JOHN COLTRANE'S TONE CIRCLE
|Er is een Nederlandse versie beschikbaar op saxwereld.nl|
I do like to mention that I am no "authority" or "expert" when it comes to Coltrane's work, or the music theory behind it and the compositions themselves. And as sax player, well, I'm still miles away from even standing in the giant shadow he cast ... not to mention his giant footsteps. Anyway, as admirer of Coltrane's work I could not resist to write this article. I wrote this article because I am fascinated by his music and have an interest in the relationship between music and math / geometry.
For an expert opinion on Coltrane you should listen to what musicians who played with him or extensively studied his work have/had to say about it.
This blog article is an addition to the article "Music and Geometry" and contains only the information about the Coltrane Tone Circle and the relationship between some of his music and geometry. Do read the mentioned article for general information about the relationship between music and geometry.
Thelonious Monk once said "All musicians are subconsciously mathematicians". Musicians like John Coltrane though have been very much aware of the mathematics of music and consciously applied it to their works. The "Coltrane Circle" is (to me) proof of it in Coltrane's case ...
SHORT INTRODUCTION ABOUT 12-TONE CIRCLES
A Tone Circle is is a geometrical representation of relationships among the 12 pitch classes (or pitch intervals) of the chromatic scale in pitch class space (circle). The most common tone circles in Western music are the "Chromatic Circle" and the "Circle of Fifths/Fourths" (image on the right).
In Western music theory there are 13 intervals from Tonic (unison) to Octave. These intervals are the: Unison, Minor Second, Major Second, Minor Third, Major Third, Fourth, Tritone, Fifth, Minor Sixth, Major Sixth, Minor Seventh, Major Seventh and Octave. When we look at these intervals (or pitch classes) and how they relate to one another in the musical tone circles, some nice geometric shapes appear.
Note: If you are interested in a more esoteric-philosophical perspective on the intervals, then read the article: "The Function of the Intervals" on Roel's World.
COLTRANE'S TONE CIRCLE
An interesting variant to the 'Circle of Fifths/Fourths' is the 'Coltrane Circle', created by saxophonist John Coltrane (perhaps influenced and based on the Joseph Schillinger System of Musical Composition and/or Nicolas Slominksy's Thesaurus of scales and musical patterns?) and was used by Yusef Lateef for his work "Repository of Scales and Melodic Patterns".
What might be good to make note of, is that it isn't clear when exactly Coltrane drew the circle and why. Did he use it to work out a particular composition? Did he try to find a new approach for his solos in that period? Perhaps it was during his explorations of Indian music in the 60s? I haven't been able to find any clear sources that can provide a clear answer to those questions.
In the drawing (on the left) there are a couple of sharps notated, they have been replaced by Corey Mwamba with their enharmonic equivalents (C♯ = D♭ and F♯ = G♭) in his drawings.
The circles above might seem a bit odd, but if we "simplify" the circle things become a lot clearer.
What we see is a circle with two concentric rings.
The outer ring displays the "Hexatonic" (6-Tone) or "Whole Tone" Scale of C (C – D – E – G♭ – A♭ – B♭ – C).
The inner ring displays the Hexatonic scale of B
(B – D♭ – E♭ – F – G - A - B).
When you "zig-zag" clockwise between the tones of these Hexatonic scales of the concentric rings it turns out to be the "Circle of Fourths" (and thus counterclockwise the "Circle of Fifths").:
C - F - B♭ - E♭ - A♭ - D♭ - G♭ - B - E - A - D - G - C
WHAT ABOUT ALL THOSE TONES IN BETWEEN?
The smaller spaces (light grey) between the larger ("main") tone spaces (darker grey) of the Hexatonic scale of C (outer ring): C–D–E–G♭–A♭–B♭–C) and B (inner ring): B–D♭–E♭–F–G-A-B contain 4 tones that - when combined with the "main" tone spaces (pitch classes) - form 6x the same Hexatonic scale within the same ring, just each shifting a tone.
