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Würschmidt family: Difference between revisions

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{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}
{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}


[[Badness]]: 0.040603
[[Badness]] (Smith): 0.040603


=== Overview to extensions ===
=== Overview to extensions ===
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== Septimal würschmidt ==
== Septimal würschmidt ==
Würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version {{Multival| 8 1 18 20 … }} which also tempers out 99/98, 176/175 and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for minerva, the 11-limit rank-3 temperament tempering out 99/98 and 176/175.
Würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version {{Multival| 8 1 18 20 … }} which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.


2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 subgroup temperament.
2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 subgroup temperament.
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{{Multival|legend=1| 8 1 18 -17 6 39 }}
{{Multival|legend=1| 8 1 18 -17 6 39 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.383
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.379
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.383


{{Optimal ET sequence|legend=1| 31, 96, 127, 285bd, 412bbdd }}
{{Optimal ET sequence|legend=1| 31, 96, 127 }}


[[Badness]]: 0.050776
[[Badness]] (Smith): 0.050776


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}
Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.447
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.441
* POTE: ~2 = 1\1, ~5/4 = 387.447


{{Optimal ET sequence|legend=1| 31, 65d, 96, 127, 223d }}
Optimal ET sequence: {{optimal ET sequence| 31, 65d, 96, 127 }}


Badness: 0.024413
Badness (Smith): 0.024413


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}
Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.626
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.469
* POTE: ~2 = 1\1, ~5/4 = 387.626


{{Optimal ET sequence|legend=1| 31, 65d, 161df }}
Optimal ET sequence: {{optimal ET sequence| 31, 65d }}


Badness: 0.023593
Badness (Smith): 0.023593


==== Worseschmidt ====
==== Worseschmidt ====
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Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}
Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.099
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.179
* POTE: ~2 = 1\1, ~5/4 = 387.099


{{Optimal ET sequence|legend=1| 3def, 28def, 31 }}
Optimal ET sequence: {{optimal ET sequence| 3def, 28def, 31 }}


Badness: 0.034382
Badness (Smith): 0.034382


== Worschmidt ==
== Worschmidt ==
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{{Multival|legend=1| 8 1 -13 -17 -43 -33 }}
{{Multival|legend=1| 8 1 -13 -17 -43 -33 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.392
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.406
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.392


{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
{{Optimal ET sequence|legend=1| 31, 96d, 127d }}


[[Badness]]: 0.064614
[[Badness]] (Smith): 0.064614


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}
Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.407
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.472
* POTE: ~2 = 1\1, ~5/4 = 387.407


{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
Optimal ET sequence: {{optimal ET sequence| 31, 65, 96d, 127d }}


Badness: 0.033436
Badness (Smith): 0.033436


== Whirrschmidt ==
== Whirrschmidt ==
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{{Multival|legend=1| 8 1 52 -17 60 118 }}
{{Multival|legend=1| 8 1 52 -17 60 118 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.881
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.853
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.881


{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}


[[Badness]]: 0.086334
[[Badness]] (Smith): 0.086334


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}
Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.882
Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.829
* POTE: ~2 = 1\1, ~5/4 = 387.882


{{Optimal ET sequence|legend=1| 34d, 65, 99e }}
Optimal ET sequence: {{optimal ET sequence| 34d, 65, 99e }}


Badness: 0.058325
Badness (Smith): 0.058325


[[Category:Temperament families]]
[[Category:Temperament families]]
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