What were these? A small note at the bottom of the acknowledgements said:

Geometric Drawings: By John Coltrane, 1960. Gifts to Yusef from John.

Over time I became fascinated by the Coltrane drawings and set about decoding them using a protractor, compass and tracing paper.

First I made a clean schematic of Coltrane’s marked-up diagram.

In thinking about it I realized it could be simplified from two rings to one without losing any of the intrinsic relationships.

Of course, from a musician’s perspective this had the surprising result of converting from a whole-tone scale in Coltrane’s original to a chromatic scale in my single-ring version. Then I realized there could be a three-ring version as well, with the intervals on each ring describing diminished triads.

This new three-ring version was visually strange and beautiful, and had a feature that wasn’t evident in either the one-ring or two ring versions: a winding pattern.

Pick a section starting with C and walk to the next C, one semitone at a time. The first four notes of the series would be C, C#, D, Eb. In the one and two ring versions D and Eb are adjacent, but in the three-ring version Eb is on the far ring.

That got me to thinking of the series as a winding banner.

And from there a 3D pattern, not a flat one.

I made a clean final version of this sketch.

From there it was natural to go on to versions with four, five and six rings.