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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, harmonization, and reharmonization?

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At least in Common Practice theory, a chromatic third relationship is two chords with roots that are either a minor or major third apart and have the same quality. C Major to E Major is an example, as is c minor to a minor. Chords with this relationship will always share a common tone (E in the first example), have one note move by step (C–B) and one note chromatically altered (G–G#). I won't post as a full answer since I can't respond to the rest of your questions, but I think the meaning is the same in relation to Coltrane. – Pat Muchmore Jul 12 '14 at 3:48
There's a fairly extensive Wiki on the Coltrane changes - See also – No'am Newman Jul 12 '14 at 7:02

Look up "Coltrane changes" in Essentially, they consist of a minor third interval followed by a perfect fourth interval (e.g. B-D-G-Bflat, etc.). Coltrane's compositions "Giant Steps" and "Countdown" are good examples, as well as his changes on the bridge of his recording of the standard "Body and Soul". The wikipedia article explains this in considerable depth.

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