Würschmidt family: Difference between revisions
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mapping generator was 8/5 for some reason instead of 5/4 |
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[[Comma list]]: 393216/390625 | [[Comma list]]: 393216/390625 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 -1 2 | 0 8 1 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.799 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.799 | ||
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[[Badness]]: 0.040603 | [[Badness]]: 0.040603 | ||
=== 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]]). | |||
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 | |||
{{Mapping|legend=1| 1 -1 2 0 | 0 8 1 14 }} | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.805 | |||
==== 2.3.5.11.23 subgroup ==== | |||
[[Subgroup]]: 2.3.5.11.23 | |||
[[Comma list]]: 243/242, 276/275, 529/528 | |||
{{Mapping|legend=1| 1 -1 2 -3 0 | 0 8 1 20 14 }} | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.690 | |||
== Septimal würschmidt == | == Septimal würschmidt == | ||