: What are the Coltrane Changes? Also known as Coltrane Matrix, Coltrane Cycle, chromatic third relations, and multi-tonic changes. What are the Coltrane Changes? What's the theory behind them? How are they used in improvisation,

The questions you've asked make for a great logical flow, so I'll structure my answer around those questions.

What Are the Coltrane Changes?

"Coltrane changes" is the name for a chord progression that jazz saxophonist John Coltrane used extensively in his playing and composing. They are generally considered to be a substitution for a 4-bar ii-V-I progression. For example, here is a ii-V-I in C maj:

| Dmin | G7 | Cmaj | Cmaj |

Here are Coltrane changes in Cmaj (which could replace the ii-V-I from above):

| Dmin E?7 | A?maj B7 | Emaj G7 | Cmaj |

The Coltrane changes contain the original ii-V-I, but between the ii (Dmin) and the V (G7), there are four additional chords. The progression still ultimately resolves to Cmaj, which is the final chord found in bar 4. That's why this progression is considered to be Coltrane changes in the key of C.

What's the Theory Behind Coltrane Changes?

Coltrane changes cycle through three tonal centers, descending by major thirds. In the example above, the Coltrane changes cycle through these three tonal centers:

| | A?maj | Emaj | Cmaj |

Then we add the V chord for each I chord, a classic maneuver with bebop reharmonizations:

| (E?7) | A?maj (B7) | Emaj (G7) | Cmaj |

There's one final step, and then we'll have completed the changes: we take the final V-I progression (G7 - Cmaj) and add the ii chord (Dmin - G7 - Cmaj), thereby completing the ii-V-I progression. But instead of putting the ii chord at the end, along with the final V-I progression, we put it at the very beginning:

| Dmin (E?7 | A?maj B7 | Emaj) G7 | Cmaj |

How Are They Used in Harmonization/Reharmonizations?

In their purest/simplest form, Coltrane changes are played as a four-bar substitution for a ii-V-I progression. Anywhere you see a ii-V-I that lasts 4 bars, you can replace the ii-V-I in Coltrane changes. The most famous example is Countdown, which is Coltrane's reharmonization of the Miles Davis song Tune Up. Here are the chord changes for Tune Up, with Coltrane changes written in parentheses in purple on top:

Three sets of Coltrane changes are seen in this song, on the first three lines:

Line 1 (m. 1-4): tonal center cycle is B?maj - G?maj - Dmaj
Line 2 (m. 5-8): tonal center cycle is A?maj - Emaj - Cmaj
Line 3 (m. 9-12): tonal center cycle is G?maj - Dmaj - B?maj
In all three cases, we see the tonal center descend by major thirds

There is a second primary use for Coltrane changes, in addition to using them as a substitution for a 4-bar ii-V-I progression. Coltrane changes can also be used as a substitution for 4 bars of a single major chord. For example, if a song contains 4 bars of Cmaj, then those 4 measures can be replaced by the Cmaj Coltrane changes (which cycle through A?maj - Emaj - Cmaj). In this case, instead of finishing the last ii-V-I, we play the start the progression with the final I chord. For example, this:

| Cmaj | Cmaj | Cmaj | Cmaj |

becomes:

| Cmaj | A?maj | Emaj | Cmaj |

which becomes:

| Cmaj (E?7) | A?maj (B7) | Emaj (G7) | Cmaj |

This second usage begins with Cmaj (the final I chord) instead of Dmin (the ii for the final V-I progression). These changes are found in Coltrane's blues called 26-2 and in Coltrane's famous song Giant Steps.

How Are They Used in Improvising?

In the songs I've shown and mentioned, the Coltrane changes are written into the tune, as a composed reharmonization. However, Coltrane changes can also be used, impromptu, by a soloist when improvising. If a horn player starts soloing over Coltrane changes, the rhythm section can either continue playing the normal ii-V-I, or they can hear the soloist and follow him/her. (Of course, this requires a good ear and proficiency with the Coltrane changes.)

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