Würschmidt family: Difference between revisions
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{{Technical data page}} | {{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 | 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 [[5/4|classical major third]], and to get to the interval class of 3 requires eight of these. In fact, (5/4)<sup>8</sup> × 393216/390625 = 6. | ||
Similar to [[meantone]], würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]], which notably is the point where it is combined with [[schismic | Similar to [[meantone]], würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]]. | ||
[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning. | |||
[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[ | |||
== Würschmidt == | == Würschmidt == | ||
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==== Subgroup extensions ==== | ==== 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]] | 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]]. | ||
With this accuracy level of [[3/2]] available, extensions that add prime [[19/1|19]] exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo), but they are very complex. | |||
The "free" higher prime for würschmidt outside the 5-limit is in fact [[23/1|23]], via tempering out [[576/575]] ({{S|24}}) and [[12167/12150]] ([[S-expression|S46<sup>2</sup>⋅S47]]). This is considered immediately below, and the no-11 restriction thereof is considered in [[#Other subgroup extensions]]. | |||
==== 2.3.5.11.23 subgroup ==== | ==== 2.3.5.11.23 subgroup ==== | ||
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== Septimal würschmidt == | == Septimal würschmidt == | ||
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 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|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]] | 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. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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== Worschmidt == | == Worschmidt == | ||
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 | ||
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Badness (Smith): 0.058325 | Badness (Smith): 0.058325 | ||
== Other subgroup extensions == | |||
=== Würschmidt (2.3.5.23) === | |||
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 | |||
Comma list: 576/575, 12167/12150 | |||
Subgroup-val mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1200.000{{c}}, ~5/4 = 387.734{{c}} | |||
* POTE: ~2 = 1200.000{{c}}, ~5/4 = 387.805{{c}} | |||
{{Optimal ET sequence|legend=0| 3, …, 28i, 31, 34, 65, 99, 164 }} | |||
Badness (Smith): 0.00530 | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
Revision as of 07:38, 9 September 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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 classical major third, and to get to the interval class of 3 requires eight of these. In fact, (5/4)8 × 393216/390625 = 6.
Similar to meantone, würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives 31edo as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are 11\34 and especially 21\65, which notably is the point where it is combined with schismic and gravity. Other edo tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the minimax tuning for the 5-odd-limit.
Mos scales may not be the best approach for würschmidt since they are even more extreme than those of magic. Rothenberg-proper scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.
Würschmidt
Subgroup: 2.3.5
Comma list: 393216/390625
Mapping: [⟨1 -1 2], ⟨0 8 1]]
- mapping generators: ~2, ~5/4
Optimal ET sequence: 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc
Badness (Smith): 0.040603
Overview to extensions
7-limit extensions
The 7-limit extensions can be obtained by adding another comma. Septimal würschmidt adds 225/224, worschmidt adds 126/125, whirrschmidt adds 4375/4374. These all use the same generator as 5-limit würschmidt.
Hemiwürschmidt adds 3136/3125 and splits the generator in two. This temperament is the best extension available for würschmidt despite its complexity. The details can be found in Hemimean clan.
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.
With this accuracy level of 3/2 available, extensions that add prime 19 exist by tempering out either 513/512 or 1216/1215 (which meet at 65edo), but they are very complex.
The "free" higher prime for würschmidt outside the 5-limit is in fact 23, via tempering out 576/575 (S24) and 12167/12150 (S462⋅S47). This is considered immediately below, and the no-11 restriction thereof is considered in #Other subgroup extensions.
2.3.5.11.23 subgroup
Subgroup: 2.3.5.11.23
Comma list: 243/242, 276/275, 529/528
Subgroup-val mapping: [⟨1 -1 2 -3 0], ⟨0 8 1 20 14]]
Optimal tuning:
- CTE: ~2 = 1200.000 ¢, ~5/4 = 387.652 ¢
- POTE: ~2 = 1200.000 ¢, ~5/4 = 387.690 ¢
Optimal ET sequence: 31, 34, 65
Badness (Smith): 0.00660
Septimal würschmidt
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 subgroup temperament.
