Mintaka
Mintaka is a temperament in the 3.7.11 subgroup where ~11/7 is a generator, and the comma 1331/1323 is tempered out, so a stack of two generators represents 27/11 in addition to 121/49, and a stack of three generators, tritave-reduced, represents 9/7. As 11/7 as a generator against the tritave produces a 5L 2s (macrodiatonic) scale, with the generator here occupying the role of a perfect fourth, it is possible to use an analogue of the chain-of-fifths notation that is standardly used for diatonic scales, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭). 9\22 is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as 7\17 and 16\39 are also useful.
As perhaps the simplest temperament of this subgroup delivering decent accuracy - and, in particular, the simplest supported by tunings such as 17edt and 22edt - Mintaka can be considered the 3.7.11 analog of 3.5.7 BPS or 2.3.5 meantone, using 7:9:11 as its fundamental consonant chord in the place of 3:5:7 or of 4:5:6.
Several extensions of this temperament are possible to incorporate additional harmonics. Off the bat, given that 1331/1323 is a [comma] with S-expression S222 * S23, one can reliably choose to temper both S22 = 484/483 and S23 = 529/528 in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to 36/23, and the interval 11/9 to 28/23. Furthermore, the tiny comma S161 = 25921/25920 can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas 253/252, 484/483, and 540/539 in the 3.7.11.20.23/4 subgroup.
Mos scales of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s, 12L 5s).
Interval chains
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 778.5 | 11/7, 36/23 |
| 2 | 1556.9 | 27/11, 69/28, 49/20 |
| 3 | 433.4 | 9/7, 77/60 |
| 4 | 1211.9 | 99/49, 243/121, 324/161, 121/60 |
| 5 | 88.4 | 81/77, 363/343, 207/196, 21/20 |
| 6 | 866.9 | 81/49, 33/20 |
| 7 | 1645.4 | 891/343, 2187/847, 207/80 |
| 8 | 521.9 | 729/539, 759/560, 27/20 |