Smate family: Difference between revisions
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The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4|just major thirds (5/4)]] with the [[8/3|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. | The '''smate family''' of temperaments tempers out [[2048/1875]], the smate comma, resulting in equation of four [[5/4|just major thirds (5/4)]] with the [[8/3|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 | == Smate == | ||
Subgroup: 2.3.5 | |||
[[Comma list]]: 2048/1875 | [[Comma list]]: 2048/1875 | ||
| Line 11: | Line 12: | ||
{{Val list|legend=1| 3, 11, 14, 17c, 20c, 37c }} | {{Val list|legend=1| 3, 11, 14, 17c, 20c, 37c }} | ||
[[Badness]]: 0. | [[Badness]]: 0.178624 | ||
== | == Septimal smate == | ||
{{see also| Mint temperaments #Smate }} | {{see also| Mint temperaments #Smate }} | ||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 36/35, 2048/1875 | [[Comma list]]: 36/35, 2048/1875 | ||
| Line 26: | Line 29: | ||
{{Val list|legend=1| 3d, 11d, 14, 17c, 37cd }} | {{Val list|legend=1| 3d, 11d, 14, 17c, 37cd }} | ||
[[Badness]]: 0. | [[Badness]]: 0.077871 | ||
=== 11-limit | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 36/35, 56/55, 243/242 | Comma list: 36/35, 56/55, 243/242 | ||
| Line 38: | Line 42: | ||
{{Val list|legend=1| 14, 17c, 37cde }} | {{Val list|legend=1| 14, 17c, 37cde }} | ||
Badness: 0. | Badness: 0.042518 | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 26/25, 36/35, 56/55, 243/242 | Comma list: 26/25, 36/35, 56/55, 243/242 | ||
| Line 50: | Line 55: | ||
{{Val list|legend=1| 14, 17c }} | {{Val list|legend=1| 14, 17c }} | ||
Badness: 0. | Badness: 0.036836 | ||
== Hemismate | == Hemismate == | ||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 256/245, 392/375 | [[Comma list]]: 256/245, 392/375 | ||
| Line 64: | Line 70: | ||
{{Val list|legend=1| 6, 11, 17c, 40bcd }} | {{Val list|legend=1| 6, 11, 17c, 40bcd }} | ||
[[Badness]]: 0. | [[Badness]]: 0.154301 | ||
=== 11-limit | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 56/55, 77/75, 256/245 | Comma list: 56/55, 77/75, 256/245 | ||
| Line 76: | Line 83: | ||
{{Val list|legend=1| 6, 11, 17c, 40bcde }} | {{Val list|legend=1| 6, 11, 17c, 40bcde }} | ||
Badness: 0. | Badness: 0.065528 | ||
=== 13-limit | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 26/25, 56/55, 77/75, 256/245 | Comma list: 26/25, 56/55, 77/75, 256/245 | ||
| Line 88: | Line 96: | ||
{{Val list|legend=1| 6, 11, 17c }} | {{Val list|legend=1| 6, 11, 17c }} | ||
Badness: 0. | Badness: 0.050472 | ||
[[Category:Theory]] | [[Category:Theory]] | ||
Revision as of 11:38, 10 December 2021
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
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
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
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
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
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
Badness: 0.050472