John Coltrane Draws a Picture Illustrating the Mathematics of Music

http://www.openculture.com/2017/04/the-tone-circle-john-coltrane-drew-to-illustrate-the-theory-behind-his-most-famous-compositions-1967.html

by qazwse_ • 9 years ago

208 64 9 years ago

openculture.com

rrherr• 9 years ago

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

pavement2• 9 years ago

The pentatonic scale. [0]

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

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

zhemao3• 9 years ago

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.

https://roelhollander.eu/en/blog-saxophone/Coltrane-Tone-Cir...

brooklyn_ashey6• 9 years ago

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.!!

anentropic2• 9 years ago

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

Nope

lucasgonze7• 9 years ago

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.

zhemao3• 9 years ago

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.

lucasgonze7• 9 years ago

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.

jesuslop2• 9 years ago

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.

lucasgonze7• 9 years ago

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.

theoh• 9 years ago

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

https://www.simonsfoundation.org/multimedia/mathematical-imp...

ams6110• 9 years ago

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?

agumonkey6• 9 years ago

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.

jedimastert• 9 years ago

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.

TheOtherHobbes• 9 years ago

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.

karmakaze• 9 years ago

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

brooklyn_ashey6• 9 years ago

He played violin.

brooklyn_ashey6• 9 years ago

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.

olivermarks• 9 years ago

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...

tyingq• 9 years ago

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.

sn9• 9 years ago

Einstein was a fairly talented amateur violinist as I recall.

DonHopkins• 9 years ago

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...

lucasgonze7• 9 years ago

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.

LostWanderer• 9 years ago

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

goodjam• 9 years ago

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!

zoul• 9 years ago

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

https://www.youtube.com/watch?v=rh6WTAHKYTc

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

https://www.youtube.com/watch?v=qypqwcrO3YE

bsaul• 9 years ago

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 !

randy909• 9 years ago

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.

meggar• 9 years ago

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

InitialLastName• 9 years ago

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

cbennett• 9 years ago

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.

andrepd• 9 years ago

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.

zengid2• 9 years ago

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

Note:

  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

fixed_carbon2• 9 years ago

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?

rdiddly• 9 years ago

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

John Coltrane Draws a Picture Illustrating the Mathematics of Music

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