Smate family: Difference between revisions
Created page with "The '''smate family''' of temperaments tempers out 2048/1875, the smate comma, resulting in equation of four just major thirds (5/4) with the 8/3|just perfect el..." |
m fix heading structure |
||
| Line 1: | Line 1: | ||
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 == | ||
[[Comma list]]: 2048/1875 | [[Comma list]]: 2048/1875 | ||
| Line 13: | Line 13: | ||
[[Badness]]: 0.179 | [[Badness]]: 0.179 | ||
= 7-limit = | == 7-limit == | ||
{{see also| Mint temperaments #Smate }} | {{see also| Mint temperaments #Smate }} | ||
| Line 28: | Line 28: | ||
[[Badness]]: 0.0779 | [[Badness]]: 0.0779 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 36/35, 56/55, 243/242 | Comma list: 36/35, 56/55, 243/242 | ||
| Line 40: | Line 40: | ||
Badness: 0.0425 | Badness: 0.0425 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 26/25, 36/35, 56/55, 243/242 | Comma list: 26/25, 36/35, 56/55, 243/242 | ||
| Line 52: | Line 52: | ||
Badness: 0.0368 | Badness: 0.0368 | ||
= Hemismate = | == Hemismate == | ||
[[Comma list]]: 256/245, 392/375 | [[Comma list]]: 256/245, 392/375 | ||
| Line 66: | Line 66: | ||
[[Badness]]: 0.1543 | [[Badness]]: 0.1543 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 77/75, 256/245 | Comma list: 56/55, 77/75, 256/245 | ||
| Line 78: | Line 78: | ||
Badness: 0.0655 | Badness: 0.0655 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 26/25, 56/55, 77/75, 256/245 | Comma list: 26/25, 56/55, 77/75, 256/245 | ||
Revision as of 21:53, 1 June 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
Comma list: 2048/1875
Mapping: [⟨1 3 2], ⟨0 -4 1]]
POTE generator: ~5/4 = 420.855
Badness: 0.179
7-limit
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: 0.0779
11-limit
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: 0.0425
13-limit
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: 0.0368
Hemismate
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: 0.1543
11-limit
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: 0.0655
13-limit
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: 0.0505