Smate family: Difference between revisions
m Removing from Category:Theory using Cat-a-lot |
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
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[[POTE generator]]: ~5/4 = 420.855 | [[POTE generator]]: ~5/4 = 420.855 | ||
{{ | {{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }} | ||
[[Badness]]: 0.178624 | [[Badness]]: 0.178624 | ||
| Line 27: | Line 27: | ||
[[POTE generator]]: ~5/4 = 422.275 | [[POTE generator]]: ~5/4 = 422.275 | ||
{{ | {{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37cd }} | ||
[[Badness]]: 0.077871 | [[Badness]]: 0.077871 | ||
| Line 40: | Line 40: | ||
POTE generator: ~5/4 = 422.217 | POTE generator: ~5/4 = 422.217 | ||
{{ | {{Optimal ET sequence|legend=1| 14, 17c, 37cde }} | ||
Badness: 0.042518 | Badness: 0.042518 | ||
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POTE generator: ~5/4 = 423.020 | POTE generator: ~5/4 = 423.020 | ||
{{ | {{Optimal ET sequence|legend=1| 14, 17c }} | ||
Badness: 0.036836 | Badness: 0.036836 | ||
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[[POTE generator]]: ~8/7 = 210.452 | [[POTE generator]]: ~8/7 = 210.452 | ||
{{ | {{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcd }} | ||
[[Badness]]: 0.154301 | [[Badness]]: 0.154301 | ||
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POTE generator: ~8/7 = 210.481 | POTE generator: ~8/7 = 210.481 | ||
{{ | {{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcde }} | ||
Badness: 0.065528 | Badness: 0.065528 | ||
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POTE generators: ~8/7 = 210.974 | POTE generators: ~8/7 = 210.974 | ||
{{ | {{Optimal ET sequence|legend=1| 6, 11, 17c }} | ||
Badness: 0.050472 | Badness: 0.050472 | ||
Revision as of 18:12, 7 May 2023
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.
Smate
Subgroup: 2.3.5
Comma list: 2048/1875
Mapping: [⟨1 3 2], ⟨0 -4 1]]
POTE generator: ~5/4 = 420.855
Optimal ET sequence: 3, 11, 14, 17c, 20c, 37c
Badness: 0.178624
Septimal smate
Subgroup: 2.3.5.7
Comma list: 36/35, 2048/1875
Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]
Wedgie: ⟨⟨ 4 -1 9 -11 3 24 ]]
POTE generator: ~5/4 = 422.275
Optimal ET sequence: 3d, 11d, 14, 17c, 37cd
Badness: 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]]
POTE generator: ~5/4 = 422.217
Optimal ET sequence: 14, 17c, 37cde
Badness: 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]]
POTE generator: ~5/4 = 423.020
Badness: 0.036836
Hemismate
Subgroup: 2.3.5.7
Comma list: 256/245, 392/375
Mapping: [⟨1 3 2 3], ⟨0 -8 2 -1]]
Wedgie: ⟨⟨ 8 -2 1 -22 -21 8 ]]
POTE generator: ~8/7 = 210.452
Optimal ET sequence: 6, 11, 17c, 40bcd
Badness: 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]]
POTE generator: ~8/7 = 210.481
Optimal ET sequence: 6, 11, 17c, 40bcde
Badness: 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]]
POTE generators: ~8/7 = 210.974
Optimal ET sequence: 6, 11, 17c
Badness: 0.050472