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The [[5-limit]] parent comma for the '''würschmidt family''' is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt | {{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. | |||
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]]/[[nestoria]] and [[gravity]]/[[larry]]. 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 fifths; this is the [[minimax tuning]]. | |||
[[ | [[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning. | ||
== Würschmidt == | |||
{{Main| Würschmidt }} | |||
[[ | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: 393216/390625 | |||
{{Mapping|legend=1| 1 -1 2 | 0 8 1 }} | |||
: mapping generators: ~2, ~5/4 | |||
* [ | [[Optimal tuning]]s: | ||
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.734 | |||
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.799 | |||
{{Optimal ET sequence|legend=1| 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 #Hemiwürschmidt|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]]; furthermore, like practically any 5-limit temperament with this accuracy level of [[3/2]] available, extensions to prime 19 exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo and [[nestoria]]). | |||
[[ | However, as discussed in the main article, the "free" higher prime for würschmidt outside the 5-limit is in fact 23, via tempering out S24 = [[576/575]] and S46<sup>2</sup> × S47 = [[12167/12150]]. Therefore, the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions. | ||
=== 2.3.5.23 subgroup === | |||
Subgroup: 2.3.5.23 | |||
Comma list: 576/575, 12167/12150 | |||
Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~5/4 = 387.734 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.805 | |||
Optimal ET sequence: {{optimal ET sequence| 3, …, 28i, 31, 34, 65, 99, 164 }} | |||
Badness (Smith): 0.00530 | |||
==== 2.3.5.11.23 subgroup ==== | |||
Subgroup: 2.3.5.11.23 | |||
Comma list: 243/242, 276/275, 529/528 | |||
Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }} | |||
Optimal tuning: | |||
* CTE: ~2 = 1\1, ~5/4 = 387.652 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.690 | |||
Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }} | |||
Badness (Smith): 0.00660 | |||
== 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 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 {[[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]], but specifically, the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap| {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap| {S8/9/10/11} }}*. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7 no matter what mapping you use (with the sharp 7 being used for this temperament); even so, it's 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, 65d + 31 = [[96edo]] is also a reasonable choice, as it has the advantage of being [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13. | ||
(<nowiki>*</nowiki> The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).) | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 225/224, 8748/8575 | |||
{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }} | |||
[[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 }} | ||
[[Badness]] (Smith): 0.050776 | |||
[[Badness]]: 0. | |||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 99/98, 176/175, 243/242 | ||
Mapping: | Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }} | ||
POTE | Optimal tunings: | ||
* CTE: ~2 = 1\1, ~5/4 = 387.441 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.447 | |||
Optimal ET sequence: {{optimal ET sequence| 31, 65d, 96, 127 }} | |||
Badness: 0. | Badness (Smith): 0.024413 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 99/98, 144/143, 176/175, 275/273 | ||
Mapping: | Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }} | ||
POTE | Optimal tunings: | ||
* CTE: ~2 = 1\1, ~5/4 = 387.469 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.626 | |||
Optimal ET sequence: {{optimal ET sequence| 31, 65d }} | |||
Badness: 0. | Badness (Smith): 0.023593 | ||
==== | ==== Worseschmidt ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Commas: 66/65, 99/98, 105/104, 243/242 | |||
Mapping: | Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }} | ||
POTE | Optimal tunings: | ||
* CTE: ~2 = 1\1, ~5/4 = 387.179 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.099 | |||
Optimal ET sequence: {{optimal ET sequence| 3def, 28def, 31 }} | |||
Badness: 0. | 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 {{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 | |||
[[Comma list]]: 126/125, 33075/32768 | |||
{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }} | |||
[[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, 96d, 127d }} | |||
[[Badness]] (Smith): 0.064614 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 126/125, 243/242, 385/384 | |||
Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~5/4 = 387.472 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.407 | |||
Optimal ET sequence: {{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|legend=1| 1 -1 2 -14 | 0 8 1 52 }} | |||
[[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 }} | ||
[[Badness]] (Smith): 0.086334 | |||
=== 11-limit === | |||
== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 243/242, 896/891, 4375/4356 | ||
Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }} | |||
Optimal tunings: | |||
* CTE: ~2 = 1\1, ~5/4 = 387.829 | |||
* POTE: ~2 = 1\1, ~5/4 = 387.882 | |||
Optimal ET sequence: {{optimal ET sequence| 34d, 65, 99e }} | |||
Badness (Smith): 0.058325 | |||
[[Category: | [[Category:Temperament families]] | ||
[[Category: | [[Category:Pages with mostly numerical content]] | ||
[[Category:Würschmidt family| ]] <!-- main article --> | [[Category:Würschmidt family| ]] <!-- main article --> | ||
[[Category:Würschmidt| ]] <!-- key article --> | |||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||