Kalismic temperaments

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These are rank-3 temperaments tempering out 9801/9800. Temperaments discussed elsewhere are:

Considered below are odin, loki, van gogh, rishi, hnoss, and gersemi, but we can begin by looking at the rank-4 temperament.

Kalismic

Subgroup: 2.3.5.7.11

Comma list: 9801/9800

Mapping: [2 0 0 0 3], 0 1 0 0 -2], 0 0 1 0 1], 0 0 0 1 1]]

Mapping generators: ~99/70, ~3, ~5, ~7

Optimal GPV sequence8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e

Odin

See also: Landscape family

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250

Mapping: [6 0 0 8 17], 0 1 0 -2 -4], 0 0 1 2 3]]

Mapping generators: ~55/49, ~3, ~5

Optimal GPV sequence12, 42, 48dee, 54c, 60e, 72, 198, 270, 342, 612, 954, 1236, 1506, 1578, 1848, 3426, 4038, 5616, 7464, 17118e, 18966e

Badness: 0.116 × 10-3

Loki

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 9801/9800

Mapping: [2 0 0 -21 -18], 0 1 0 4 2], 0 0 1 3 4]]

Mapping generators: ~99/70, ~3, ~5

Optimal GPV sequence12, 22, 34d, 56d, 74d, 84de, 96d, 118, 130, 152, 248, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c

Badness: 0.493 × 10-3

Van Gogh

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 199297406/199290375

Mapping: [2 0 8 0 11], 0 1 1 2 1], 0 0 -9 -1 -10]]

Mapping generators: ~99/70, ~3, ~9/7

Optimal GPV sequence22, 58, 80, 138cde, 204cde, 226ce, 240d, 262d, 284, 320, 342, 742, 764, 1084, 1106, 1448, 1506, 1848, 4038, 4802, 5144, 6992

Badness: 0.297 × 10-3

Rishi

The 7-limit comma [65 -84 10 16 ~ 0.13¢ has the ratio of the exponents of 3 and 2 that is close to the one in 81/8. The square root of the latter is close to 35/11. This suggests tempering out (81/8)(35/11)-2, which is the kalisma.

Apart from 35/11, 35/33, and the equivalents of their squares, 81/8 and 9/8, another equave that comes to mind is 3/2, especially after tempering out the chalmersia. When 3/2 is chosen as the equave, Fokker blocks of 34 notes can be used that are close to 34edf and 58edo.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 572145834917888/571919811374025

Mapping: [2 0 3 -10 -4], 0 1 2 4 4], 0 0 8 -5 3]]

Mapping generators: ~99/70, ~3, ~17364375/14172488

Optimal GPV sequence24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e

Badness: 2.10 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 10648/10647, 371293/371250

Mapping: [2 0 3 -10 -4 2], 0 1 2 4 4 3], 0 0 8 -5 3 7]]

Mapping generators: ~99/70, ~3, ~364/297

Optimal GPV sequence: 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e

Badness: 0.505 × 10-3

Hnoss

To the wizma [-6 -8 2 5 = 420175/419904, the kalisma is a natural complement, as their product is the tinge.

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 41503/41472

Mapping: [2 0 1 2 6], 0 1 4 0 2], 0 0 -5 2 -3]]

Mapping generators: ~99/70, ~3, ~144/77

Optimal GPV sequence22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee

Badness: 0.368 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 17303/17280

Mapping: [2 0 1 2 6 -3], 0 1 4 0 2 1], 0 0 -5 2 -3 4]]

Optimal GPV sequence: 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef

Badness: 0.867 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1089/1088, 1225/1224, 2025/2023

Mapping: [2 0 1 2 6 -3 0], 0 1 4 0 2 1 6], 0 0 -5 2 -3 4 -6]]

Optimal GPV sequence: 22f, 54cffgg, 72, 166g, 198g, 270, 364, 436, 634g, 706f

Badness: 0.862 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199

Mapping: [2 0 1 2 6 -3 0 13], 0 1 4 0 2 1 6 2], 0 0 -5 2 -3 4 -6 -6]]

Optimal GPV sequence: 72, 94, 166g, 198g, 270, 436, 634g, 706f

Badness: 0.901 × 10-3

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 2530/2527, 2737/2736

Mapping: [2 0 1 2 6 -3 0 13 19], 0 1 4 0 2 1 6 2 -2], 0 0 -5 2 -3 4 -6 -6 -2]]

Optimal GPV sequence: 72, 94, 166g, 270, 342f, 436, 706fi

Badness: 1.14 × 10-3

Gersemi

The extension to 13-limit with 4225/4224 is weak but facilitates the use of 18/7 as the equave. Fokker blocks of 128 notes are available for the latter, corresponding to 94edo. 18/7 is split into 4 parts that become ~19/15 in 19-limit. Also, (18/7)3 ~ 17/1 via the chlorisma. However, the tones 9/8 and (19/15)/(9/8) = 152/135 have distinct mappings.

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 9801/9800, 41503/41472

Mapping: [2 0 1 2 6 9], 0 1 9 -2 5 -6], 0 0 -10 4 -6 7]]

Mapping generators: ~99/70, ~3, ~154/65

Optimal GPV sequence: 44, 50, 94, 144, 176, 220, 270, 590, 684, 954

Badness: 1.06 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 1089/1088, 1225/1224, 2025/2023, 4225/4224

Mapping: [2 0 1 2 6 9 0], 0 1 9 -2 5 -6 12], 0 0 -10 4 -6 7 -12]]

Optimal GPV sequence: 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684

Badness: 1.46 × 10-3

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757

Mapping: [2 0 1 2 6 9 0 1], 0 1 9 -2 5 -6 12 11], 0 0 -10 4 -6 7 -12 -11]]

Mapping generators: ~99/70, ~3, ~45/19

Optimal GPV sequence: 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h

Badness: 1.11 × 10-3

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925

Mapping: [2 0 1 2 6 9 0 1 7], 0 1 9 -2 5 -6 12 11 3], 0 0 -10 4 -6 7 -12 -11 -3]]

Optimal GPV sequence: 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi

Badness: 1.23 × 10-3