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THE GEOMETRY OF JOHN COLTRANE’S MUSIC

Er is een Nederlandse versie beschikbaar op saxwereld.nl

Note: this article is a “work in progress”. There is a lot more to “figure out” about Coltrane’s work, so the content of this article might change.

In this article I will use and refer to the “Coltrane Tone circle”. I would suggest you read the Roel’s World article “John Coltrane’s Tone Circle” (if you haven’t done so already) before you proceed with this article.

FOREWORD

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 … or to follow in his giant steps. 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.

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 his works.

In this article I will write shortly about:

THE THREE TONIC SYSTEM
“GIANT STEPS”
THE FOUR TONIC SYSTEM
“CENTRAL WEST PARK”
“EQUINOX”

NOTE: All sheet music presented in this article are the Concert Pitch (piano) versions.
(not only sax players love his work, right?)

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. 

“GIANT STEPS”, “CENTRAL WEST PARK” & “EQUINOX”

I will not write about how to improvise on the Coltrane’s compositions mentioned in this article.

First of all, I do not consider myself developed enough as soloist to “teach” others. I have still so much more to learn and master myself that I don’t feel it is appropriate to do so.

Secondly, the purpose of this article is to share something about the geometry of some of Coltrane’s work. Not to “teach” the reader how to improvise, there are nice websites and blog articles available online already about that. 



COLTRANE CHANGES

3 TONIC SYSTEM – (MAJOR 3RDS CYCLE)

In jazz harmony, the “Coltrane changes (Coltrane Matrix or cycle, also known as chromatic third relations and Three Tonic System) are a harmonic progression variation using (repeatedly) modulating up or down a major third. With a 3 Tonic System there are essentially 3 different tonal centers present in the cycle: Each key center descends in Major 3rds

3 “stacked” Major 3rds form an Augmented Triad, visualized geometrically with a Trigon (triangle). With the Augmented Triad you cycle the Tone Circle in 3 steps.

2 “stacked” Augmented Triads an Augmented Second or Minor Third apart form an Augmented Scale or Whole Tone Scale.

Coltrane wasn’t the first using a Three Tonic System descending in Major 3rds in Jazz though. The bridge of the composition Have You Met Miss Jones (1937) as written by Richard Rodgers:

Bridge of "Have You Met Miss Jones"

The tonal centers in the bridge of this piece are B♭Maj7G♭Maj7 and DMaj7, all a Major Third apart from each other. You can hear the chord progression in the J.R. Monterose youtube below at 00:27 …

So, maybe these changes should have been called “Rodgers Changes” instead of “Coltrane Changes” …

Of course without Coltrane’s further explorations, development and implementation of this tonic system (for his in 1960 released album “Giant Steps“, a real masterpiece and Jazz album “Must Have”) it might never have become as known and popular among Jazz musicians and fans as it was and is. He was after all the first that composed a complete piece based on it, not “just a bridge” (as far as I know).

“GIANT STEPS”

Below a Concert Pitch (piano) sheet with the chord progression of Giant Steps. 
The tonal centers are colored Blue-Green

Giant Steps - Chord Progressions

The movement in Major 3rds down (BGE♭) isn’t the only movement in 3rds in this piece, there is also the I-V relationship (B → D7, G → B♭7 and E♭ → F♯7), a Minor 3rd (3 semitones) movement.

And there are more “3‘s” in this piece. We also find the interval movement of a “Tritone” (3 whole tones) in bars 3, 7, 9, 11, 13 and 15 between the Tone Centers and the IIs of the II-V’s to the next Tone Center. Here are the first 4 bars of this piece with it’s movements:

First 4 bars of Giant Steps ...

Note: Coltrane did not list any minor tonalities in his Tone Circle. In case you want to make the chord progressions of this piece match the tones in the original Coltrane Tone Circle, or with a single row (Major only) Circle of Fifths, then theoretically we could use C6, or better C/A the relative Major/minor substitution for Am7 instead. In that case this is what we get, a triadic reduction of a Coltrane Cycle:

Relative Major/minor substitution example

Naturally you could replace the other minor chords with their relative Major/minor substitutes in this piece as well. This altered version might look different ‘on paper’, but sounds the same. 

Note: the geometric shapes created with the relative Major substitute of the Minor chords will be different though.

