Module 11: Harmony II: substitution and reharmonisationLesson 5advanced⏱ 90 min + three weeks of daily practice

Coltrane changes and chromatic mediants

Understand the major-third cycle behind Coltrane changes, play the Giant Steps matrix and the Countdown-style substitution over a ii–V–I as chord shapes and lines, and use chromatic-mediant moves as a modern colour.

Why three keys instead of one

Every device so far has stayed in one key and decorated it. In 1959 and 1960 John Coltrane recorded a set of tunes built on something different: three tonics a major 3rd apart, each approached by its own V7, cycling round so fast that no single key wins. The octave divides into exactly three major 3rds (four frets each), so the three keys are evenly spaced and the cycle closes on itself. Players call the result Coltrane changes, the major-third cycle, or, after the tune that made it famous, the Giant Steps matrix. Whether or not you ever call the tune on a gig, the cycle teaches you three things no other progression does: how to move between distant keys with chord tones alone, how to voice-lead chords whose roots are a major 3rd apart, and how to play fast changes without falling back on licks. This lesson treats the harmony; the melodies of the Coltrane tunes are not notated here, and you should learn them from the records.

Chromatic mediants: the building block

Two major chords whose roots are a major 3rd apart (C and E, C and Ab) are chromatic mediants. They share exactly one note, so they are neither closely related nor completely foreign, and moving directly between them has a bright, slightly unreal quality that film composers and Coltrane both exploited. Look at the three tonic chords that divide the octave starting from C.

C and E share E; E and Ab share G# (Ab); Ab and C share C. Each pair is a chromatic-mediant relationship, and C–E–Ab–C (or in the other direction, C–Ab–E–C) is a closed cycle. Here it is as a constant-structure progression, one drop-2 shape on the top four strings moved down four frets at a time, which is the simplest possible way to play the cycle and a sound worth having in its own right.

\tempo 72
.
\ts 4 4
:1 (9.4 9.3 8.2 8.1){ch "Cmaj7"} | (5.4 5.3 4.2 4.1){ch "Abmaj7"} | (1.4 1.3 0.2 0.1){ch "Emaj7"} | (9.4 9.3 8.2 8.1){ch "Cmaj7"} |

The matrix: each key gets its dominant

Coltrane's step was to put a V7 in front of each tonic so that the cycle becomes a chain of V–I cadences: Cmaj7 Eb7 | Abmaj7 B7 | Emaj7 G7 | Cmaj7. Read it slowly. Eb7 is the V of Ab; B7 is the V of E; G7 is the V of C. The tonics descend by major 3rds (C, Ab, E) and every dominant is a minor 3rd above the tonic it follows and a 5th above the tonic it leads to. In Roman numerals from C the whole thing is I–bIII7–bVImaj7–VII7–IIImaj7–V7–I, which looks baroque on paper but is simply three V–I cadences in three keys.

The famous tune uses the keys B, G and Eb, starting on the upper one: Bmaj7 D7 | Gmaj7 Bb7 | Ebmaj7, then a ii–V back up into G, then the same descent from G to Eb to B, then a ii–V back up into Eb. Its second half is the same three keys approached from below with ii–Vs. Nothing in it is outside the matrix; the difficulty is only the speed and the fact that each key lasts two beats. Here are the first three bars of the matrix as chord shapes in the original keys, with the F#7 that leads back to B added at the end of bar 3 so that the loop closes.

\tempo 80
.
\ts 4 4
:2 (1.4 3.3 0.2 2.1){ch "Bmaj7"} (0.4 2.3 1.2 2.1){ch "D7"} | (0.4 0.3 0.2 2.1){ch "Gmaj7"} (3.4 3.3 3.2 4.1){ch "Bb7"} | (1.4 3.3 3.2 3.1){ch "Ebmaj7"} (2.4 3.3 2.2 2.1){ch "F#7"} | :1 (1.4 3.3 0.2 2.1){ch "Bmaj7"} |

Study the voice leading, because it is the reason the cycle is playable at all. From each tonic to the dominant that follows it, one note stays and three move by half step: the 5th of the tonic is the 3rd of the dominant (compare Bmaj7 and D7 in the table: the F# is common, while D#, A# and B each move one fret). From each dominant to the next tonic, it is an ordinary V–I. If you learn the loop as three ordinary V–I resolutions with a three-half-step slip between each tonic and the next dominant, your hands stop panicking.

The same loop in C, on the middle string set, so that you have it in two registers:

\tempo 80
.
\ts 4 4
:2 (3.5 5.4 4.3 5.2){ch "Cmaj7"} (4.5 5.4 3.3 4.2){ch "Eb7"} | (3.5 5.4 1.3 4.2){ch "Abmaj7"} (2.5 4.4 2.3 4.2){ch "B7"} | (2.5 2.4 1.3 4.2){ch "Emaj7"} (2.5 3.4 0.3 3.2){ch "G7"} | :1 (2.5 2.4 0.3 1.2){ch "Cmaj7"} |

The cycle also runs upwards (C to E to Ab to C), with each key again preceded by its V7: Cmaj7 B7 | Emaj7 Eb7 | Abmaj7 G7 | Cmaj7. The dominants now sit a half step below each tonic they follow and resolve down a 5th as usual. Both directions are used; the descending one is the one in the famous tune and the one you should learn first.

