Think:color wheel. Every improvising musician (and some classical musicians as well)holds a similar diagram in their heads when practicing common chord progressions like ii-V-Is and rhythm changes, and the blues, etc... and on and on. Each note/chord in the western scale pulls against others next and opposite to it in the same way, regardless of the key one inhabits in any given moment. These relationships are dynamic; variables, in a way, pulling against one another in an interconnected system. The tonic(I, or "c" in c major, for example) has a "home" quality, the dominant (V) has a cadential quality in relation to the tonic, etc., and each chord has its role. Within this system, one can create the illusion that one is "in a key" that one is not actually fully in for the overall work-- this is like nesting methods in code. Loops are loops and the idea remains the same no matter how big or small, no matter the coding language. Chordal relations in music are the same. This diagram may be what Coltrane holds in his head as he plays patterns and repeats motives in a given solo to imply chordal relationships and "transportation" from one realm to another. These realms have anticipatory meaning we all understand unconsciously, even if we are not professional jazz musicians. (for example, we all feel the need for "do" if it is left off the end of "do re mi fa sol la ti...." in "Do, a deer".) There is nothing magical about Coltrane's diagram that I can see that is beyond what most improvisers hold in their heads as they practice patterns over chord changes. And yet, there is something wonderful about how these chords and notes create inherent meaning and character roles relative to one another-- like the color wheel. It is still cool to see the way Coltrane represented these for himself in his own mind, (in other words- how simple his ingredients- they are the same as ours, and the result!)implying how he practiced improvising, and the path those patterns may have taken.

When improvising over changes, it is easy to get lost in a compositional idea and lose one's place in the form. Maintaining a simple "relational data structure" like this circle is way way to maintain one's balance a place in the overall form and structure. Jazz musicians work to internalize this in practice, so that in performance, they can be free and expressive while still within the form. And then...there is Dolphy.!!

The pentatonic scale. [0]

Once you tune your ear into it, it's absolutely everywhere.

[0] https://en.wikipedia.org/wiki/Pentatonic_scale