Coltrane's Tone Circle

John Coltrane once sketched a diagram for his friend and mentor Yusef Lateef: the chromatic scale arranged in circles, lines and circles identifying relationships. Lateef later referenced it in his Repository of Scales and Melodic Patterns (see the original). As far as I know, Coltrane never explained the diagram in writing, and there is no consensus on exactly what it was meant to show.

At its core, the drawing is the circle of fifths, familiar to most musicians. Coltrane's version fills in the circle with chromatic scales, separated into two circles of whole steps, that join the fifths. Lines then trace different harmonic relationships between the notes, uncovering shapes that reveal the chord structures behind "Giant Steps", arguably Coltrane's most famous harmonic experiment.

In this post, we'll rebuild the diagram as an interactive widget and attempt to decode its harmonic secrets.

§ Reading the Circle

The large notes around the outside form the circle of fourths: moving clockwise from C, each note is a perfect fourth above the last (C, F, B♭, E♭, and so on). Between each pair sit four smaller notes that climb chromatically, alternating between an outer and inner ring. Following them clockwise from C, you'll pass C♯, D, E♭, and E before arriving at F. Twelve fourths add up to sixty semitones, or five octaves, so every note appears on the ring five times.

The circular band inside the chromatic scale contains scale degrees for the fourths, measured from the note at the top. The controls below allow for changing key with the arrow buttons: the circle rotates to bring the new key to the top, while the degrees stay in place. The buttons below highlight the shape of a relationship within the circle. We'll explore those shapes later in the post.

Throughout the post, descriptions of the circle assume C as the root, the starting point when the page loads.

§ A Note on Notation

Instead of Roman numerals (I, ii, V), this post uses the Nashville number system to express diatonic relationships. It grew out of the Nashville studio scene, where session players needed charts they could transpose on the spot: 1-4-5 means C-F-G in the key of C, and G-C-D in the key of G. A superscript minus marks a minor chord, so 2⁻ in C is D minor. Labels like ♭9 and ♯11 are extensions rather than chords.

§ The Circle of Fifths

If you start on any note and keep stacking perfect fifths (C, G, D, A, E, and so on), after twelve steps you will land back on C, having visited every note exactly once. The sequence forms a perfect circle. On this tone circle, the sequence of fifths runs counter-clockwise. Read it the other way and each step is a perfect fourth instead (C, F, B♭, E♭, ...), because a fifth up and a fourth down arrive at the same note. The circle of fourths and fifths is the same circle.

The circle allows us to quickly see how closely two keys are related. Neighboring keys share six of their seven notes: C major and G major differ only in G's F♯, and C major and F major differ only in F's B♭. Each step away swaps one more note, until you reach the opposite side of the circle. C major and F♯ major, a tritone apart, have just two notes in common, which makes them as distant as two keys can be.

Jazz harmony often works by stacking fourths (quartal harmony), as opposed to the more traditional approach of stacking thirds. McCoy Tyner's piano voicings in Coltrane's quartet are a famous example, along with the "So What" chord from Miles Davis's Kind of Blue. That openness and ambiguity is an essential component of the modern, 50's and 60's, jazz sound.

Stacking fourths is also why the circle has twelve positions. A fourth is five semitones, which shares no factor with the twelve in an octave, so a chain of fourths visits every note before returning to where it started. Intervals that do divide the octave evenly close the loop sooner: stack minor thirds and you're back after four notes, or major thirds and you're back after three. Those shortcuts are the shapes inside the circle.

§ Symmetric Shapes

Many chords are asymmetric. A major triad stacks a major third and then a minor third, and that asymmetry is how your ear determines the root of the chord. The shapes on the tone circle establish an opposing idea: divide all the notes of the octave into equidistant parts to produce different harmonic symmetries.

There are only a few unique ways to do this. The circle draws the first three, with the fourth as an imaginary line across the circle to the tritone (substitution):

Because every note is equidistant from the others, each has an equally strong claim to the root. You can hear that ambiguity: symmetric chords sound unsettled and suspended, like a question waiting for an answer, rather than open in the way quartal voicings are. That's what makes them so interesting harmonically: they can resolve in several directions, which makes them natural pivots for moving between keys. The repeating nature of these equidistant groups means the unique combinations are limited.

