The Coltrane Matrix
Coltrane Matrix: a reharmonization of chord substitutions of a basic ii-V-I with movement in major thirds creating an augmented triad. This is also know as "Coltrane Substitution," "Coltrane Changes," etc.
Coltrane first introduced the "Coltrane Matrix" on his album Blue Train on tunes such as Moment's Notice and Lazy Bird, and later on took things further on Giant Steps.
The B section from Have You Met Miss Jones served as inspiration for Coltrane because of the major 3rd modulation from D to Gb to Bb.
1) Here is a normal ii-V-I in C major:
| ii | V | I ||
| dmin7 | G7 | Cmaj7 ||
2) Now with the Major 3rds Cycle:
| ii V** | I* V** | I* V** | I* |
| dmin7 Eb7 | Abmaj7 B7 | Emaj7 G7 | Cmaj7 |
This progression moves from Ab, down a major 3rd to E, then down another major 3rd to C.
Every modulation to a new key is tonicized by a preceding V chord, indicated my the ** in the diagram.
A Visual Example
Here is the cycle of 4ths, and as you can see, the triangle connects the tones that are a major 3rd apart.
In this diagram, you can rotate the triangle and figure out all the possibilities for the major 3rds cycle.