To not over-complicate things, I will only use the original chords (not these Major/minor substitutes) for this article. I have therefor altered Coltrane’s Tone Circle in this article, adding the Minor chords used in the example compositions to it.

AUGMENTED TRIADS

All tonics of the chords used in Giant Steps can be found back at the Circle of Fifths/Fourths within 3 of the 4 Augmented triads within the octave.

The first Augmented Triad is formed by 
tonics of the Tonal Centers of this piece: B – E♭ – G.

The second Augmented Triad is formed by the tonics of the Dominant 7 chords
B, D and F (G♭).

The 3rd Augmented Triad is formed by the tonics of the Minor 7th chords: A – C♯ (D♭) – F.

When combining these Augmented triads we can form several Hexatonic (6-tone) scales.

If you are familiar with (sacred) geometry you might have noticed that the circle above does not only shows 3 Trigons, but a Hexagram as well if you merge the triangles of the Minor 7th and Dominant 7th chords (the II‘s and V‘s). 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”. To create a (3D) Merkaba you have to “merge” 2 (3D) Tetrahedrons. And to create a Tetrahedron you need 4 (2D) Trigons. 1 Trigon is formed by the (3) tones of an Augmented Triad

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). 

If this is why Coltrane explored the 3 Tonic System and applied geometry to his compositions I don’t know. His interest in spiritual growth and for the occult could very well have been the reason, some of the titles of his compositions (“Ascension“, “Ascent“, “Sun Ship“, “Cosmos”, “Interstellar Space“, “Spiritual”, et cetera) do suggest he did. 

HEXATONIC SCALES

In music theory a Hexatonic Scale is a scale with 6 pitches or notes per octave. The most commonly used Hexatonic scales are the Whole Tone Scale, the Augmented Scale and the Blues Scale (the Minor Pentatonic Scale plus a 5th). Hexatonic scales can be divided in 3 x 2 “pitch pairs” (“dyad“), Augmented Scale example: (3 -1) – (3 -1) – (3-1).

THE AUGMENTED SCALE

The (symmetrical) Augmented Scales can be thought of as an interlocking combination of two augmented triads an augmented second or minor third apart and contains two interval types: the Minor Third (m3) and the Semitone (s).

There are two modes:

Mode 1: m3 – s – m3 – s – m3 – s also written as 3 – 1 – 3 – 1 – 3 – 1
Mode 2: s – m3 – s – m3 – s – m3  also written as 1 – 3 – 1 – 3 – 1 – 3

THE WHOLE TONE SCALE

Even though the Augmented Scales are related to the Augmented relationships between all the chords types used in Giant Steps, they do not “work” everywhere equally well. There is though another Hexatonic scale that can be created with the tonics of the chords of this composition. When you combine the Augmented Triad of the tonics of the Dominant 7 Chords and the Minor 7th Chords the Whole Tone Scale.

The Whole Tone (T) Scale is written as: T – T – T – T – T – T or 2 – 2 – 2 – 2 – 2 – 2“.

Below you can see how the Augmented and Whole Tone Scales relate to the Augmented Triads based on the Tonics of the Chords used in Giant Steps (BE♭ , G / B♭7D7, F7 / Am7C♯m7Fm7):

AUGMENTED SCALE

1: B – D – E♭ – G♭ – G – B♭B
2: D – E♭ – G♭ – G – B♭ – B – D

AUGMENTED SCALE

1:  B♭ – C# – D – – F♯ – A – B♭
2: A – B♭ – C# – D – 
– F♯ – A

WHOLE TONE SCALE

A – B – C# – E♭ – F – – A


CHORD PROGRESSION (HARMONIC MOVEMENT) “GIANT STEPS”

Below a “visualization” of the relationship between the chords used in Giant Steps:

 


5TH ↓

TRITONE

4TH 

MINOR 3RD 


When looking at the circles above you might think that it is draw wrong. A Tritone is normally drawn as a line between opposite (180 degrees) tones and a 4th would be one step (30 degrees) counterclockwise (in the Circles above), as the 5th is one step (30 degrees) clockwise.

Well, yes, that would be the case if the movements took place between (the tonics of) two Major or two Minor tonalities/chords. Tritones between (the tonics of) one Major and one Minor key are connected at a 90 degree angle and the 4th at 60 degrees (instead of 30).

