I'm trying to parse out the logic, but it looks like the outer ring and inner rings are moving clockwise in whole steps, with the outer ring starting on C and the inner starting on C#. If we took the rings and straightened them out we'd have (its hard to tell some of the notes so correct me if I'm wrong):

  [1]                     [2]                     [3]                     [4]                     [5]                     [1]
   C   D   E   F#  Ab  Bn? C   D   E   F#  An? Bb  C   D   E   Gb  Ab  Bb  C   D   E   F#  G#  Bb  C   D   E   F#  Ab  Bb  C
     C#  Eb  F   G   A   B   C#  Eb  F   G   A   B   Dn  Eb  F   G   A   B   C#  Dn  F   G   A   B   C#  En? F   G   A   B  

Note:

  n = (natural sign)
  ? = its hard to read the scribbles.

It looks like he augments some of the intervals in the middle of the scale and diagramming how to elide from one to the other. Note that Bn could be considered distinct from B. Coltrane was familiar with the overtone series that generate what are called harmonics. Harmonics could be thought of as the sine waves that sum together to create the timbre of a pitch. If you've played guitar and you pluck the string while gently pressing the string down at the 1/2, 1/3, 1/4, 1/5 (etc) division, you'll force the string to vibrate only at the frequency that corresponds to that harmonic (ie 1/2 would force it to vibrate only at 2f where f = the lowest vibration mode of the string, sorry If I'm not conveying the physics perfectly). Anyway, the overtone-series closely approximates a major scale, but it gets tricky when you want to shift from one series to the next because the series is generated from integer ratios, and prime numbers generate similar but distinct pitches. Long story short, you can have several genus of the same note that are all mathematically distinct.

..I've got a better description here https://music.stackexchange.com/a/33787