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Würschmidt family: Difference between revisions

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== Würschmidt ==
== Würschmidt ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 393216/390625
[[Comma list]]: 393216/390625


[[Mapping]]: [{{Val|1 7 3}}, {{Val|0 -8 -1}}]
{{Mapping|legend=1| 1 7 3 | 0 -8 -1 }}


[[POTE generator]]: ~5/4 = 387.799
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.799


{{Optimal ET sequence|legend=1|3, 28, 31, 34, 65, 99, 164, 721c, 885c }}
{{Optimal ET sequence|legend=1|3, 28, 31, 34, 65, 99, 164, 721c, 885c }}
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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.


Subgroup: 2.3.5.7
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.


[[Comma list]]: [[225/224]], 8748/8575
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 7 3 15 }}, {{Val| 0 -8 -1 -18 }}]
[[Comma list]]: 225/224, 8748/8575
 
{{Mapping|legend=1| 1 7 3 15 | 0 -8 -1 -18 }}


{{Multival|legend=1| 8 1 18 -17 6 39 }}
{{Multival|legend=1| 8 1 18 -17 6 39 }}


[[POTE generator]]: ~5/4 = 387.383
[[Optimal tuning]] ([[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, 285bd, 412bbdd }}
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Comma list: 99/98, 176/175, 243/242
Comma list: 99/98, 176/175, 243/242


Mapping: [{{val| 1 7 3 15 17 }}, {{val| 0 -8 -1 -18 -20 }}]
Mapping: {{mapping| 1 7 3 15 17 | 0 -8 -1 -18 -20 }}


POTE generator: ~5/4 = 387.447
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.447


{{Optimal ET sequence|legend=1| 31, 65d, 96, 127, 223d }}
{{Optimal ET sequence|legend=1| 31, 65d, 96, 127, 223d }}
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Comma list: 99/98, 144/143, 176/175, 275/273
Comma list: 99/98, 144/143, 176/175, 275/273


Mapping: [{{val| 1 7 3 15 17 1 }}, {{val| 0 -8 -1 -18 -20 4 }}]
Mapping: {{mapping| 1 7 3 15 17 1 | 0 -8 -1 -18 -20 4 }}


POTE generator: ~5/4 = 387.626
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.626


{{Optimal ET sequence|legend=1| 31, 65d, 161df }}
{{Optimal ET sequence|legend=1| 31, 65d, 161df }}
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Commas: 66/65, 99/98, 105/104, 243/242
Commas: 66/65, 99/98, 105/104, 243/242


Mapping: [{{val| 1 7 3 15 17 22 }}, {{val| 0 -8 -1 -18 -20 -27 }}]
Mapping: {{mapping| 1 7 3 15 17 22 | 0 -8 -1 -18 -20 -27 }}


POTE generator: ~5/4 = 387.099
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.099


{{Optimal ET sequence|legend=1| 3def, 28def, 31 }}
{{Optimal ET sequence|legend=1| 3def, 28def, 31 }}
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Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo]], [[34edo]], or [[127edo]] as a tuning. If 127 is used, note that the val is {{val| 127 201 295 '''356''' }} (127d) and not {{val| 127 201 295 '''357''' }} as with würschmidt. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.
Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo]], [[34edo]], or [[127edo]] as a tuning. If 127 is used, note that the val is {{val| 127 201 295 '''356''' }} (127d) and not {{val| 127 201 295 '''357''' }} as with würschmidt. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 33075/32768
[[Comma list]]: 126/125, 33075/32768


[[Mapping]]: [{{val| 1 7 3 -6 }}, {{val| 0 -8 -1 13 }}]
{{Mapping|legend=1| 1 7 3 -6 | 0 -8 -1 13 }}


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


[[POTE generator]]: ~5/4 = 387.392
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.392


{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
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Comma list: 126/125, 243/242, 385/384
Comma list: 126/125, 243/242, 385/384


Mapping: [{{val| 1 7 3 -6 17 }}, {{val| 0 -8 -1 13 -20 }}]
Mapping: {{mapping| 1 7 3 -6 17 | 0 -8 -1 13 -20 }}


POTE generator: ~5/4 = 387.407
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.407


{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}
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[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with 7 mapped to the 52nd generator step.  
[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with 7 mapped to the 52nd generator step.  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 393216/390625
[[Comma list]]: 4375/4374, 393216/390625


[[Mapping]]: [{{val| 1 7 3 38 }}, {{val| 0 -8 -1 -52 }}]
{{Mapping|legend=1| 1 7 3 38 | 0 -8 -1 -52 }}


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


[[POTE generator]]: ~5/4 = 387.881
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.881


{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
{{Optimal ET sequence|legend=1| 34d, 65, 99 }}
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Comma list: 243/242, 896/891, 4375/4356
Comma list: 243/242, 896/891, 4375/4356


Mapping: [{{val| 1 7 3 38 17 }}, {{val| 0 -8 -1 -52 -20 }}]
Mapping: {{mapping| 1 7 3 38 17 | 0 -8 -1 -52 -20 }}


POTE generator: ~5/4 = 387.882
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.882


{{Optimal ET sequence|legend=1| 34d, 65, 99e }}
{{Optimal ET sequence|legend=1| 34d, 65, 99e }}
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Badness: 0.029545
Badness: 0.029545
== Relationships to other temperaments ==
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 temperament.


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt family| ]] <!-- main article -->
[[Category:Würschmidt| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Würschmidt|#]] <!-- list on top of cat -->
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