The S-expression-based comma list of the 11-limit würschmidt discussed here is {S8/S10, S9/S11, S15}. Tempering out S9 or S11 results in 31edo, and in complementary fashion, tempering out S8 or 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 superparticulars 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.
Subgroup: 2.3.5.7
Comma list: 225/224, 8748/8575
Mapping: [⟨1 -1 2 -3], ⟨0 8 1 18]]
Optimal ET sequence: 31, 96, 127
Badness (Smith): 0.050776
11-limit
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 243/242
Mapping: [⟨1 -1 2 -3 -3], ⟨0 8 1 18 20]]
Optimal tunings:
- CTE: ~2 = 1200.000 ¢, ~5/4 = 387.441 ¢
- POTE: ~2 = 1200.000 ¢, ~5/4 = 387.447 ¢
Optimal ET sequence: 31, 65d, 96, 127
Badness (Smith): 0.024413
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 144/143, 176/175, 275/273
Mapping: [⟨1 -1 2 -3 -3 5], ⟨0 8 1 18 20 -4]]
Optimal tunings:
- CTE: ~2 = 1200.000 ¢, ~5/4 = 387.469 ¢
- POTE: ~2 = 1200.000 ¢, ~5/4 = 387.626 ¢
Badness (Smith): 0.023593
Worseschmidt
Subgroup: 2.3.5.7.11.13
Commas: 66/65, 99/98, 105/104, 243/242
Mapping: [⟨1 -1 2 -3 -3 -5], ⟨0 8 1 18 20 27]]
Optimal tunings:
- CTE: ~2 = 1200.000 ¢, ~5/4 = 387.179 ¢
- POTE: ~2 = 1200.000 ¢, ~5/4 = 387.099 ¢
Optimal ET sequence: 3def, 28def, 31
Badness (Smith): 0.034382
Worschmidt
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 ⟨127 201 295 356] (127d) and not ⟨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
Comma list: 126/125, 33075/32768
Mapping: [⟨1 -1 2 7], ⟨0 8 1 -13]]
Optimal ET sequence: 31, 96d, 127d
Badness (Smith): 0.064614
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 243/242, 385/384
Mapping: [⟨1 -1 2 7 -3], ⟨0 8 1 -13 20]]
Optimal tunings:
- CTE: ~2 = 1200.000 ¢, ~5/4 = 387.472 ¢
- POTE: ~2 = 1200.000 ¢, ~5/4 = 387.407 ¢
Optimal ET sequence: 31, 65, 96d, 127d
Badness (Smith): 0.033436
Whirrschmidt
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
Comma list: 4375/4374, 393216/390625
Mapping: [⟨1 -1 2 -14], ⟨0 8 1 52]]
Optimal ET sequence: 34d, 65, 99
Badness (Smith): 0.086334
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 896/891, 4375/4356
Mapping: [⟨1 -1 2 -14 -3], ⟨0 8 1 52 20]]
Optimal tunings:
- CTE: ~2 = 1200.000 ¢, ~5/4 = 387.829
- POTE: ~2 = 1200.000 ¢, ~5/4 = 387.882
Optimal ET sequence: 34d, 65, 99e
Badness (Smith): 0.058325
Other subgroup extensions
Würschmidt (2.3.5.23)
Extensions to harmonics 47 and 49 are also available at +11 and +5 generator steps respectively, equalizing the 45::50 segment of the harmonic series. If we derive a mapping of prime 7 from this mapping of 49, then we get the weak extension hemiwürschmidt.
Subgroup: 2.3.5.23
Comma list: 576/575, 12167/12150
Subgroup-val mapping: [⟨1 -1 2 0], ⟨0 8 1 14]]
Optimal tunings:
- CTE: ~2 = 1200.000 ¢, ~5/4 = 387.734 ¢
- POTE: ~2 = 1200.000 ¢, ~5/4 = 387.805 ¢
Optimal ET sequence: 3, …, 28i, 31, 34, 65, 99, 164
Badness (Smith): 0.00530