Divisions of the Octave

After reading Dmitri Tymoczko's groundbreaking paper in SCIENCE (The Geometry of Musical Chords), which maps chords on Moebius strips, toruses, based on their coordinates -- and resolves that important harmonies are close to equal divisions of the octave -- I got to thinking about this in a tuning perspective.

Take both the geometric and arithmetic division of the octave into three or four. For three, we have (0,4,8) and (0,5,9) and for four
we have (0,3,6,9) and (0,4,7,10) (Also harmonic divisions would just be the mirror inverse of the arithmetic ones). Now I thought it would
be interesting to find the differences between these in the "sweet spot" which would be between steps 3 and 10 (for arithmetic/geometric) and between steps 2 and 9 (for geometric/harmonic). both represent comparing linear and logarithmic divisions of the octave.

Now, it's kind of interesting that you can take 12:13:14:...23:24 in both kinds of divisions --- and in the sweet spot you get these ratios
when comparing them, always roughly a semitone:

1.0394
1.0511
1.0583
1.0614
1.0607
1.0564
1.0500
1.0406

Kind of a bell curve, going up and down. Now cleaning up a bit by
capitalizing on harmonic divisions we can throw out 3 and also add a 2 step, and for symmetry get

1.0406
1.0500
1.0564
1.0607
1.0614
1.0607
1.0564
1.0500
1.0406

Or going the other way

1.0226
1.0394
1.0511
1.0583
1.0614
1.0583
1.0511
1.0394
1.0226

which is not as good.

Now to just fill in 1 and 11, which can be done easily when comparing
0,4,7,10 and 1,4,7,10 or 0,5,9 and 1,5,9 and 2,5,8,0 and 2,5,8,11 or
3,7,0 and 3,7,11, where one can use 1.0583 or 1.0511, I prefer the first because it is so close to a semitone. Finally, obtain

1.0583
1.0406
1.0500
1.0564
1.0607
1.0614
1.0607
1.0564
1.0500
1.0406
1.0583

which I guess in some way shows that in 12-tET this ratio is pretty stable. This is 7-limit, so I guess taking only 5-limit steps (which would be steps 3,4,5,7,8,9 (leave out 6, which is 17) and find that
the error all around is

1.0500
1.0511 or 1.0564
1.0583 or 1.0607

etc

which is always within .001, about 1 percent, roughly the error of the major third (5/4) in 12-tET. (14 cents)