
John Coltrane Draws a Picture Illustrating the Mathematics of Music - qazwse_
http://www.openculture.com/2017/04/the-tone-circle-john-coltrane-drew-to-illustrate-the-theory-behind-his-most-famous-compositions-1967.html
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lucasgonze
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.

~~~
jesuslop
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.

~~~
theoh
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...](https://www.simonsfoundation.org/multimedia/mathematical-impressions-
making-music-with-a-mobius-strip/)

~~~
jesuslop
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.

------
DonHopkins
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diagram in Illustrator, for an unpublished paper that another musician friend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](http://imgur.com/gallery/oJKln)

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

~~~
lucasgonze
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.

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

[https://www.youtube.com/watch?v=rh6WTAHKYTc](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](https://www.youtube.com/watch?v=qypqwcrO3YE)

~~~
bsaul
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
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.

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andrepd
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.

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

~~~
fixed_carbon
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?

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

~~~
fixed_carbon
"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.

~~~
lucasgonze
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.

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

------
ams6110
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?

~~~
agumonkey
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
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.

~~~
tchaffee
I'm surprised people think that's unusual. [https://teamgaslight.com/blog/why-
are-so-many-software-devel...](https://teamgaslight.com/blog/why-are-so-many-
software-developers-also-musicians)

~~~
zengid
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.

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

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

~~~
brooklyn_ashey
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.!!

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

Nope

~~~
brooklyn_ashey
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.

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LostWanderer
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
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!

