JOHN COLTRANE'S TONE CIRCLE
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.
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.
An interesting variant to the 'Circle of Fifths/Fourths' is the 'Coltrane Circle', created by saxophonist John Coltrane (perhaps based on Nicolas Slominksy's Thesaurus of scales and musical patterns?) and was used by Yusef Lateef for his work "Repository of Scales and Melodic Patterns".
In the drawing (on the left) there are a couple of sharps notated, they have been replaced by Corey Mwamba with their enharmonic equivalents (C♯ = D♭ and F♯ = G♭) in his drawings.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 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 large tone spaces (pitch classes) - form 6x the same Hexatonic scale within the same ring, just each shifting a tone.
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?
Perhaps the circled tones it outlines 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
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:
A Pentagram & Pentagon appear when you draw lines between the same tones in the Coltrane Circle" (in the example with the tone C).
Note: the lines, numbers and Pentagram in the Coltrane Circle (on the left) were not drawn by John Coltrane. Who did is not clear.
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.