Smate family: Difference between revisions
Normalize mappings. Review optimal ET sequences. |
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Comma list: 26/25, 36/35, 56/55, 243/242 | Comma list: 26/25, 36/35, 56/55, 243/242 | ||
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Comma list: 56/55, 77/75, 256/245 | Comma list: 56/55, 77/75, 256/245 | ||
{{Mapping|legend=0| 1 -5 4 2 1 | 0 8 -2 1 3 }} | |||
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Comma list: 26/25, 56/55, 77/75, 256/245 | Comma list: 26/25, 56/55, 77/75, 256/245 | ||
{{Mapping|legend=0| 1 -5 4 2 1 7 | 0 8 -2 1 3 -4 }} | |||
Optimal tunings: | Optimal tunings: | ||
Latest revision as of 12:33, 24 September 2026
- 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
- 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]]
- 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
- 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