No-twos subgroup temperaments: Difference between revisions
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[[Support]]ing [[ET]]s: {{EDs|b131, b22, b153, b284, b415, b109, b437, b175, b546, b87c, b699, b240, b590, b721|equave=t}} | [[Support]]ing [[ET]]s: {{EDs|b131, b22, b153, b284, b415, b109, b437, b175, b546, b87c, b699, b240, b590, b721|equave=t}} | ||
=== Adhara === | |||
Adhara cleaves the step of Mebsuta in three to produce a remarkable Don Page temperament for the chord 7:9:11:13:17 (that is, setting [[13/11]] to two-thirds of 9/7, and [[17/13]] to four-thirds of 11/9). It can be extended to even higher subgroups fairly naturally, and encompasses several prominent tunings within its structure (such as [[65edt]]~[[41edo]], [[131edt]], and [[197edt]]). | |||
[[Subgroup]]: 3.7.11.13.17 | |||
[[Comma list]]: 14161/14157, 107811/107653, 1108809/1108723 | |||
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2}}, {{val| 0 -15 12 22 38}}] | |||
Sval mapping generators: ~3, ~119/117 | |||
[[Optimal tuning]]s: | |||
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[119/117]] = 28.979 | |||
* [[CTE]]: ~3 = 1\1ed3, ~[[119/117]] = 28.970 | |||
[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247f]], [[3216edt|b3216defff]] | |||
==== 3.7.11.13.17.19 subgroup ==== | |||
This includes the natural extension of Mebsuta to prime 19. | |||
[[Subgroup]]: 3.7.11.13.17.19 | |||
[[Comma list]]: 3213/3211, 3971/3969, 14161/14157, 41553/41503 | |||
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -15 12 22 38 -21}}] | |||
Sval mapping generators: ~3, ~119/117 | |||
[[Optimal tuning]]s: | |||
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[119/117]] = 28.973 | |||
* [[CTE]]: ~3 = 1\1ed3, ~[[119/117]] = 28.970 | |||
[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[525edt|b525]], [[722edt|b722]], [[919edt|b919]], [[2035edt|b2035df]] | |||
==== 3.7.8.11.13.17.19 subgroup ==== | |||
This sets two-thirds of 11/9 to [[8/7]]. | |||
[[Subgroup]]: 3.7.8.11.13.17.19 | |||
[[Comma list]]: 513/512, 729/728, 833/832, 969/968, 3971/3969 | |||
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -15 -7 12 22 38 -21}}] | |||
Sval mapping generators: ~3, ~64/63 | |||
[[Optimal tuning]]s: | |||
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[64/63]] = 28.978 | |||
* [[CTE]]: ~3 = 1\1ed3, ~[[64/63]] = 28.975 | |||
[[Optimal ET sequence]]: [[65edt|b65]], [[66edt|b66]], [[131edt|b131]], [[197edt|b197]], [[328edt|b328]], [[525edt|b525]], [[722edt|b722]], [[1247edt|b1247âf]], [[1969edt|b1969ââf]] | |||
(â is the wart for 8.) | |||
==== 3.5.7.8.11.13.17.19.23 subgroup ==== | |||
At slightly lower accuracy, [[Procyon]] can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit. | |||
[[Subgroup]]: 3.5.7.8.11.13.17.19.23 | |||
[[Comma list]]: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125 | |||
[[Sval]] [[mapping]]: [{{val| 1 2 2 2 2 2}}, {{val| 0 -35 -15 -7 12 22 38 -21 -75}}] | |||
Sval mapping generators: ~3, ~64/63 | |||
[[Optimal tuning]]s: | |||
* PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~[[64/63]] = 29.032 | |||
* [[CTE]]: ~3 = 1\1ed3, ~[[64/63]] = 29.041 | |||
[[Optimal ET sequence]]: [[65edt|b65i]], [[66edt|b66i]], [[131edt|b131]] | |||
= Other tritave-based subgroups = | = Other tritave-based subgroups = | ||