Breedsmic temperaments: Difference between revisions
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{{Main| Hemififths }} | {{Main| Hemififths }} | ||
Hemififths | Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}. | ||
By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice. | By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice. | ||
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[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = | * [[CTE]]: ~2 = 1200.0000, ~49/40 = 351.4464 | ||
* [[POTE]]: ~2 = | : [[error map]]: {{val| 0.0000 +0.9379 -0.1531 -0.0224 }} | ||
* [[POTE]]: ~2 = 1200.0000, ~49/40 = 351.4774 | |||
: error map: {{val| 0.0000 +0.9999 +0.6221 +0.0307 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
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{{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }} | {{Optimal ET sequence|legend=1| 41, 58, 99, 239, 338 }} | ||
[[Badness]]: 0.022243 | [[Badness]] (Smith): 0.022243 | ||
=== 11-limit === | === 11-limit === | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = | * CTE: ~2 = 1200.0000, ~11/9 = 351.4289 | ||
* POTE: ~2 = | * POTE: ~2 = 1200.0000, ~11/9 = 351.5206 | ||
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }} | |||
Badness: 0.023498 | Badness (Smith): 0.023498 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = | * CTE: ~2 = 1200.0000, ~11/9 = 351.4331 | ||
* POTE: ~2 = | * POTE: ~2 = 1200.0000, ~11/9 = 351.5734 | ||
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }} | |||
Badness: 0.019090 | Badness (Smith): 0.019090 | ||
=== Semihemi === | === Semihemi === | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~99/70 = | * CTE: ~99/70 = 600.0000, ~49/40 = 351.4722 | ||
* POTE: ~99/70 = | * POTE: ~99/70 = 600.0000, ~49/40 = 351.5047 | ||
{{Optimal ET sequence|legend=0| 58, 140, 198 }} | |||
Badness: 0.042487 | Badness (Smith): 0.042487 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~99/70 = | * CTE: ~99/70 = 600.0000, ~49/40 = 351.4674 | ||
* POTE: ~99/70 = | * POTE: ~99/70 = 600.0000, ~49/40 = 351.5019 | ||
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }} | |||
Badness: 0.021188 | Badness (Smith): 0.021188 | ||
=== Quadrafifths === | === Quadrafifths === | ||
This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense. | This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = | * CTE: ~2 = 1200.0000, ~243/220 = 175.7284 | ||
* POTE: ~2 = | * POTE: ~2 = 1200.0000, ~243/220 = 175.7378 | ||
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }} | |||
Badness: 0.040170 | Badness (Smith): 0.040170 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
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Optimal tunings: | Optimal tunings: | ||
* CTE: ~2 = | * CTE: ~2 = 1200.0000, ~72/65 = 175.7412 | ||
* POTE: ~2 = | * POTE: ~2 = 1200.0000, ~72/65 = 175.7470 | ||
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }} | |||
Badness: 0.031144 | Badness (Smith): 0.031144 | ||
== Tertiaseptal == | == Tertiaseptal == | ||