Smate family

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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]]

POTE generator: ~5/4 = 420.855

Optimal GPV sequence3, 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]]

POTE generator: ~5/4 = 422.275

Optimal GPV sequence3d, 11d, 14, 17c, 37cd

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]]

POTE generator: ~5/4 = 422.217

Optimal GPV sequence14, 17c, 37cde

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]]

POTE generator: ~5/4 = 423.020

Optimal GPV sequence14, 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]]

POTE generator: ~8/7 = 210.452

Optimal GPV sequence6, 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]]

POTE generator: ~8/7 = 210.481

Optimal GPV sequence6, 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]]

POTE generators: ~8/7 = 210.974

Optimal GPV sequence6, 11, 17c

Badness: 0.050472