Smate family: Difference between revisions
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m Spell 1\1 in decimal; note the badness metric being used; misc. cleanup |
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{{Mapping|legend=1| 1 3 2 | 0 -4 1 }} | {{Mapping|legend=1| 1 3 2 | 0 -4 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | : mapping generators: ~2, ~5/4 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 420.855 | |||
{{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }} | {{Optimal ET sequence|legend=1| 3, 11, 14, 17c, 20c, 37c }} | ||
[[Badness]]: 0.178624 | [[Badness]] (Smith): 0.178624 | ||
== Septimal smate == | == Septimal smate == | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 25: | Line 25: | ||
{{Multival|legend=1| 4 -1 9 -11 3 24 }} | {{Multival|legend=1| 4 -1 9 -11 3 24 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 422.275 | ||
{{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37ccdd }} | {{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37ccdd }} | ||
[[Badness]]: 0.077871 | [[Badness]] (Smith): 0.077871 | ||
=== 11-limit === | === 11-limit === | ||
| Line 38: | Line 38: | ||
Mapping: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }} | Mapping: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 422.217 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 3de, 14, 17c, 37ccddee }} | ||
Badness: 0.042518 | Badness (Smith): 0.042518 | ||
=== 13-limit === | === 13-limit === | ||
| Line 51: | Line 51: | ||
Mapping: {{mapping| 1 3 2 6 7 3 | 0 -4 1 -9 -10 2 }} | Mapping: {{mapping| 1 3 2 6 7 3 | 0 -4 1 -9 -10 2 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 423.020 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 3de, 14, 17c }} | ||
Badness: 0.036836 | Badness (Smith): 0.036836 | ||
== Hemismate == | == Hemismate == | ||
| Line 66: | Line 66: | ||
{{Multival|legend=1| 8 -2 1 -22 -21 8 }} | {{Multival|legend=1| 8 -2 1 -22 -21 8 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 210.452 | ||
{{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcd }} | {{Optimal ET sequence|legend=1| 6, 11, 17c, 40bcd }} | ||
[[Badness]]: 0.154301 | [[Badness]] (Smith): 0.154301 | ||
=== 11-limit === | === 11-limit === | ||
| Line 79: | Line 79: | ||
Mapping: {{mapping| 1 3 2 3 4 | 0 -8 2 -1 -3 }} | Mapping: {{mapping| 1 3 2 3 4 | 0 -8 2 -1 -3 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 210.481 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 6, 11, 17c, 40bcde }} | ||
Badness: 0.065528 | Badness (Smith): 0.065528 | ||
=== 13-limit === | === 13-limit === | ||
| Line 92: | Line 92: | ||
Mapping: {{mapping| 1 3 2 3 4 3 | 0 -8 2 -1 -3 4 }} | Mapping: {{mapping| 1 3 2 3 4 3 | 0 -8 2 -1 -3 4 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 210.974 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 6, 11, 17c }} | ||
Badness: 0.050472 | Badness (Smith): 0.050472 | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Smate family| ]] <!-- main article --> | [[Category:Smate family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Revision as of 11:33, 1 January 2025
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]]
- 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]]
Wedgie: ⟨⟨ 4 -1 9 -11 3 24 ]]
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]]
Wedgie: ⟨⟨ 8 -2 1 -22 -21 8 ]]
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