Würschmidt family: Difference between revisions
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Comma list: 243/242, 5632/5625 | Comma list: 243/242, 5632/5625 | ||
{{Mapping|legend=2| 1 -1 2 -3 | 0 8 1 20 }} | |||
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Comma list: 243/242, 276/275, 529/528 | Comma list: 243/242, 276/275, 529/528 | ||
{{Mapping|legend=2| 1 -1 2 -3 0 | 0 8 1 20 14 }} | |||
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Septimal 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. | Septimal 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 | 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|2.3.7.11-subgroup]] temperament. | ||
The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13. | The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13. | ||
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Comma list: 99/98, 176/175, 243/242 | Comma list: 99/98, 176/175, 243/242 | ||
{{Mapping|legend=0| 1 -1 2 -3 -3 | 0 8 1 18 20 }} | |||
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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|legend=0| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }} | |||
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Commas: 66/65, 99/98, 105/104, 243/242 | Commas: 66/65, 99/98, 105/104, 243/242 | ||
{{Mapping|legend=0| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }} | |||
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Comma list: 126/125, 243/242, 385/384 | Comma list: 126/125, 243/242, 385/384 | ||
{{Mapping|legend=0| 1 -1 2 7 -3 | 0 8 1 -13 20 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 243/242, 896/891, 4375/4356 | Comma list: 243/242, 896/891, 4375/4356 | ||
{{Mapping|legend=0| 1 -1 2 -14 -3 | 0 8 1 52 20 }} | |||
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== Other subgroup extensions == | == Other subgroup extensions == | ||
=== Würschmidt (2.3.5.23) === | === Würschmidt (2.3.5.23) === | ||
Extensions | Extensions for 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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Comma list: 576/575, 12167/12150 | Comma list: 576/575, 12167/12150 | ||
{{Mapping|legend=2| 1 -1 2 0 | 0 8 1 14 }} | |||
Optimal tunings: | Optimal tunings: | ||