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 tunings:

• WE: ~2 = 1191.896 ¢, ~5/4 = 418.013 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 421.637 ¢

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

Badness (Sintel): 4.190

Septimal smate

Subgroup: 2.3.5.7

Comma list: 36/35, 2048/1875

Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]

Optimal tunings:

• WE: ~2 = 1191.900 ¢, ~5/4 = 419.425 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 423.046 ¢

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

Badness (Sintel): 1.971

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 tunings:

• WE: ~2 = 1191.914 ¢, ~5/4 = 419.372 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 423.010 ¢

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

Badness (Sintel): 1.406

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 tunings:

• WE: ~2 = 1194.045 ¢, ~5/4 = 420.921 ¢
• CWE: ~2 = 1200.000 ¢, ~5/4 = 423.402 ¢

Optimal ET sequence: 3de, 14, 17c

Badness (Sintel): 1.522

Hemismate

Subgroup: 2.3.5.7

Comma list: 256/245, 392/375

Mapping: [⟨1 3 2 3], ⟨0 -8 2 -1]]

Mapping generators: ~2, ~8/7

Optimal tunings:

• WE: ~2 = 1192.090 ¢, ~8/7 = 209.065 ¢
• CWE: ~2 = 1200.000 ¢, ~8/7 = 210.778 ¢

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

Badness (Sintel): 3.905

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 tunings:

• WE: ~2 = 1192.679 ¢, ~8/7 = 209.197 ¢
• CWE: ~2 = 1200.000 ¢, ~8/7 = 210.780 ¢

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

Badness (Sintel): 2.166

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 tunings:

• WE: ~2 = 1194.135 ¢, ~8/7 = 209.943 ¢
• CWE: ~2 = 1200.000 ¢, ~8/7 = 211.116 ¢

Optimal ET sequence: 6, 11, 17c

Badness (Sintel): 2.086