Würschmidt family: Difference between revisions
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[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning. | [[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning. | ||
== Würschmidt == | == Würschmidt == | ||
{{ | {{Main| Würschmidt }} | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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{{Mapping|legend=1| 1 -1 2 | 0 8 1 }} | {{Mapping|legend=1| 1 -1 2 | 0 8 1 }} | ||
: mapping generators: ~2, ~5/4 | |||
{{Optimal ET sequence|legend=1| 3, 28, 31, 34, 65, 99, 164, 721c, 885c }} | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.734 | |||
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.799 | |||
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }} | |||
[[Badness]]: 0.040603 | [[Badness]]: 0.040603 | ||
=== Subgroup extensions === | === Overview to extensions === | ||
==== 7-limit extensions ==== | |||
The 7-limit extensions can be obtained by adding another comma. Septimal würschmidt adds [[225/224]], worschmidt adds [[126/125]], whirrschmidt adds [[4375/4374]]. These all use the same generator as 5-limit würschmidt. | |||
Hemiwürschmidt adds [[3136/3125]] and splits the generator in two. This temperament is the best extension available for würschmidt despite its complexity. The details can be found in [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]. | |||
==== Subgroup extensions ==== | |||
Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to [[11/9]], that is by tempering out [[5632/5625]] in addition to [[243/242]]; furthermore, like practically any 5-limit temperament with this accuracy level of [[3/2]] available, extensions to prime 19 exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo and [[nestoria]]). | Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to [[11/9]], that is by tempering out [[5632/5625]] in addition to [[243/242]]; furthermore, like practically any 5-limit temperament with this accuracy level of [[3/2]] available, extensions to prime 19 exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo and [[nestoria]]). | ||
However, as discussed in the main article, the "free" higher prime for würschmidt outside the 5-limit is in fact 23, via tempering out S24 = [[576/575]] and S46<sup>2</sup> × S47 = [[12167/12150]]. Therefore, the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions. | However, as discussed in the main article, the "free" higher prime for würschmidt outside the 5-limit is in fact 23, via tempering out S24 = [[576/575]] and S46<sup>2</sup> × S47 = [[12167/12150]]. Therefore, the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions. | ||
=== 2.3.5.23 subgroup === | |||
Subgroup: 2.3.5.23 | |||
Comma list: 576/575, 12167/12150 | |||
Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~5/4 = 387.734 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.805 | |||
{{ | Optimal ET sequence: {{optimal ET sequence| 3, …, 28i, 31, 34, 65, 99, 164 }} | ||
Badness (Smith): 0.00530 | |||
==== 2.3.5.11.23 subgroup ==== | ==== 2.3.5.11.23 subgroup ==== | ||
Subgroup: 2.3.5.11.23 | |||
Comma list: 243/242, 276/275, 529/528 | |||
Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }} | |||
Optimal tuning: | |||
* CTE: ~2 = 1\1, ~5/4 = 387.652 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.690 | |||
{{ | Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }} | ||
Badness (Smith): 0.00660 | |||
== Septimal würschmidt == | == Septimal würschmidt == | ||