With other words, the angle between a Major and Minor tonics/tonalities/chords is half or double the angle between two Major or two Minor tonics/tonalities/chords.

3 – 1 PATTERN

Besides the Augmented Scales (that can be compiled by with the Tonics of the chords used in Giant Steps) there is yet another 3 – 1 pattern embedded in bar 1-4 and bar 5-8. Not as scale tones but in chord progressions. You can see this pattern appear when you look at the geometric division of the Circle of Fourths. In this case we are not going around the circle with a 3 Semitone – 1 Semitone sequence, but with a 3 Fourths – 1 Fourth sequence (see “closed cycle” below).

The chord progression of Giant Steps in Bar 1-4 starting at the 1st Tone Center of B generates a 3 – 1 – 3 – 1 pattern in bar 1-3. Coltrane did not finish the full cycle (with the 3rd 3 – 1) though, but changes direction via a II-V-I (A minor – D Dominant 7) to the 2nd Tone Center G. This II-V-I pattern generates a Tritone (TT) – 1 – 1 sequence instead.

What is interesting though is when you are using the alternate version (earlier mentioned in this acticle) with the C6 (relative Major/Minor) substitute for Am7 (and skip the VD7), another 3 – 1 sequence can be created to complete the 3 – 1 – 3 – 1 – 3 – 1 pattern:

CLOSED CYCLECYCLE TO G (ORIGINAL)CYCLE TO G ALT.
B – D – G – B♭ – E♭ – G♭ – BB – D – G – B♭ – E♭ – A – D – GB – D – G – B♭ – E♭ – C – G
3 – 1 – 3 – 1 – 3 – 13 – 1 – 3 – 1 – TT – 1 – 13 – 1 – 3 – 1 – 3 – 1

In Bar 5-8 the same pattern repeats, but then from Tone Center G:

CLOSED CYCLECYCLE TO E (ORIGINAL)CYCLE TO E ALT.
G – B♭ – E – G♭ – B – D – GG – B – E – G♭ – B – F – B – GG – B – E – G♭ – B – A – E
3 – 1 – 3 – 1 – 3 – 13 – 1 – 3 – 1 – TT – 1 – 13 – 1 – 3 – 1 – 3 – 1

In bar 9-16 the 3 – 1 pattern / sequence does no longer appear, Coltrane switches to standard II-V-I progressions passing the 3 Tone Centers


4 TONIC SYSTEM – (MINOR 3RDS CYCLE)

Coltrane might have been most known for his implementation of the Three Tonic System, but that wasn’t the only geometric system he used. For some pieces he used the Four Tonic System. There are differences in the implementation of this concept, but for this article I will use the method Coltrane used for his piece “Central West Park“. 

There are essentially 4 different tonal centers present in this composition. Each key center ascends or descends in Minor 3rds or their Tritone Substitudes

4 “stacked” Minor 3rds form an Diminished 7 Chord, visualized geometrically with a Quadragon (Square). With the Diminished 7th Chord you cycle the Tone Circle in 4 steps.

2 “merged” Diminished 7 Chords form a Diminished Scale.

“CENTRAL WEST PARK”

Below the first part of “Central West Park” (Concert Pitch – Piano – sheet). The tonal centers are in Blue-Green.

Below the first 4 bars of this piece with the tonal centers and their minor 3rd and tritone movements:  

BMaj7→ DMaj7→ AMaj7→ FMaj7→ BMaj7:

Not only do the tonics of the 4 Major 7th Chords form a Diminished 7 Chord, so do the rest of the chords. The tonics of the Dominant 7th Chords form a Diminished 7 Chord, as do the tonics of the Minor 7th Chords


DIMINISHED 7 CHORDS

Diminished Seventh Chord is a four note chord that comprises a diminished triad plus the interval of a diminished seventh (alternatively regarded enharmonically as a major sixth) above the tonic. These tones are the 1st, 3rd, 5th and double flat7th (or enharmonically the 1st, 3rd, 5th and 6th) of any minor scale.

Another way to look at this chord is to see it as 4 minor 3rds “stacked” on top of each other. With the Diminished 7 Chord you make one cycle around the Tone Circle in 4 steps.

Diminished 7 Chord can be visualized geometrically by drawing a Quadragon (square) between it’s tones in the Circle of 5ths.

