Smate family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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]]

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]]

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