§ Diminished 7th Chords (Squares)

A diminished 7th chord is created by stacking minor thirds: C, E♭, F♯ (G♭), A. Select C°7 above and you'll see these four notes form a square. The other two squares, F°7 and B♭°7, round out the set of diminished 7th chords. Shifting a diminished 7th chord up by a minor third yields the same four notes again, so there are only three distinct combinations. The 'root' of the chord can change based on context, but it can only be one of these three groupings.

Diminished chords are often used in jazz as a substitute for dominants. Take a G7♭9 chord (G, B, D, F, A♭) and drop the root, and what's left is B, D, F, A♭, a diminished 7th chord. Since any note of that square could be the "missing" root's third, the same chord works as a rootless G7♭9, B♭7♭9, D♭7♭9, or E7♭9. Those four roots form a square of their own.

§ Augmented Triads (Triangles)

An augmented triad stacks major thirds: C, E, G♯. Select C+ and you'll see a triangle, one of four. Like the diminished chords, any of its notes could be the root, so C+, E+, and G♯+ are the same chord.

Augmented triads are less common as chords in their own right, but the major-third relationship they're built on is the foundation of Coltrane changes. Select E♭+ and look at the three notes it connects: E♭, G, and B. Those are the three keys of "Giant Steps".

§ The Whole-Tone Scale (Hexagon)

Dividing the octave into whole steps gives six notes: C, D, E, F♯, G♯, B♭. This is the whole-tone scale, and its floating, unresolved sound can be heard in the opening bars of Stevie Wonder's "You Are the Sunshine of My Life". There are only two whole-tone scales, and the circle draws the one containing the current key's tonic. Over a dominant chord, it supplies the ♯11 and ♭13, which makes it a common choice for a 7♯5 chord.

§ Octaves (Pentagon)

Lastly, the pentagon connects the five tonics on the ring. Each side of the pentagon spans one octave of the chromatic sequence, providing contextual landmarks when using the circle mid-solo.

§ Applications of the Tone Circle

§ 2⁻-5-1 Progression

The 2⁻-5-1 is the backbone of jazz harmony. In C, it's Dm7, G7, Cmaj7. Find those roots on the circle, D (2⁻), G (5), and C (1), and you'll see that they're neighbors, stepping clockwise into the tonic. Each chord's root falls a fifth to the next, the strongest resolution there is, so the progression pulls toward home on each change.

Voice leading follows the same scheme. The 7th of each chord slips down a half step to become the 3rd of the next: the C in Dm7 falls to the B in G7, and the F in G7 falls to the E in Cmaj7. These two notes, the 3rd and 7th, are called guide tones: they do almost all the work of defining each chord, and here they move as little as possible.

Extending the cadence further, 3⁻-6⁻-2⁻-5-1 (Em7, Am7, Dm7, G7, Cmaj7) walks five steps clockwise around the circle. Chains like this provide that forward-moving feel of many jazz standards.

§ Tritone Substitution

The tritone splits the octave exactly in half, so it's the only interval that works out the same is either direction: a tritone up from G is D♭, and so is a tritone down. On the tone circle, it's the line straight across.

That symmetry is useful for substitutions on dominant chords. G7's 3rd and 7th are B and F, a tritone apart. D♭7 has the tritone of G as its root, F as its 3rd, and B as its 7th: the same two notes with their roles swapped. Since the 3rd and 7th are what make a dominant chord sound like a dominant, D♭7 can stand in for G7. Substituting it into our 2⁻-5-1 gives Dm7, D♭7, Cmaj7. The guide tones resolve exactly as before, but now the bass descends chromatically, D, D♭, C, giving a more fluid, less orchestral, feel.

§ Coltrane Changes

"Giant Steps" (1960) takes these all these ideas and melds them into a seamless form that is now the rite of passage for all aspiring jazz soloists.