What is also interesting, is that the Hexatonic scales formed with the tones in the smaller spaces progress in opposite direction as the Hexatonic scales from the larger tone spaces of the rings do. The Hexatonic Scale from C going clockwise is C-D-E-G♭-A♭-B♭-C. If you start from C and go counterclockwise you get the same scale in "reverse": C-B♭-A♭-G♭-E-D-C.
All Hexatonic scales within the same ring use exactly the same 6 tones but any of these tones could be used as the tonic of a hexatonic scale. See the table below:
|THE 6 HEXATONIC (6-TONE) SCALES OF THE OUTER RING|
|THE 6 HEXATONIC (6-TONE) SCALES OF THE INNER RING|
WHY HAVE TONES BEEN CIRCLED?
It's not completely clear why Coltrane circled those tones, he never made note of it. The tones that have been circled are the Minor 2nd or "Supertonic" and the Major 7th or "Leading Tone".
Perhaps Coltrane wanted to visualize how chromatic neighbor tones move in Fourths/Fifths to adjacent neighbor tones? All tones are centers that move to other centers.
Perhaps the circled tones outline the relationship between Diminished 7th Chords within the Diminished Scale? An example:
The C Diminished 7th Chord is C - E♭ - G♭ - A. To turn this into a Diminished scale, you need to add another Diminished 7th Chord a semitone higher:
D♭ - E - G - B♭ or lower: B - D - F - A♭. Results:
C - D♭ - E♭ - E - G♭ - G - A - B♭ - C
C - D - E♭ - F - G♭ - A♭ - A - B - C
It is commonly known that Coltrane did like using the Diminished Scale (or "Double Diminished" as it was called because it is build from two Diminished 7th Chords). An example of that is his solo in “Moment’s Notice” (in measure 74 where he plays a Bb7 diminished scale pattern). Another example is his solo in "Epistrophy" during the live perfomance at Carnegie Hall with Thelonious Monk.
ALTERED DOMINANT (ALT DOM) CHORDS
Jazz guitarist, composer and music theorist Mark Rossi shared another way of looking at the circled tones.
An Alt Dom chord is a dominant chord (centered around the 5th of the key) but with a minor 7th on top (hereby creating a Dominant 7th) and the 5th and 9th of the chord either lowered or raised by one half step. This in turn gives us either a b5 or a #5 instead of a natural 5 as well as a b9 and #9.
When you add the 3 Diminished 7th Chords to a table you get the following result:
|5||←||b6 or 7||→||8||3||←||4 or 2||→||b3|
NATABHAIRAVI-CHARUKESI (NATURAL MINOR + MELODIC MAJOR) "COMPOUND" SCALE
Corey Mwamba shared an alternative interpretation about the meaning of the circled tones, he thinks they might form what he calls a "compound scale". This compound scale is formed my combining the "Natural Minor" (Natabhairavi) and the "Melodic Major" (Charukesi) scales, characteristic for North Indian music (something Coltrane developed an interest for in the 60s (see "John Coltrane and the integration of Indian concepts in Jazz improvisation" by Carl Clements).
Corey writes: "We can see that the two scales have two enharmonic points; one at the third degree of each scale, and one at the sixth. If we transliterate Natabhairavi to d♭ and combine it with Charukesi mapped from c, we can see an intersection that contains e and a♭. Natabhairavi is the top line, circled in blue; Charukesi is circled in red."
He continues: "Arranged in chromatic order, the first, fourth and seventh degrees of Natabhairavi are aligned with the degrees from Charukesi in a way that matches the segment 3–4 on the original diagram." With the "original diagram" Corey referes to the Coltrane Circle with the Pentagram drawn into it. In that version the Circle the 5 segments are numbered.
For additional information and images, read Corey's article "Way of Seeing Coltrane (IV)".
