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 9/7's rather than the sharp 5/4's used here.

Smate

Subgroup: 2.3.5

Comma list: 2048/1875

Mapping: [⟨1 -1 3], ⟨0 4 -1]]

Mapping generators: ~2, ~8/5

Optimal tunings:

• WE: ~2 = 1191.8960 ¢, ~8/5 = 773.8834 ¢

error map: ⟨-8.104 +1.683 +15.491]

• CWE: ~2 = 1200.000 ¢, ~8/5 = 778.3632 ¢

error map: ⟨0.000 +11.498 +35.323]

Optimal ET sequence: 3, 11, 14, 17c, 20c, 37cc, 57bccc

Badness (Sintel): 4.19

Septimal smate

Subgroup: 2.3.5.7

Comma list: 36/35, 2048/1875

Mapping: [⟨1 -1 3 -3], ⟨0 4 -1 9]]

Optimal tunings:

• WE: ~2 = 1191.8999 ¢, ~8/5 = 772.4750 ¢

error map: ⟨-8.100 -3.955 +16.911 +7.749]

• CWE: ~2 = 1200.0000 ¢, ~8/5 = 776.9542 ¢

error map: ⟨0.000 +5.862 +36.732 +23.762]

Optimal ET sequence: 3d, 14, 17c, 37ccdd, 54cccdd

Badness (Sintel): 1.97

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 243/242

Mapping: [⟨1 -1 3 -3 -3], ⟨0 4 -1 9 10]]

Optimal tunings:

• WE: ~2 = 1191.9136 ¢, ~8/5 = 772.5419 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/5 = 776.9899 ¢

Optimal ET sequence: 3de, 14, 17c, 37ccddee, 54cccddee

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 243/242

Mapping: [⟨1 -1 3 -3 -3 -5], ⟨0 4 -1 9 10 -2]]

Optimal tunings:

• WE: ~2 = 1194.0453 ¢, ~8/5 = 773.1246 ¢
• CWE: ~2 = 1200.000 ¢, ~8/5 = 776.5983 ¢

Optimal ET sequence: 3de, 14, 17c

Badness (Sintel): 1.52

Hemismate

Subgroup: 2.3.5.7

Comma list: 256/245, 392/375

Mapping: [⟨1 -5 4 2], ⟨0 8 -2 1]]

Mapping generators: ~2, ~7/4

Optimal tunings:

• WE: ~2 = 1192.0901 ¢, ~7/4 = 983.0251 ¢

error map: ⟨-7.910 +1.795 +15.997 -1.620]

• CWE: ~2 = 1200.0000 ¢, ~7/4 = 989.2222 ¢

error map: ⟨0.000 +11.823 +35.242 +20.396]

Optimal ET sequence: 6, 11, 17c, 40bccd, 57bcccd

Badness (Sintel): 3.90

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 256/245

Mapping: [⟨1 -5 4 2 1], ⟨0 8 -2 1 3]]

Optimal tunings:

• WE: ~2 = 1192.6793 ¢, ~8/7 = 983.4825 ¢
• CWE: ~2 = 1200.0000 ¢, ~8/7 = 989.2203 ¢

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

Badness (Sintel): 2.17

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 77/75, 256/245

Mapping: [⟨1 -5 4 2 1 7], ⟨0 8 -2 1 3 -4]]

Optimal tunings:

• WE: ~2 = 1194.1349 ¢, ~7/4 = 984.1921 ¢
• CWE: ~2 = 1200.0000 ¢, ~7/4 = 988.8836 ¢

Optimal ET sequence: 6, 11, 17c

Badness (Sintel): 2.09