If we draw lines between the 4 tonal centers, as well as the other chords, we get the following squares / Diminished 7 Chords:

BDFA♭
CE♭G♭A
C♯m – Em – Gm – B♭m
 

DIMINISCHED SCALES

The Diminished Scale (an Octatonic Scale) is an eight-note symmetrical scale which the notes ascend (or decent) in alternating intervals of a whole step and a half step. The twelve tones of the chromatic scale are covered by 3 disjoint Diminished Seventh Chords. The notes from exactly 2 such Diminished 7th Chords combination form an Octatonic Scale. Because there are exactly 3 ways to select 2 from 3, there are exactly 3 Octatonic Scales in the 12-Tone system. Octatonic Scales are the only Scales that can be “split” into 4 “pitch pairs” (groups of 2 tones). There are two modes: the first begins its ascent with a whole step between its first two notes, while the second begins its ascent with a half step (semitone). What mode to use does depend on the chords used in the composition.

Mode 1: WHOLE-STEP/HALF-STEP:
B♭ – C – D♭ – E♭ – E – G♭ – G – A – B♭
B – C♯ – D – E – F – G – G♯ – A♯ – B
C – D – E♭ – F – G♭ – A♭ – A – B – C

Mode 2: HALF-STEP/WHOLE-STEP:
B – C – D – E♭ – F – G♭ – A♭ – A – B
C – C♯ – D♯ – E – F♯ – G – A – A♯ – C
C♯ – D – E – F – G – A♭ – B♭ – B – C♯

The Whole-Step/Half-Step Diminished scale is mostly used on Diminished 7 Chords. The Half-Step/Whole-Step is mostly used on Dominant 7 Chords with possible extensions ♭9, ♯9, 13 and ♭5/♯11.”

Diminished Scales can be transposed by 3, 6, or 9 semitones (up or down) without changing the tone “collection”.

Octatonic Scales are the only Scales that can be “split” into 4 “pitch pairs” (groups of 2 tones).

The twelve tones of the chromatic scale are covered by 3 disjoint Diminished Seventh Chords. The notes from exactly 2 Diminished 7th Chords combination form an Octatonic Scale. Because there are exactly 3 ways to select 2 from 3, there are exactly 3 Octatonic Scales in the 12-Tone system:

– – – A♭  +  – E♭ – G♭ – A– – – A♭  +  B♭ – D♭ – – G– E♭ – G♭ – A  +  B♭ – D♭ – – G


– – – E♭ – – G♭ – A♭ – A – B– D – D – E – F – G – A♭ – B♭ – BC – D – E – E – G – G – A – B♭ – C
THE MODES:

The 1st begins its ascent with a whole step, while the 2nd begins its ascent with a half step.

WHOLE-STEP/HALF-STEP EXAMPLES:HALF-STEP/WHOLE-STEP EXAMPLES:

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

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

CHORD PROGRESSION (HARMONIC MOVEMENT) “CENTRAL WEST PARK”
MEASURE 1 – 5  MEASURE 6 – 10HARMONIC POLYGRAM 

In the tone circle above you see the exact “movement” between the harmonic progression of “Central West Park” in the first 5 bars.

The black lines are the II-V-I‘s. The Blue lines represent Fourths

Above the tone circle with the “movement” between the harmonic progression of “Central West Park” in bar 6 to 10.

The lines are: FifthFourthMinor SecondMajor 7th / Minor Second.

This tone circle shows all connections between the chords used in “Central West Park. 

The movement (above) from bar 1-5 looks pretty interesting geometrically.

Coltrane “swapped” 2 of the 4 Tone Centra (of the Four Tonic System) for his composition.

He goes in II-V-I‘s around the Circle passing each Tone Centrum like this: 

 D  A♭  F  B 

… in stead of the standard geometrically more symmetric: 

 D  F  A♭  B 

… that you can see on the right. 



“EQUINOX”

The reason I like to write something about Equinox is that it is a Minor Blues with a “hidden reference” to the Autumnal Equinox. I found out about that thanks to a post by Facebook friend Robert Aguirre, who shared his finding online around the 23rd of September.

But, before we look into that, first a couple of words about what an Equinox is, in case you are not familiar with that term:

An Equinox is an astronomical event in which the plane of Earth‘s equator passes through the Sun’s Ecliptic. The Sun’s Ecliptic, the Earth’s Equator and Axis all intersect.

