I think I've mentioned it before, but there is one microtemperament 768 does a particularly good job with. Though the obvious et for it is 171, 768 does just fine.
I'm calling it "mutt", for Midi UniT Temperament. Its generators can be taken to be 2^(1/3) and a near-just, very slightly narrow 5/4, or else the 12-et major third, and the 14 cent error of a 12-et major third.
It looks like one of those temperaments [that it's] up to me to explore since no one else would be interested, but it is not a bad one if you want to go to microland. If I do write something in it, for sure I will tune it to 768-equal.
Mutt [2^(1/3), 5/4] generators
<<21 3 -36 -44 -116 -92|| {65625/65536, 250047/250000}
[<3 5 7 8|, <0 -7 -1 12|]
err 0.04784
| family name: | mutt |
| period: | 1/3 octave |
| generator: | 5/4 |
| name: | mutt |
| comma: | |-44 -3 21>, the mutt comma |
| mapping: | [<3 5 7|, <0 -7 -1|] |
| poptimal generator: | 9/771 |
| TOP period (cents): | 400.023 |
| TOP generator (cents): | 386.016 or 14.007 |
| MOS (cardinality): | 84, 87, 171, 429, 600, 771 |
| name: | mutt |
| wedgie: | <<21 3 -36 -44 -116 -92|| |
| mapping: | [<3 5 7 8|, <0 -7 -1 12|] |
| 7-limit poptimal generator: | 21/1794 |
| 9-limit poptimal generator: | 2/171 |
| TOP period (cents): | 400.025 |
| generator (cents): | 385.990 or 14.035 |
| TM basis (ratios): | {65625/65536, 250047/250000} |
| MOS (cardinality): | 84, 87, 171 |
The mutt temperament has two remarkable properties. In the 5-limit, the mutt comma reduces the lattice of pitch classes to three parallel strips of major thirds. The strips are three fifths (or three minor thirds, if you prefer) wide. In other words, tempering via mutt reduces the 5-limit to monzos of the form |a b c>, where b is -1, 0 or 1. In the 7-limit, the landscape comma 250047/250000 reduces the entire 7-limit to three layers of the 5-limit; everything in the 7-limit can be written |a b c d>, where d is -1, 0, or +1. Putting these facts together, we discover that mutt reduces the 7-limit to nine infinite chains of major thirds. In mutt, everything in the 7-limit can be written |a b c d> where both b and d are in the range from -1 to 1, so that |b|<=1 and |d|<=1.
The other remarkable property explains its name: it is supported by the standard val for 768 equal. Since dividing the octave into 768 = 12*64 parts is what some systems use for defining pitch (using the coarse, but not the fine, conceptual "pitch wheel" of midi) mutt is a temperament which accords to this kind of midi unit, hence the acronym Midi Unit Tempered Tuning, or "mutt".
The fact that the smallest MOS is 84 and the generator is about the 14 cent difference between the 400 cent third of equal temperament and a just third of 386 cents limits the applicability of mutt. If we tune 84 notes in 768 equal to mutt, we divide 400 cents by a step of 9 repeated 27 times, followed by a step of 13. If we now use this to tune seven rows, each of which divides the octave into twelve parts, we have rows with the pattern [63 63 63 67 63 63 63 67 63 63 63 67], a modified version of 12EDO.
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