It is not completely clear if the lines and numbers drawn in the Coltrane Circle version below on the left were drawn by Coltrane himself. If he drew the lines or not will be impossible to check, but we could compare the numbers drawn in the Coltrane Circle with those from copies of his scores. For this comparison I have used the score of Love Supreme and several scores displayed at recordmecca.com
A Pentagram & Pentagon appear when you draw lines between the same tones in the Coltrane Circle" (in the example with the tone C).
Below you see the numbers found in various scores side by side with the numbers of the Coltrane Circle image:
Now, I'm no expert in graphoanalysis, so I will just describe what I noticed:
In all scores as well in the tone circle we see a certain inconsistency in the wring of the numbers. The "1" is sometimes written as a single line, sometimes with additional horizontal lines. The 7 is sometimes written with a horizontal line in the center, sometimes without. The "4" is open sometimes and closed at the top at other times. The "2" has a little "loop" in some cases but others not. It seems though that the writing in the scores was done quicker, more like scribbling then seems to be the case with the Coltrane Circle. This is not a surprise though, specially with last minutes arrangements scores often look like scribbles.
To my eyes though all numbers look somewhat similar and perhaps could have been written by the same person, I'm not convinced about it though. Perhaps an expert in graphoanalysis might be able to shed some light on it? Feel free to contact me if graphoanalysis is in your field of expertise and have something to share about what you see above.
Below links to the used scores to compare with the Coltrane Tone Circle:
WHAT DO THOSE NUMBERS MEAN?
"What do those numbers mean?" is a question I have received via mail several times.
Well, to me it seems that the 5 numbers outside the circle 1-5 are the easiest to explain. They mark the 5 octaves this tone circle covers.
Not per say related or intended, but 5 octaves = 5 x 12 tones = 60 tones. There are 60 seconds in a minute and 60 minutes in an hour. Perhaps that's why some would refer to this circle as a "clock". There is nothing in this drawing though that suggests this to be one of the reasons for the design of this circle.
Inside the circle you notice a sequence of numbers 7-6-5-4-3-2-1-2-3-4-5-6-7 and reversed 1-2-3-4-5-6-7-6-5-4-3-2-1, apparently showing you the chromatic (semitone) relationship between the tones listed in both inner and outer ring when combined in one. The 1's (C) and 7's (F#) are a Tritone (six "spaces" between the lines) apart from each other. This might suggest a so called "Tritone Substitution".
A Tritone substitution is one of the most common chord substitutions used in Jazz and is the foundation for more complex substitution patterns like Coltrane changes. Other examples of the tritone substitution (known in the classical world as an augmented sixth chord) can be found in classical music since the Renaissance period. The Tritone substitution can be performed by exchanging a dominant seven chord for another dominant seven chord which is a Tritone away from it.
In the Coltrane Circle you see a sequence from 1-7 starting from C (top of the Circle) to F# both clockwise and counterclockwise. Could that suggest a substitution of C7 by F#7?
If you have another (perhaps better) idea about this sequence, please do contact me.
THE COLTRANE CIRCLE & POLYGONS / POLYGRAMS
The 12 tones of the Circle of Fifths/Fourths are normally placed in one ring instead of two but the result is same:
|6 LINES||4 TRIGONS||3 SQUARES||2 HEXAGONS||1 DODECAGON||1 DODECAGRAM|
For more details about the polygons/polygrams and tone circles do read Roel's World article "Music & Geometry".
There are two geometric shapes though that can be created with the Coltrane Tone Circle, but that can not be drawn into the standard Circle of Fifths/Fourths or Chromatic Circle:
To finish this article with I like to share a "music video" of Coltrane's piece "11383" with the Coltrane Tone Circle used as base/inspiration for the visualization. Note: the visualization of the Coltrane Circle does not accurately follows the music - as becomes obvious later on in the video - but is nonetheless a nice 'work of art'.
JOHN COLTRANE'S MUSIC & GEOMETRY
If you find this article interesting, you might like to read the Roel's World article "John Coltrane's Music & Geometry" as well. In this article I write a bit more about the relationship between Coltrane's music and it's mathematical / geometrical interpretation.