Countdown-style substitution over a ii–V–I

Coltrane's other great trick was to pour the whole cycle into a single ii–V–I. Take Dm7 G7 | Cmaj7, two bars. Keep the Dm7 on the first two beats and the Cmaj7 at the end, and fill the space between them with the descending matrix aimed at C: Dm7 Eb7 | Abmaj7 B7 | Emaj7 G7 | Cmaj7. The G7 that was always there is now the last dominant of the cycle, and the two beats before it have been replaced by two V–I cadences in the "wrong" keys. This is the substitution underneath the tune that reworks a well-known ii–V–I ballad at a fast tempo, and it is the most useful thing in this lesson because ii–V–Is are everywhere. In Roman numerals it is ii7–bIII7–bVImaj7–VII7–IIImaj7–V7–Imaj7, spread over the bars the original ii–V–I occupied.

\tempo 80
.
\ts 4 4
:2 (10.6 10.4 10.3 10.2){ch "Dm7"} (9.6 8.4 8.3 8.2){ch "Eb7"} | (8.6 6.4 8.3 8.2){ch "Abmaj7"} (7.6 7.4 8.3 7.2){ch "B7"} | (7.6 6.4 8.3 5.2){ch "Emaj7"} (7.6 5.4 7.3 6.2){ch "G7"} | :1 (7.6 5.4 5.3 5.2){ch "Cmaj7"} |

A note on the first move. Dm7 to Eb7 is the ii chord sliding up a half step into a dominant, and it is the only "unexplained" step in the whole substitution: everything after it is a V–I. Practise that one change on its own until it sounds like a decision and not an accident. In a slower tune you can put the full three-key cycle into a two-bar ii–V–I exactly as written; in a fast tune you have to choose, because six chords in two bars at 250 bpm is a lot for everybody. Most players use the Countdown-style substitution for one ii–V–I per chorus, or on the last one, and leave the rest alone.

Lines through the cycle

Coltrane's own solution to improvising on the matrix was almost entirely arpeggios and four-note scale cells: 1–2–3–5 on the tonics, chord tones on the dominants. The reason is in the time available: with a new key every two beats there is no room for enclosures or long scale runs, and chord tones are the only thing that outlines three keys unmistakably. The line below goes through the first three bars of the matrix using nothing but chord tones, one four-note arpeggio per chord, in one position. It will sound mechanical at first. That is fine: the point is to put the changes in your ear and your hand, and the music comes afterwards.

\tempo 100
.
\ts 4 4
:8 9.4{ch "Bmaj7"} 8.3 7.2 6.1 7.3{ch "D7"} 7.2 10.2 8.1 | 8.2{ch "Gmaj7"} 7.1 10.1 7.2 8.4{ch "Bb7"} 7.3 6.2 9.2 | 8.3{ch "Ebmaj7"} 8.2 6.1 7.3 :2 8.3 |

Once this is comfortable, do three things to it:

1. Change the direction of alternate arpeggios: up on the tonics, down on the dominants. This gives you contrary motion between the chords and gets rid of the sawtooth sound.
2. Replace the 7th with the 9th on the tonic chords (1–3–5–9 instead of 1–3–5–7) and the 5th with the 13th on the dominants. The sound opens up immediately.
3. Start each arpeggio from a different chord tone (3rd, then 5th, then 7th) so that the first note of each new chord is the nearest chord tone to where you were. This is the Module 7 targeting principle at speed.

When you can do all three in time at 120 bpm, start adding the 1–2–3–5 cell on the tonic chords. Those four notes (root, 9th, 3rd, 5th) are the simplest statement of a major key, and they are the basis of most of what Coltrane played on these tunes.

Using the cycle as a colour

Away from the Coltrane repertoire, the major-third cycle survives as a vocabulary of short moves:

• A single chromatic-mediant substitution. In a I–vi–ii–V turnaround, replace the vi with IIImaj7 or bVImaj7 (Cmaj7 Emaj7 | Dm7 G7 or Cmaj7 Abmaj7 | Dm7 G7). One foreign major seventh chord in a diatonic turnaround is enough to make it sound modern.
• A three-cadence ending. Instead of a plain ii–V–I, play bIII7 bVImaj7 | VII7 IIImaj7 | V7 Imaj7 as a three-bar tag: the whole matrix, aimed at the final tonic.
• Constant-structure maj7 chords by major 3rds under a pedal bass, which is the opening example of this lesson, for intros and vamps.
• Reharmonising a sustained melody note with maj7 chords a major 3rd apart: any note is the root of one maj7 chord and the 3rd of the maj7 chord a major 3rd below it (E is the root of Emaj7 and the 3rd of Cmaj7, as the first table shows), so a long note can be harmonised by two chromatic mediants in turn. The next lesson uses this.

The reason these work is the reason the whole cycle works: a major 3rd is far enough from the tonic to sound new and close enough (one shared note) to sound related.

Exercises

1. The three keys. For every key, name the other two keys that divide the octave with it (C: E and Ab; F: A and Db; and so on). Say them going down and going up.
2. Constant structure. Play the Cmaj7 | Abmaj7 | Emaj7 | Cmaj7 example starting on each of the twelve maj7 chords, on the top four strings, then with the drop-2 shapes on strings 5–4–3–2.
3. The matrix as a loop. Play Bmaj7 D7 | Gmaj7 Bb7 | Ebmaj7 F#7 | as a continuous loop at 60 bpm until you can keep it going for three minutes without looking. Then the C version. Then start the loop on each of the twelve maj7 chords.
4. Countdown-style ii–V–I. Play Dm7 Eb7 | Abmaj7 B7 | Emaj7 G7 | Cmaj7 in C, F, Bb, Eb, G and D at 60 bpm, then 80, then 100. Always follow it with the plain ii–V–I so that you hear what was substituted.
5. Lines. Play the chord-tone line, then apply the three changes (direction, 9ths and 13ths, starting notes). Write out your version in C and in F.
6. Colour moves. Take a turnaround in a tune you know and replace the vi with IIImaj7 or bVImaj7 once per chorus. Record yourself and decide whether you like it.

Quiz