The song cycles through three keys a major third apart: B, G, and E♭. Select E♭+ under the circle and you'll see the 'augmented triad' triangle connecting them.

Here are the changes:

| Bmaj7   D7  | Gmaj7   Bb7 | Ebmaj7  . | Am7   D7   |
| Gmaj7   Bb7 | Ebmaj7  F#7 | Bmaj7   . | Fm7   Bb7  |
| Ebmaj7  .   | Am7     D7  | Gmaj7   . | C#m7  F#7  |
| Bmaj7   .   | Fm7     Bb7 | Ebmaj7  . | C#m7  F#7  |

In the second half, the key changes every two bars instead of every two beats, cycling through E♭, G, and B. Each of those 'roots' is approached by a full 2⁻-5-1: Fm7, B♭7 to E♭, Am7, D7 to G, and C♯m7, F♯7 to B. In the first half, we see an abbreviated version of this, cycling through the 'augmented triad' roots, using 5-1 movement into each root. The result is a tune that feels like it's constantly arriving and never settling, which is a big part of why it's such a gauntlet among jazzers. Throw in a high tempo and we've got a stew going.

§ Modes

A mode is a type of scale, often relative to a major scale, that is associated with melodic and harmonic characteristics. C Dorian, for example, is the B♭ major scale played from C — C is the second degree of B♭ major. From the perspective of C major, this scale has ♭3 and ♭7, which gives that darker rock and blues vibe. I often reach for Dorian, Mixolydian, and even Lydian, when a song contains chords borrowed from adjacent keys, especially dominant chords.

The circle helps visualize the relationships between modes. A major scale is seven neighboring notes on the circle: C major is F, C, G, D, A, E, and B, all in a row. Slide that window one step clockwise and you get F major, which swaps B for B♭. Played from C, that's C Mixolydian. Every further step clockwise swaps in one more flat and gives a darker mode, while a step counter-clockwise swaps in a sharp and gives the brighter Lydian:

We can also use the circle as a reference for the set of modes that contain the same notes. In the case of C, we move the root instead of the window of notes, and we rotate counter-clockwise:

These two selections mirror each other on the circle. C Phrygian comes from A♭ major, four steps clockwise from C, and E Phrygian, the Phrygian mode of C major, sits four steps counter-clockwise. For each pair, we can draw a horizontal line through the circle to see the correspondence.

§ More Modal

Not every case fits cleanly on the circle, though. Take E7 in the key of C — part of a common C E7 Am progression. The key's own chord on E is Em7 (3⁻), but E7 raises its G to G♯, which provides a leading tone to A. We can think of E7 as the 5 of A minor (the 6⁻), a secondary dominant that resolves one step clockwise. The matching scale here is E Phrygian dominant (E, F, G♯, A, B, C, D), the 5th mode of A harmonic minor, which is C major with a single change, G to G♯. And, an interesting note: the 'single change' in C Phrygian dominant is E♭ to E.

§ Extensions

Extensions — the notes stacked above a chord's 7th — don't seem to follow as clear a pattern on the circle. Extensions are found by stacking thirds (9, 11, 13), and the circle is ordered by fourths and fifths. But, it's interesting to see where some of them fall on the circle.

§ Closing Thoughts

We might never know Coltrane's true intentions behind the tone circle. Even from an unenlightened view, the circle succinctly captures many fundamentals of jazz harmony. A 2⁻-5-1 is a few steps clockwise, a tritone substitution is a jump straight across, Coltrane changes are a triangle of augmented triads, and major modes are sliding windows around the circle. Visualizing these relationships helps reinforce these mechanics of harmony.

So, grab your instrument of choice and explore. Listen to other Coltrane tunes, other jazz greats, and see what you can find. Perhaps the biggest secrets of the circle are yet to be uncovered.

§ References

• Coltrane's Tone Circle
• Coltrane Geometry
• Open Culture - The Tone Circle John Coltrane Drew
• The Jazz Piano Site - Coltrane Changes
• Coltrane's Sheets of Sound