Würschmidt family: Difference between revisions
m →Septimal würschmidt: note 96edo as the smallest patent tuning above 31edo, move explanation of shortened form to avoid diverting focus |
→2.3.5.23 subgroup: note further extensions |
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{{Technical data page}} | |||
The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. The [[generator]] is a classic major third, and to get to the interval class of fifths requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6. | The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. The [[generator]] is a classic major third, and to get to the interval class of fifths requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6. | ||
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=== 2.3.5.23 subgroup === | === 2.3.5.23 subgroup === | ||
Extensions to harmonics [[47/1|47]] and [[49/1|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]]. | |||
Subgroup: 2.3.5.23 | Subgroup: 2.3.5.23 | ||
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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 | 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 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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{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }} | {{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }} | {{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }} | {{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||