User:VIxen/Sandbox
Wizmic family
These are rank-3 temperaments where the wizma 420175/419904 is tempered out. For the clan of rank-2 temperaments with this comma, see Wizmic microtemperaments.
Wizmic
Subgroup: 2.3.5.7
Comma list: 420175/419904
Mapping: [⟨1 0 3 0], ⟨0 1 4 0], ⟨0 0 -5 2]]
Mapping generators: ~2, ~3, ~648/245
Optimal ET sequence: 5, 17, 22, 27, 45, 49, 50, 72, 99, 171, 441, 612, 2812, 3082, 3253, 3424, 3694, 3865, 4306, 4477d, 4918d, 5089d, 5701d
Badness: 0.0864 × 10-3
Gersemi
To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the tinge.
18/7 is a possible equave. Fokker blocks of 128 notes are available for it, 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
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: 10c, 18bcd, 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: 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]]
Mapping generators: ~99/70, ~3, ~154/65
Optimal ET sequence: 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684
Badness: 1.46 × 10-3
19-limit
Subgroup: 2.3.5.7.11.13.19
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 ET sequence: 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h
Badness: 1.11 × 10-3
23-limit
Subgroup: 2.3.5.7.11.13.19.23
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]]
Mapping generators: ~99/70, ~3, ~45/19
Optimal ET sequence: 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi
Badness: 1.23 × 10-3
13-limit (ibnsinmic)
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]]
Mapping generators: ~99/70, ~3, ~144/77
Optimal ET sequence: 10c, 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef
Badness: 0.867 × 10-3
13-limit (schisminic)
Subgroup: 2.3.5.7.11.13
Comma list: 4096/4095, 9801/9800, 41503/41472
Mapping: [⟨2 0 1 2 6 21], ⟨0 1 4 0 2 -6], ⟨0 0 -5 2 -3 3]]
Mapping generators: ~99/70, ~3, ~144/77
Optimal ET sequence: 22, 72f, 94, 176, 248, 270, 612, 882, 1152, 2034
Badness: 1.56 × 10-3
Skeetsmic family
These are rank-3 temperaments where the skeetsma is tempered out.
Skeetsmic
Subgroup: 2.3.5.7
Comma list: 14348907/14336000
Mapping: [⟨1 0 0 -14], ⟨0 1 0 15], ⟨0 0 1 -3]]
Mapping generators: ~2, ~3, ~5
Optimal ET sequence: 5, 7d, 12d, 39ddd, 44cdd, 46d, 48, 53, 111, 118, 171, 742, 795, 913, 966, 1137, 1308, 2050, 2221, 2392, 3358, 3529, 7229b, 10758bbd
Badness: 0.333 × 10-3
Skald
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 1240029/1239040
Mapping: [⟨2 0 1 -28 -25], ⟨0 1 0 15 13], ⟨0 0 1 -3 -2]]
Mapping generators: ~99/70, ~3, ~5
Optimal ET sequence: 10c, 12de, 44cddeee, 46de, 48, 58, 106, 118, 224, 342, 742, 966, 1084, 1308, 1650, 2392, 2734, 5126bdb, 6776bd
Badness: 3.20 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 59535/59488
Mapping: [⟨2 0 1 -31 -27 -22], ⟨0 1 0 15 13 11], ⟨0 0 2 -6 -4 -3]]
Mapping generators: ~99/70, ~3, ~220/117
Optimal ET sequence: 12de, 46def, 58, 166, 224, 460, 684, 1426, 2334, 3076
Badness: 1.31 × 10-3
Other from the kalismic family
Rishi
The 7-limit comma [65 -84 10 16⟩ ~ 0.13c 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 pitches per equave 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 ET sequence: 24, 34d, 58, 150cdee, 208ccddee, 252ccddeee, 262ccdee, 286ccdee, 310cddee, 320ccee, 344cdee, 378ce, 402de, 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 ET sequence: 24, 34d, 58, 150cdeef, 208ccddeeff, 252ccddeeefff, 262ccdeefff, 286ccdeeff, 310cddeeff, 320cceeff, 344cdeef, 378cef, 402def, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e
Badness: 0.505 × 10-3
Odin (harmonismic)
(Equave 3/2: q63ef & q70p & q95p)
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 105644/105625
Mapping: [⟨6 0 1 10 20 34], ⟨0 1 0 -2 -4 -6], ⟨0 0 2 4 6 7]]
Mapping generators: ~55/49, ~3, ~325/154
Optimal ET sequence: 12f, 108cddeeefff, 120cdeeff, 126, 132de, 144, 258ef, 270, 552, 684, 954, 1236, 1506, 2190, 3696, 5886, 6156, 8346, 12042
Badness: 0.418 × 10-3