Smate family
The smate family of temperaments tempers out 2048/1875, the smate comma, resulting in equation of four just major thirds (5/4) with the just perfect eleventh (8/3). It therefore requires an extremely sharp tuning of the just major third. 17edo and 20edo provide it and make for good tunings.
Smate
Subgroup: 2.3.5
Comma list: 2048/1875
Mapping: [⟨1 3 2], ⟨0 -4 1]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 420.855
Optimal ET sequence: 3, 11, 14, 17c, 20c, 37c
Badness: 0.178624
Septimal smate
- See also: Mint temperaments #Smate
Subgroup: 2.3.5.7
Comma list: 36/35, 2048/1875
Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]
Wedgie: ⟨⟨4 -1 9 -11 3 24]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.275
Optimal ET sequence: 3d, 11d, 14, 17c, 37ccdd
Badness: 0.077871
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 243/242
Mapping: [⟨1 3 2 6 7], ⟨0 -4 1 -9 -10]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217
Optimal ET sequence: 3de, 14, 17c, 37ccddee
Badness: 0.042518
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 56/55, 243/242
Mapping: [⟨1 3 2 6 7 3], ⟨0 -4 1 -9 -10 2]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020
Optimal ET sequence: 3de, 14, 17c
Badness: 0.036836
Hemismate
Subgroup: 2.3.5.7
Comma list: 256/245, 392/375
Mapping: [⟨1 3 2 3], ⟨0 -8 2 -1]]
Wedgie: ⟨⟨8 -2 1 -22 -21 8]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.452
Optimal ET sequence: 6, 11, 17c, 40bcd
Badness: 0.154301
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 256/245
Mapping: [⟨1 3 2 3 4], ⟨0 -8 2 -1 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.481
Optimal ET sequence: 6, 11, 17c, 40bcde
Badness: 0.065528
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 56/55, 77/75, 256/245
Mapping: [⟨1 3 2 3 4 3], ⟨0 -8 2 -1 -3 4]]
Optimal tuning (POTE): ~2 = 1\1, ~8/7 = 210.974
Optimal ET sequence: 6, 11, 17c
Badness: 0.050472