Kalismic temperaments
These are rank-3 temperaments tempering out 9801/9800. Temperaments discussed elsewhere are:
- Jubilee (+50/49 or 99/98) → Jubilismic family
- Fantastic (+225/224) → Marvel family
- Bisector (+121/120 or 245/243) → Sensamagic family
- Julius or varda (+176/175) → Diaschismic rank three family
- Hagrid (+243/242) → Cataharry family
- Uniwiz (+385/384) → Keenanismic temperaments
- Varuna (+441/440) → Werckismic temperaments
- Hades (+540/539) → Swetismic temperaments
- Dimcomp (+1375/1372) → Dimcomp family
- Baldur (+2401/2400) → Breed family
- Thor (+3025/3024 or 4375/4374) → Ragismic family
- Semiporwell (+6144/6125) → Porwell family
- Semicanou (+14641/14580) → Canou family
Considered below are odin, lycoris, van gogh, hnoss, sif, loki, pessoal and rishi. For the rank-4 temperament, see Rank-4 temperament #Kalismic (9801/9800).
Odin
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 ET 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
Lycoris
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 1771561/1771470
Mapping: [⟨2 0 4 -6 1], ⟨0 1 1 3 2], ⟨0 0 -6 5 -1]]
- mapping generators: ~99/70, ~3, ~81/70
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.9439, ~81/70 = 252.6028
Optimal ET sequence: 152, 328, 342, 836, 1178, 1354, 1506, 1848, 2684, 4038, 4190, 4532, 11254, 15786e
Badness: 0.249 × 10-3
Higanbana
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 1399680/1399489
Mapping: [⟨2 1 5 -3 3 8], ⟨0 2 2 6 4 1], ⟨0 0 -6 5 -1 4]]
- mapping generators: ~99/70, ~1458/1001, ~81/70
Optimal tuning (CTE): ~99/70 = 1\2, ~1458/1001 = 650.9715, ~81/70 = 252.6037
Optimal ET sequence: 166, 190, 304d, 328, 494, 684, 1012, 1178, 1506, 2190, 2684, 4190, 6380, 9064, 15938
Badness: 0.416 × 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 ET 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
Hnoss
To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the argyria, [-9 -4 0 3 2⟩ = 41503/41472. VIxen named it and gersemi after Odr's and Freya's daughters since in 11-limit, they both temper out one comma of odin (9801/9800) and one of freya (41503/41472). There is also a word play on "semi-" that hints at the split of the octave in half.
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 ET 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 ET 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 ET 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 ET 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 ET 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 ET 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 ET 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]]
Optimal ET 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 ET sequence: 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi
Badness: 1.23 × 10-3
Sif
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 2097152/2096325
Mapping: [⟨2 0 1 7 11], ⟨0 1 0 1 -1], ⟨0 0 4 -5 -1]]
- mapping generators: ~99/70, ~3
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.9492, ~48/35 = 546.6041
Optimal ET sequence: 22, 46, 68, 114, 134, 156, 178, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2742, 3236, 3506, 4742d, 10990ccdde, 15732ccdddee
Badness: 0.423 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4096/4095, 6656/6655, 9801/9800
Mapping: [⟨2 0 1 7 11 16], ⟨0 1 0 1 -1 -3], ⟨0 0 4 -5 -1 1]]
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.9902, ~48/35 = 546.6064
Optimal ET sequence: 22, 46, 156f, 178, 202f, 224, 270, 494, 764, 1012, 1236, 1506, 3236, 3506, 4742df, 6248cdf, 8248cdef, 12990ccddeff, 14496ccdddeeff
Badness: 0.339 × 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 ET 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
Pessoal
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 131072/130977
Mapping: [⟨2 0 1 10 14], ⟨0 1 0 -1 -3], ⟨0 0 3 -1 2]]
- mapping generators: ~99/70, ~3, ~32/21
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.0759, ~32/21 = 728.7910
Optimal ET sequence: 36, 46, 84, 94, 130, 224, 270, 494, 764, 1164, 1658, 3586cd, 5244cdde, 6008bcdde
Badness: 0.499 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095
Mapping: [⟨2 0 1 10 14 13], ⟨0 1 0 -1 -3 -1], ⟨0 0 3 -1 2 -2]]
Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 702.0637, ~32/21 = 728.7786
Optimal ET sequence: 36, 46, 84, 94, 130, 224, 270, 494, 764, 1258, 1882d, 2152d
Badness: 0.391 × 10-3
Linus
Linus tempers out the linus comma, [11 -10 -10 10⟩ in the 7-limit and can be described as the 80 & 130 & 270 temperament. It tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.
7-limit
Subgroup: 2.3.5.7
Comma list: 578509309952/576650390625
Mapping: [⟨10 0 0 -11], ⟨0 1 0 1], ⟨0 0 1 1]]
- mapping generators: ~15/14, ~3, ~5
Optimal tuning (CTE): ~15/14 = 1\10, ~3/2 = 702.0437, ~5/4 = 386.5040
Optimal ET sequence: 50, 60, 80, 130, 270, 1270, 1540, 1810, 1940, 2080, 2210c, 2480c
Badness: 1.12 × 10-3
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 391314/390625
Mapping: [⟨10 0 0 -11 4], ⟨0 1 0 1 -1], ⟨0 0 1 1 2]]
Optimal tuning (CTE): ~15/14 = 1\10, ~3/2 = 702.0010, ~5/4 = 386.5927
Optimal ET sequence: 50, 60e, 80, 130, 190, 270, 670, 940, 1130, 1400, 1800c, 2070c, 2340c
Badness: 1.12 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 4225/4224, 4459/4455
Mapping: [⟨10 0 0 -11 4 37], ⟨0 1 0 1 -1 0], ⟨0 0 1 1 2 0]]
Optimal tuning (CTE): ~15/14 = 1\10, ~3/2 = 702.0010, ~5/4 = 386.5928
Optimal ET sequence: 50, 60e, 80, 130, 190, 270, 590, 730, 860, 1130, 1590df, 1860def
Badness: 0.771 × 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.
When VIxen was naming this temperament, sruti was its only named rank-2 subtemperament. Rishis (sages) are believed to have created shruti texts.
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 ET 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]]
Optimal ET sequence: 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e
Badness: 0.505 × 10-3