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♭) and that all intervals are extremely stretched, though the 5L 7s macrochromatic scale is suggested for musical use due to the hardness of the macrodiatonic and the increased breadth of the tritave. 9\22edt is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as 7\17edt and 16\39edt 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.
Mos scales of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s).
For technical data, see No-twos subgroup temperaments#Mintaka.
Extensions of Mintaka
Several extensions of this temperament are possible to incorporate additional harmonics.
Add 20 and 23/4
Off the bat, given that 1331/1323 is a lopsided 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.
Add 19
There are two reasonable ways to incorporate prime 19 into the subgroup. For tunings of the generator sharper than 9\22edt, the step 81/77 approaches or exceeds 20/19 in quality, and therefore can be identified with 20/19 by tempering out 1540/1539, equating 19/9 to (77/81)(20/9), 13 generators down (or alternatively, if one refuses to admit the even number 20 into the subgroup, by tempering out 16929/16807).
Minalzidar
The alternative extension to include prime 19, known as Minalzidar, works better for tunings flatter than 9\22edt, where it is the most accurate to find 19/9 at (9/7)^3, 9 generators up, tempering out the comma 6561/6517. The two representations meet at 22edt.
In this range, the optimal representation of 5 is that obtained by tempering out 120285/117649, which equates 5 with (529/243)2, placing it 16 generators down. However, as soon as prime 20 is inserted, this also equates 5 with (20/9)2, tempering 81/80 in the 3.4.5 subgroup. Furthermore, this then equates 4/3 to 27/20, 8 generators up, therefore creating a square root of 4 at 4 generators up and making this an insane restriction of meantone. Therefore, as soon as prime 5 is incorporated, this temperament folds into Eshurizel, an elaborate extension of 11-limit squares.
Add 4 and 5
For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma), it is reasonable to combine this temperament with BPS, and additionally temper out 245/243, thereby equating 5/3 to 81/49 at 6 generators up. This is mintra temperament, which splits the BPS generator in three. With the inclusion of 20 in the subgroup above, 4/3 would therefore also appear, at the position of (20/9)/(5/3), 14 generators down.
If we combine all of the above, using the sharper representation of 5 and 19, we find the complete 3.4.5.7.11.19.23 temperament with commas 100/99, 133/132, 253/252, 484/483, and 540/539.
Interval chains
One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121; in this temperament, it is equated to 99/49 and placed four generators up. In flatter tunings of the generator, this is closer to a true octave. This interval is meriting of special treatment in terms of consonance and dissonance.
Tritave-reduced harmonics below 243 are marked in bold.
| # | Cents* | Approximate Ratios (3.7.11) | Approximate ratios (3.7.11.20.23/4 extension) |
|---|---|---|---|
| -4 | 690.0 | 49/33, 121/81 | 161/108, 180/121 |
| -3 | 1468.5 | 7/3 | 180/77 |
| -2 | 345.0 | 11/9 | 28/23, 60/49 |
| -1 | 1123.5 | 21/11 | 23/12 |
| 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 |
Tuning spectrum
| Edt Generator |
Eigenmonzo (Unchanged-interval) |
Generator (¢) |
Comments |
|---|---|---|---|
| 7\17 | 783.158 | ||
| 11/7 | 782.492 | 0-comma | |
| 23\56 | 781.160 | ||
| 39\95 | 780.803 | ||
| 28/23 | 780.702 | ||
| 363/343 | 780.405 | 1/5-comma | |
| 16\39 | 780.289 | ||
| 57\139 | 779.938 | ||
| 49/33 | 779.883 | 1/4-comma | |
| 41\100 | 779.802 | ||
| 25\61 | 779.490 | ||
| 59\144 | 779.273 | ||
| 34\83 | 779.114 | ||
| 9/7 | 779.013 | 1/3-comma | |
| 43\105 | 778.896 | ||
| 52\127 | 778.753 | ||
| 33/20 | 778.478 | ||
| 81/77 | 778.317 | 2/5-comma | |
| 27/20 | 778.177 | ||
| 778.124 | DR 7:9:11, close to 18/43-comma | ||
| 9\22 | 778.073 | ||
| 21/20 | 777.675 | ||
| 11/9 | 777.274 | 1/2-comma | |
| 38\93 | 777.143 | ||
| 29\71 | 776.855 | ||
| 20\49 | 776.308 | ||
| 31\76 | 775.797 | ||
| 49/20 | 775.669 | ||
| 23/12 | 775.636 | ||
| 11\27 | 774.871 | ||
| 121/63 | 772.055 | Full comma |
Other tunings
- DKW (3.7.11): ~3 = 1\1, ~11/7 = 778.466
Audio examples
short composition by Wensik in 22edt (without using the temperament's MOS scales), based on the 7:9:11 chord and its inversion, 63:77:99.