Last Updated on March 1, 2020

Approx. reading time: 16 minutes

JOHN COLTRANE’S TONE CIRCLE

Er is een Nederlandse versie beschikbaar op saxwereld.nl

FOREWORD

January 22, 2016

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

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” (1981).

Stephon Alexander wrote in his book “The Jazz of Physics: The Secret Link Between Music and the Structure of the Universe ” that he tried calling Jusef Lateef and asked when he was told that Jusef Lateef was not availabe: “Could I leave him a message about the diagram that John Coltrane gave him as a birthday gift in ’61?“

The year 1961 would “date” the tone circle one year after the release of the groundbreaking album “Giant Steps”, in the same year as the release of the albums: “Coltrane Jazz”, “My Favorite Things”, “Olé Coltrane” and “Africa/Brass” and several years earlier then unique albums like for example “A Love Supreme” (1965) and “Ascension” (1966).

According to Lucas Gonze Jusef Lateef mentioned: “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.” This is an intriguing thought, if Coltrane was always drawing “things like that”, could that mean that there might be more versions of this tone circle (or other) somewhere in a box or folder in a museum, at the residence of one of his relatives or in the archive of musicians John Coltrane worked with as well?

Did John Coltrane drew it to work out a particular composition? Did he try to find a new approach for his solos in that period? I haven’t been able to find any clear sources that can provide a clear answer to those questions.

A Pentagram & Pentagon appears between the same tones in the Coltrane Circle” (in the original and reproduction with the tone C) when connected by a line.



Click on the Coltrane drawing to enlarge it.

CIRCLE VERSIONS

There are two versions of the circle shared online. A “detailed” version and a “clean” version with only the circled tones. What is good to point out is that these are two are separately drawn circles, as you can see in the “overlay” in the middle below. I have aligned the letter “C” of both drawings. The blue-overlay is the “clean” version (first one displayed below):

When you look closer, you can see two more differences:

(img.1) On the left a cut from the “clean” version, on the right a cut from the “detailed version”. (img.1): In both versions the [A] (tone center) has been “squared”. In the “clean” version A♭ is notated, in the “detailed” version G♯ is notated. The “detailed” version also shows a mistake. Instead of circling both tones siding the [A] chromatically, the tones siding the G♯ chromatically have been circled. (img.2) On the left a cut from the “clean” version, on the right a cut from the “detailed version”. (img.2): In both versions the [E] has been “squared”. In the “clean” version E♭ is notated, in the “detailed” version D♯ is notated.

You might wonder, which circle was drawn first?

Well, most logically is to presume the more “detailed” version was draw first. Why? It contains a mistake in the circling of the neighboring tones of tone center [A] and it seems logical that this mistake would have been corrected in a next version, thus the “clean” version (without the mistake) would have come second, only displaying the most important aspect of the circle, the 12 “tone centers” and circled neighbor tones. But, I am guessing here.

What about those numbers and lines?

There has been some speculating going on about if the lines and numbers drawn in the Coltrane Circle “detailed” version were drawn by Coltrane himself or perhaps if they were added later by someone else?. 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

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 writing 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.

Below links to the used scores to compare with the Coltrane Tone Circle:

One thought shared among musicians online is that the writing of the numbers (and lines) in the “detailed” version of the tone circle could perhaps have been drawn by someone else. Jusef Lateef seems to be the #1 “suspect”, after all, he shared the drawing in his book “Repository of Scales and Melodic Patterns “. I have not been able though to find any handwritten music sheets by Jusef Lateef to compare his handwriting.



In my search for answers I send a message to the Yusef Lateef Facebook page, hoping someone could shed some light on it. I am grateful I got a reply, Ayesha Lateef wrote:

“Brother John gifted the “circle” to Yusef Lateef while the content of both repositories is the result of Yusef’s own research.“

When I specifically asked if the numbers and lines in the circle might have been drawn by Yusef Lateef, she replied:

“From what I understand the whole thing is from Brother John.“

More about the numbers and their possible meaning/function later on in this article.

Below on the left you see a scanned copy of an original drawing of the “Coltrane Circle”. On the right an better readable (by Roel modified) image by Corey Mwamba from his article “Coltrane’s Way Of Seeing“:

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 (the 12 “Tone Centers”) 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 Center” 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 “Tone Center” 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 “Tone Center” 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 C D E G♭ A♭ B♭ C B♭ C D E G♭ A♭ B♭ A♭ B♭ C D E G♭ A♭ G♭ A♭ B♭ C D E G♭ E G♭ A♭ B♭ C D E C D D E G♭ A♭ B♭

THE 6 HEXATONIC ( 6 -TONE) SCALES OF THE INNER RING B D♭ E♭ F G A B A B D♭ E♭ F G A G A B D♭ E♭ F G F G A B D♭ E♭ F E♭ F G A B D♭ E♭ B D♭ D♭ E♭ F G A

“Double Power”

We know Coltrane had an investigative mind and a deep interest in mathematics, philosophy, the occult and religion.

Earlier in this article you probably noticed the Pentagram Coltrane drew in his circle. Now we have looked at the Hexatonic nature of the tone circle, we can also draw a Hexagram between the tones of the Hexatonic scale of the outer ring.