This occurs twice each year, around 20 March (Vernal Equinox) and between 22 and 23 September (Autumnal Equinox). 

Why is the 23th of September of importance? 

John Coltrane was born on 23 September  1926, around 17:00 .

On the 23 rd of September 1926 the Equinox took place at 15:26:22 EDT, duration = 12 Hours, 08 Minutes and 09 Seconds. With other words: John Coltrane was born during the Autumnal Equinox!


EQUINOX (Concert Pitch – REAL BOOK version):

For the Tone-Circle visualization I have used the commonly used Real Book music sheet version in the key of C minor. The original though as recorded in the studio (and used for the embedded Youtube) is in the key of C# minor!

Equinox

What we see above, is a 12-bar Minor Blues starting after the intro at [1]. In the tone circle (on the right) you see with the black lines and arrows the chord progressions and their geometric relationship.

The only non “standard” chord in this basic 3-chord (IIVV) Minor Blues, is the A♭9 chord in the 9th bar [9].

What the numbers 9 (bar) and 12 (bars Blues) have in common with the Autumnal Equinox, is that the Autumnal Equinox takes place in the 9th month (September) each year (12 months).

The A♭9 chord is the relative Major/Minor Substitute for the Fm7 chord (the IV, as used in Equinox at the usual places for a Minor Blues).

But why specifically the A♭9 substitute?

The reason for this (I think) can be found when we look at the melody of this composition. In bar [9] we find 3 tones, all D‘s. If we use Fm7 – the 4th degree (IV) – in the 9th bar, then the D would be the 6th of the chord (as it is in bar 10). D could also have been the 5th of the G Dominant 7/9 chord if that would have been the chord of choice for bar [9].

But, by using the A♭9 as substitute chord, the D in the melody of bar [9] is given a different role, not the 5th or 6th, but it becomes the Tritone of A♭

The tones that form the Tritone interval are geometrically located on the opposite sides of the Tone Circle and divide the tone circle in two equal halves, as the Equator divides Earth into two Hemispheres.

At the moment of the Equinoxes, Earth’s Equator passes the center of the Sun’s Ecliptic. If you imagine the Tone Cirlce to be Earth and the Tritone to be the Equator, then the result of this merger you see on the right.

It looks like Coltrane intended to “accentuate” the 9th bar with the A♭9 substitute chord and with it the 9th month and the Autumnal Equinox.

FUNNY FOOTNOTE: the “standard” Blues progressions / chords of this composition (the I, IV and V‘s) are all located in the tone circle above the A♭9  D tritone. With other words, the Blues is played above the “Equator”, in the “Northern Hemisphere” (North America, the “birthplace” of the Blues).

EQUINOX (Concert Pitch – recorded version):

Above I have used the commonly used Real Book music sheet version in the key of C minor in combination with the “Coltrane Cirle”, to visualize the “hidden references” to the Equinox both harmonically and melodically withing the composition. The original version though as recorded in the studio (and used for the embedded Youtube) is in the key of C# minor!

I have therefor modified the Coltrane Circle (transposed it a semitone up) to visualize the audio (recorded version) as well.

The following chords are used in the original (audio) version:
‖:  C#m7  |  C#m7  |  C#m7  |  C#m7  |
|  F#m7  |  F#m7  |  C#m7  |  C#m7  |
|   A7   |   G#7   |   C#m7   |   C#m7   :‖


This idea might seem a bit “far-fetched” at first, perhaps I’m “seeing things” there that were not “orchestrated” by Coltrane as such, but since Coltrane was intellectually and spiritually developed and interested in the “occult” this might have been the reason …

In this article I have used and referred to the “Coltrane Tone circle”. I would suggest you read the Roel’s World article “John Coltrane’s Tone Circle” (if you haven’t done so already) for more information about the “Coltrane Cirle”.



MORE COLTRANE COMPOSITIONS?

Dear visitor … If you have are aware of / come across other compositions of Coltrane with interesting geometric ‘properties’ and/or other hidden features and references, then please do feel free to contact me, so I can add your findings to this article (with a reference of course).

REFERENCES MENTIONED IN THIS ARTICLE:


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The album (first advert) and books about John Coltrane’s “Giant Steps”: