet video, I think in the end they show the moebius strip obtained by glueing opposite points, opposite meaning the two points pinned by a needle perpendicular to the donut.

The actual shape in this diagram contains five octaves. That's 60 semitones. The 60 semitones correspond to minutes on a clock, of course. The correspondence between music theory and time is not accidental. Both came out of Babylonian numerology, which had a thing with the integers that pop out of trigonometry - 12, 60, 360.

Thanks for pointing out the "two circles" way of factoring a torus. I hadn't thought of that.

could you elaborate on the toroidal structure for people like me who don't know music theory please ? what are the two fundamental cycles that make ZxZ ?

I see semitones going around the main circle - but how are they connected back to themselves the other way ?

Here is a cross-section drawing I did of the toroidal structure. This is for minor thirds instead of whole tones, meaning that there are three semitones per winding instead of two.


An artist friend of mine, Howard Penner, made a cleaned-up version of this diagram in Illustrator, for an unpublished paper that another musician friend wrote about it. [1]

I can't find these papers by Hafez Modirzadeh [2] online, but there are some other papers with references to them:

"Spiraling Chinese Cyclic Theory and Modal Jazz Practice Across Millenia", Journal for Music In China. (II:2). 2000

Book Review for In The Course of Performance: Studies in the World of Musical Improvisation. Journal of the Society for Ethnomusicology. (XLIV:1). 2000

"Spiraling Cyclic Theory and Modal Jazz Practice: The 60-Tone Case of John Coltrane and Ching Fang", SEM Conference Paper (Nov. 20), Austin. 1999

[1] http://imgur.com/gallery/oJKln

[2] http://radiodecibel.nl/index.php?page=artist&id=Hafez%20Modi...

This drawing shows a closed cycle covering five octaves. But musical pitches are not a closed cycle, they keep going down and up from there. How can you take this drawing and extend it to an arbitrarily high or low pitch? Make it a spiral.

What you're seeing is a spiral at an angle orthogonal to the angle of view. Imagine a spiral curling around the Z-axis, and a viewer looking straight down Z. The arms of the spiral aren't visible because they are occluded.

Much more interesting (IMHO) is the Giant Steps animation by Michal Levi:


It perfectly shows the structure of the music. The later music video One (set to music of Jason Lindner) is also beautiful:


Thats insanely beautiful. Thank you for sharing!!

I have a mild case of synesthesia and this is one of the closest things I have seen/heard (:) ) that comes close-ish to capturing how I see/feel when I experience great music.

Of course, I was not surprised to see the creator of the video list a similar motivation in his caption for the upload.

Don't understand the downvote. The first video may seem arbitrary to people not familiar with jazz theory but it's a really good representation of the structure of one of clotrane's most emblematic song. Thanks for sharing !

The downvote wasn't me but I'll agree that this doesn't seem much more related to the music than any visual synced to the tempo. Coltrane's drawing captures a system that could be used by other musicians to create new musical ideas or theories. They're very different things so the video seems off-topic to me.

I still haven't figured out how to play over Giant Steps changes, although shredding through augmented scales has about a 50% success rate.

The fun thing about Giant Steps is that if you hold any note for long enough, eventually it will resolve correctly.

So... What's the deal with the diagram then? The article explains nothing. Some vague ramblings about the geometry of Einstein's quantum theory (?) and no discussion of Coltrane's drawing at all.

I'm trying to parse out the logic, but it looks like the outer ring and inner rings are moving clockwise in whole steps, with the outer ring starting on C and the inner starting on C#. If we took the rings and straightened them out we'd have (its hard to tell some of the notes so correct me if I'm wrong):

  [1]                     [2]                     [3]                     [4]                     [5]                     [1]
   C   D   E   F#  Ab  Bn? C   D   E   F#  An? Bb  C   D   E   Gb  Ab  Bb  C   D   E   F#  G#  Bb  C   D   E   F#  Ab  Bb  C
     C#  Eb  F   G   A   B   C#  Eb  F   G   A   B   Dn  Eb  F   G   A   B   C#  Dn  F   G   A   B   C#  En? F   G   A   B  

  n = (natural sign)
  ? = its hard to read the scribbles.
It looks like he augments some of the intervals in the middle of the scale and diagramming how to elide from one to the other. Note that Bn could be considered distinct from B. Coltrane was familiar with the overtone series that generate what are called harmonics. Harmonics could be thought of as the sine waves that sum together to create the timbre of a pitch. If you've played guitar and you pluck the string while gently pressing the string down at the 1/2, 1/3, 1/4, 1/5 (etc) division, you'll force the string to vibrate only at the frequency that corresponds to that harmonic (ie 1/2 would force it to vibrate only at 2f where f = the lowest vibration mode of the string, sorry If I'm not conveying the physics perfectly). Anyway, the overtone-series closely approximates a major scale, but it gets tricky when you want to shift from one series to the next because the series is generated from integer ratios, and prime numbers generate similar but distinct pitches. Long story short, you can have several genus of the same note that are all mathematically distinct.

..I've got a better description here https://music.stackexchange.com/a/33787

I'm pretty sure both rings are normal whole-tone scales and all the places you have marked as Xn are simply Xb.

Curious though if you have a take on the boxed numbers over the C's. Perhaps calling out how the original tonic position changes in relation to everything else as you move around the circle?

The boxed numbers are octaves. This graph stretches the 12 semitone octave-image that we're used to seeing out to 60 semitones.

"out to 60 semitones" -- does this mean you're interpreting the diagram as a pitch-map covering five octaves? I was interpreting it as a tone-map comprising 12 semitones grouped into five distinct tonal regions with the center web indicating specific movements from one tonal region to another.

EDIT: Just saw your other comment interpreting it as a spiral, which effectively answers my question.

Yes, that's the way I interpret the diagram.

I think the importance of there being five tonal regions is that this enables the multiples of 12 to line up at 60. These numbers are significant because they connect back to the Babylonian sexagesimal ("base 60") system used in both music theory and trigonometry.


Yeah, a little disappointed there's no explanation. They couldn't find a single musician to interview for the article?

The article opens with the claim that "Albert Einstein and John Coltrane had quite a lot in common" but then hardly addresses that further.

Did Einstein have any musical talent/interests?

This Psychology Today article goes pretty deep on that front: https://www.psychologytoday.com/blog/imagine/201003/einstein...

Talks about musical talents, but also some other things where you might see similarities if you read it, then stories about Coltrane.

I think the author is alluding to Einstein and Coltrane having quite a lot in common conceptually around envisaging the universe, sacred geometry and spirituality http://upliftconnect.com/spiritual-wisdom-of-albert-einstein...

There's a quote about music and Einstein, don't know more.

Also feynmann liked percussions a lot and maybe music in general.

What I find in learning music alone, is that it's highly relative, there's no real absolute, so the space of combination is huge, yet very regular in principles; what's not to like for a scientist.

I'm both a musician and developer, and people usually look at me weird when I say that. But really, there's a great overlap if you just look.

I'm surprised people think that's unusual. https://teamgaslight.com/blog/why-are-so-many-software-devel...

This fills me with joy; I was itching to write a similar blog post.

Quoting article above:

  Music is an abstract medium — the printed note requires 
  interpretation and execution. Like the written line of 
  code, there is often much more than meets the eye.

That's why I dislike classical education, which put notation and theory first before intuition.

You should definitely try Overtone (http://overtone.github.io) and Sonic Pi (http://sonic-pi.net), if you haven't already.

Sonic Pi looks great! Thanks for the suggestion! Overtone looks interesting, but Sonic Pi being ruby seems more approachable/easier to install and run.

I see you are using emacs :) Setting up Overtone isn't that difficult and Overtone + Emacs Live(https://overtone.github.io/emacs-live/) makes for a great live coding setup :) Both of them are by Sam Aaron, I find his work inspiring. Anyways, have fun with live coding!

I have always said that when I'm deep into composing, the brain activity seems extremely similar to when I'm working out an algebra problem, or a big code refactor.

fitting pieces together and not breaking the invariant and relationship between them ? #harmony

Programming can be very "artistic" too, but I bet not in your day to day job. Most people want war-like efficacy.

Same for music. There are a handful of famous composers who get paid to just do their thing, but for everyone else the gigs are "give me a 70 second song that sounds like X and has a big dramatic peak at the 55 second mark, due Friday".

Very good point. I tend to only think about music as timeless pieces of harmonic bliss :)

What I love about music is that whenever you think you can reduce it to a formula or algorithm, it slips through your fingers.

I've been studying theory for a while now, and it's become obvious we know a lot less about music than we think we do.

I've read maybe, say, 2 chapters of theory. A bit of names and patterns.

In a way I agree with you, the more you dig the less you know. But at the same time, I have this feeling that .. I do know music. I know because when I'm on it, the few times it happened, it wasn't a theory, but more like a sense of spaces. A sense that fills you in a way that is impossible not to notice. Things are simple, borderline automatic, a very thin closed loop of sensation and action. A bit like an ODE ;) If you're tuned, you can now "see" that space [1] path, lines and curves and glide on them through the movement inertia.

It's not a static space, it's more like a slightly viscous fluid that also responds to your changes of momentum. If you decide not to glide smoothly anymore, you're welcome to do so, but the ground, the space will now change shape, so you have to know how to stay tuned and react to that. Then you can keep on gliding and it will feel musical.

My most used word when speaking about music is sensitivity.

[1] seeing in the abstract geometrical sense, a notion of direction, distance, scale, that we use to move, but can be used to organize any kind of sensory inputs I believe.

Yes people today treat music as if the sound were an end product (consumerism), but music is a medium through which major areas of experience are communicated.

E.g. scales could be thought of buildings. One can hardly make one without inflecting it in some way that reflects values. To the point where evading this inevitability invokes meta-communication.

He played violin.

Perhaps the writer meant that while they both studied structures, they both had a personal style that was powerful and served to advance the thinking in their fields. Both were more concerned with leaving those structure to adventure toward philosophical comment through their work.

I have read that Einstein lied to improvise on violin iirc when coming up with new ideas

Einstein was a fairly talented amateur violinist as I recall.

Fascinating topic. I wish I knew how to interpret Coltrane's diagram. Any ideas?

Think:color wheel. Every improvising musician (and some classical musicians as well)holds a similar diagram in their heads when practicing common chord progressions like ii-V-Is and rhythm changes, and the blues, etc... and on and on. Each note/chord in the western scale pulls against others next and opposite to it in the same way, regardless of the key one inhabits in any given moment. These relationships are dynamic; variables, in a way, pulling against one another in an interconnected system. The tonic(I, or "c" in c major, for example) has a "home" quality, the dominant (V) has a cadential quality in relation to the tonic, etc., and each chord has its role. Within this system, one can create the illusion that one is "in a key" that one is not actually fully in for the overall work-- this is like nesting methods in code. Loops are loops and the idea remains the same no matter how big or small, no matter the coding language. Chordal relations in music are the same. This diagram may be what Coltrane holds in his head as he plays patterns and repeats motives in a given solo to imply chordal relationships and "transportation" from one realm to another. These realms have anticipatory meaning we all understand unconsciously, even if we are not professional jazz musicians. (for example, we all feel the need for "do" if it is left off the end of "do re mi fa sol la ti...." in "Do, a deer".) There is nothing magical about Coltrane's diagram that I can see that is beyond what most improvisers hold in their heads as they practice patterns over chord changes. And yet, there is something wonderful about how these chords and notes create inherent meaning and character roles relative to one another-- like the color wheel. It is still cool to see the way Coltrane represented these for himself in his own mind, (in other words- how simple his ingredients- they are the same as ours, and the result!)implying how he practiced improvising, and the path those patterns may have taken.

When improvising over changes, it is easy to get lost in a compositional idea and lose one's place in the form. Maintaining a simple "relational data structure" like this circle is way way to maintain one's balance a place in the overall form and structure. Jazz musicians work to internalize this in practice, so that in performance, they can be free and expressive while still within the form. And then...there is Dolphy.!!

"Every improvising musician ... holds a similar diagram in their heads when practicing common chord progressions"


Is your problem with diagram, similar, or improvising? I can understand a problem with improvising, in that I should have been more specific. Specifically, every improvising musician working in the area of any type of jazz that involves playing over chord changes. The diagram discussed here is really very fundamental, open and "big picture". Certainly, musicians have their own "circle of" diagrams in their heads if they ever plan on doing ii-v-1s etc. If they are playing anything resembling chord changes, it would be hard to avoid thinking in patterns of chord progressions, unless the person was aiming to subvert that system (certainly an acceptable strategy) and create their own tonal/chordal language while other band members do changes. Many avant-garde players do that, and then many don't bother with chord changes at all. But when one is expected to produce a bass line or a harmonic line, one does need some kind of a diagram that (because of the nature of the harmonic system itself) will be very very similar to this one. I'm a professional jazz musician. I play traditional and experimental jazz, and I seriously have yet to meet a single colleague who doesn't have their own circular or loop-like diagram that looks a heck of a lot like this one. But I'm open to hearing about a new way of thinking-- perhaps you could tell me about that, and I could share ti with my colleagues. We all do tend to get trapped in our own ways.

The pentatonic scale. [0]

Once you tune your ear into it, it's absolutely everywhere.

[0] https://en.wikipedia.org/wiki/Pentatonic_scale

Not quite. It's actually a hexatonic scale, AKA the whole tone scale. Each note is a whole step up from the last one. The outer circle is a C whole tone scale and the inner circle is a B whole tone scale (i.e. they're offset by a half step). The circles and lines are used to highlight other harmonic relations, like the circle of fifths. The following blog post explains it in more detail.


Nice blog post citation. Jazz musicians absolutely love to dig into these structures from https://www.worldcat.org/title/thesaurus-of-scales-and-melod...

McCoy Tyner has said that Coltrane carried Slonimsky's book with him almost everywhere he went for several years.

I have a copy but I'm way too stupid/musically uneducated to grok it. Back when I started with music at the age of 12 I thought it was "cool" to be autodidact and not know theory and I have regretted it ever since, because try as I might learning it now, it doesn't really stick. You need those 8-10 hours of practice a day I suppose, and they're hard to come by in your adult life.

You only need to be great at the thing YOU do to be a great musician. Ornette and Chet Baker did a lot with "not knowing theory". Ornette has his own theories, as did Monk. Chet Baker was notoriously bad at playing changes from a sheet- but he was great at hearing them in tunes and hearing what he wanted to play. Ella had no idea about music theory. I'd say you are in great company. The best musicians develop their own things, and the good news is: it is never too late for you. You go ahead and make something you love- you will be the expert at that. :)

Is it me, or did he screw up the V highlight on the left hand side of the wheel? He has a G# on the outer ring partnered with its neighboring G and A on the inner ring - but to fit with the pattern it should be the A on the inner ring partnered with the G#/Bb on the outer ring.

I noticed that straight away---and I hardly know anything about music! It just breaks the symmetry pattern, so it couldn't be right.

yes it must be a mistake

it seems in other drawings which appear to also be by Coltrane they are drawn as you suggest, A + Ab/Bb preserving the symmetry


Huh, yeah you're right, there's a mistake in Coltrane's original version.

Is there a chance it was on purpose. To give the music a particular sound?

Well fancy that! I stand corrected.

Thanks for pointing that out!

Is it possible to paint by playing music?

Not exactly thinking of a visualiser but it would be interesting to see how one could actually do that

Let's imagine one note as a point on a plane, then an interval (2 notes) as a line connecting those notes. Next we have a chord (3 notes) and we get a triangle. By inverting the notes of the chord (135, 351 etc) we get a ring that representd different triangle shapes. Let's go into third dimension by adding one more note, thus a 4 chord can represent a pyramid.. you could also stay in 2D and represent a square etc.. by modulating and changing the tonal center you can move around the canvas or in space.. representing different data structures.. the horizontal movement would animate the structure as well!

Guidelines | FAQ | Support | API | Security | Lists | Bookmarklet | Legal | Apply to YC | Contact