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 sequence: 8d, 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 sequence: 12, 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 sequence: 12, 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 sequence: 22, 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 sequence: 24, 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 sequence: 22, 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

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