What appears when we combine the Pentagram and Hexagram, is the symbol of “Double Power“. As symbol of ‘double power’ or the unity of the Pentagram and Hexagram, it symbolizes the “mystical marriage” of the micro and macrocosms. The Inner and Outer Worlds. “As Above, So Below“.

Below you see on the left the Coltrane Circle. In this case instead of single tones I approach it as chords. Those of you familiar with Coltrane’s music will directly notice that the colored sections of the circle form the chord progressions of Giant Steps (in “concert pitch“).

When you merge the triangles formed with the Minor 7th and Dominant 7th chords (the II‘s and V‘s), a Hexagram is formed. The Hexagram can be seen as a 2D version of the 3D Star Tetrahedron, also known as “Merkaba“.

HEXAGRAM (GIANT STEPS) HEXAGRAM STAR TETRAHEDRON (MERKABA)

“Mer-ka-ba” means “light-spirit-body”. The Merkaba represent the innermost law of the physical world: the inseparable relationship between the two complementary halves – the positive and negative, the manifest and the unmanifest – which form a perfect equilibrium. In creation they rule as two opposite laws: the law of spirit and the law of matter. The Merkaba is also been called the “divine light vehicle” allegedly used by ascended masters to connect with and reach those in tune with the higher realms, the spirit/body surrounded by counter-rotating fields of light, (wheels within wheels).

I understand for those of you reading this with no interest in mathematics, philosophy, the occult and religion this might all seem a bit far-fetched. But, if you look at some of the titles of his compositions (“Ascension“, “Ascent“, “Sun Ship“, “Cosmos”, “Interstellar Space“, “Spiritual”, et cetera) then is seems more then clear that music, the occult / religion and geometry / math were all connected for Coltrane.

More about Giant Steps you can read in the article “The Geometry of John Coltrane’s Music“.



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 Major 7th or “Leading Tone“, the Tonic and the Minor 2nd or “Supertonic” (see image below).



Perhaps Coltrane wanted to visualize how chromatic neighbor tones lead to adjacent neighbor tones / Tone Centers?



Every Major 7th (mentioned above) is the Major Third of the key (tone center) a Fifth higher (next tone center counterclockwise) as well. [suggestion by Mark Rossi]

Example: the B circled along with the C (tone center) is the Major Third of G (next tone center counterclockwise in the Coltrane Circle).

Every Minor 2nd is also the Major Third of the parallel Major of the Relative Minor key of the by circle connected tone center. [suggestion by Mark Rossi]

Example: the D♭ circled along with the C (tone center) is the Major Third of A Major, the parallel Major key of A Minor, the relative minor key of C Major (tone center).



DIMINISHED SCALES

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:

G A♭ A B♭ B C D♭ D Eb E F G♭ B♭ B C D♭ D E♭ E F G♭ G A♭ A D♭ D E♭ E F G♭ G A♭ A B♭ B C E F G♭ G A♭ A B♭ B C D♭ D E♭

5 ← ♭ 6 or 7 → 8 3 ← 4 or 2 → ♭ 3 B♭ ← B → C B♭ ← B → C D♭ ← D → E♭ D♭ ← D → E♭ E ← F → Gb E ← F → G♭ G ← A♭ → A G ← A♭ → A

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” scale (Natabhairavi) and the “Melodic Major” scale (Charukesi) a semitone lower, 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)“.

ALL-INTERVAL TETRACHORD

Stephon Alexander wrote in his book “The Jazz of Physics: The Secret Link Between Music and the Structure of the Universe ” that it has been argued by Australian pianist Sean Wayland that the All-Interval Tetrachord can be used as a method to play through the chord changes of “Giant Steps” (see video: https://www.youtube.com/watch?v=sQGWAnYd7Iw by Marc Hannaford).

An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords when expressed in prime form. In set theory notation, these are {0146} and {0137} (their inversions: {0256} and {0467}).

From the Tonic C we would get: C-Db-E-Gb {0146} and C-Db-Eb-G {0137} (their inversions: C-D-F-Gb {0256} and C-E-Gb-G {0467}). As you can see, the {0146} sets contain only tones circled on the Coltrane Circle (C-Db-E-Gb) if you follow the Circle clockwise from C.

ALL-TRICHORD HEXACHORD

This though made me wonder if another “tone-series” would align better with or include more tones of the series of circled tones: the All-Trichord Hexachord.

The all-trichord hexachord is a unique hexachord that contains all twelve trichords, or from which all twelve possible trichords may be derived. The prime form of this set class is {012478}

From the Tonic C we would get: C-Db-D-E-F-Gb. All but the 2nd pitch class (D) used in this All-Trichord Hexachord are circled at the Coltrane Circle if you follow the Circle clockwise from C.

WHAT DO THE NUMBERS IN THE DRAWING MEAN?

“What do those numbers mean?” is a question I have received via mail several times. Well, 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.

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.

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’.

You can watch this video on Facebook as well.

REFERENCES MENTIONED IN THIS ARTICLE:

Books mentioned and/or quoted from in this article and other interesting (related) books: