@johncarlosbaez
3/n
Of course one could also try to fill in the rest tones of the 4 twelvetone octaves, in order to get towards a full „charismatic twelvetone tuning“ . This seems however not so straight forward. In order to fill in quite some „omitted tones“ one could tune first the rest of the domain e.g. meander „from above within the domain“ , ie. start at the just fifth above , i.e. at G and go to g‘‘. One obtains: C→7→[G-Bb-db-e<-5<-B-d-f-ab→7→e‘b-g‘b-a‘-c‘‘<-5<-g‘-b‘b-d‘‘b-e‘‘-g‘‘]→-5→C. One sees that e, c‘‘and e‘‘ are double. There one probably has to make a decision, which to take, or take something between the respective two. Some hexagons have a double layering, like being the A in the first tune path and the B in the second. Taking the fifth below one gets still not all notes of the 4 octaves „charismatically tuned“, but one could probably use the other fifths to fill in the gaps. Or one could simply go up and down octaves from the above from the above „charismatic scale“. a.s.o.

@johncarlosbaez
2/n
Now follows a scale, that is a bit more interesting than going up the thirds and then jump by a fifth: If one denotes Ab as A flat etc. and denotes a jump by a minor third (3 halftones, factor 6/5) by „-“ and denoted a jump by a just fifth (7 halftones, factor 3/2) by „– > 7 – >“ and denoted a jump back by a just fourth „< - 5 < - „ (5 halftones back, factor ¾) then the tuning goes e.g. C-Eb-Gb-A - >7 - > e-g-bb-d‘b < - 5 < - ab-b-d‘-f‘ - >7 - > c‘‘-e‘‘b-g‘‘b-a‘‘-c‘‘‘. So between C and c‘‘‘ there is the charismatic schisma, which is connected with the fact that the pure 4 octave jump has to be „distorted“, by multiplying the frequency of C with 16.0489, instead of 16. I would like to call this scale a „charismatic scale“. As one can see this scale meanders through the 3 thirdcycles D-F-Ab-B-D, C-Eb-Gb-A-C, and Db-E-G-Bb-Db that define the usual Tonnetz domain. So one could also think about „flattening“ this scale to one octave, by omitting either C-Eb-Gb-A at the beginning or c‘‘-e‘‘b-g‘‘b-a‘‘ at the end, or by finding a „compromise“. But unfortunately one looses a lot of „charisma“ in that process, that is I could imagine that the step from B to a higher octave C could sound strange.

@johncarlosbaez Yeah I reply there and post some copied content here.
1/n
OK now thinking a bit further one sees that the charismatic schisma allows for an interesting tuning that gives a different tuning per octave. This comment here gets now a bit out of proportion but let me cast that out more. I actually, just this week, wanted to learn a bit more about Gregorian modes, as a free time project, which shall explain my sudden interest in this topic.
The charismatic schisma defines a small 2 rows-14 column fundamental domain.
One can get tuning scales into that domain, which could eventually -with a grain of salt- be seen as „modes“, as, with 4 octaves, they are roughly within the usual singing range.
A tuning scale meanders in the fundamental domain.