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.

These temperaments resemble skwares, which also splits 8/3 into four major thirds, but those major thirds are flattish 9/7s rather than the sharp 5/4s used here.

Smate

Subgroup: 2.3.5

Comma list: 2048/1875

Mapping: [⟨1 3 2], ⟨0 -4 1]]

mapping generators: ~2, ~5/4

Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 420.855

Optimal ET sequence: 3, 11, 14, 17c, 20c, 37c

Badness (Smith): 0.178624

Septimal smate

Subgroup: 2.3.5.7

Comma list: 36/35, 2048/1875

Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]

⟨⟨4 -1 9 -11 3 24]]

Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 422.275

Optimal ET sequence: 3d, 11d, 14, 17c, 37ccdd

Badness (Smith): 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 = 1200.000, ~5/4 = 422.217

Optimal ET sequence: 3de, 14, 17c, 37ccddee

Badness (Smith): 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 = 1200.000, ~5/4 = 423.020

Optimal ET sequence: 3de, 14, 17c

Badness (Smith): 0.036836

Hemismate

Subgroup: 2.3.5.7

Comma list: 256/245, 392/375

Mapping: [⟨1 3 2 3], ⟨0 -8 2 -1]]

⟨⟨8 -2 1 -22 -21 8]]

Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 210.452

Optimal ET sequence: 6, 11, 17c, 40bcd

Badness (Smith): 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 = 1200.000, ~8/7 = 210.481

Optimal ET sequence: 6, 11, 17c, 40bcde

Badness (Smith): 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 = 1200.000, ~8/7 = 210.974

Optimal ET sequence: 6, 11, 17c

Badness (Smith): 0.050472