Although the author puts a disclaimer up front ("no expert") it is still astounding how wrong one can be (taking all his honest effort into account). It's not that he´s wrong "by choice", but if your priors are wrong there's just no other possible outcome.
He is using the right "words" or "concepts" (tetrachord, hexachord) but these words have been mutilated by "musicology" and "music theory" to the point that they don't have any "useful" meaning at all anymore. The tetrachords and hexachords he mentions just are no tetra-/hexachords. If this fundamental building block is wrong, everything else down the line has to be, too.
How do I know?
Well, a colleague of mine (fellow professor) has found the Bach manuscript and I have found the Coltrane manuscript where BOTH of them lay out the whole of "music theory" in half a page.
If the two greatest masters of music used the SAME system, it should be ok for you and me. Just stay away from all the crap coming out of "music theory" or "musicology" or YouTube videos or websites made by "experts" or "non-experts". They just don't know.
Instead take any piece of classical music (e.g. WTK 1) or any Coltrane improvisation (from 1960 on) and look for the smallest "building block" - WITHOUT any priors!
That's difficult, I know, but that separates the men from the boys. These priors are "scales" (there is no C-C seven-note scale), or "harmonies" (there is no "dominant", Bach wouldn't know what you were talking about, and Coltrane just used it to be able to communicate with the rest of the world), or wrong tetra- and hexachords.
So just LOOK at the page or transcription! No priors! Then it will become obvious to you, too, especially in Coltrane's later improvisations, e.g. Live at the Half Note, and Bach made it abundantly clear in the first two fugues of the WTK 1.
That's why he wrote it in the first place (read the title page).
It's there for all of us to see and understand and - much more importantly - to use!
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