Würschmidt family: Difference between revisions

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)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 [[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.

{{Main| Würschmidt }}

[[Comma list]]: 393216/390625

{{Mapping|legend=1| 1 -1 2 | 0 8 1 }}

[[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

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]].  

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 S46² × S47 = [[12167/12150]]. Therefore, the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions.

Subgroup: 2.3.5.23

Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}

* 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

Subgroup: 2.3.5.11.23

Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}

* POTE: ~2 = 1\1, ~5/4 = 387.690

Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }}

Badness (Smith): 0.00660

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.

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 }}

* [[CTE]]: ~2 = 1\1, ~5/4 = 387.379

{{Optimal ET sequence|legend=1| 31, 96, 127 }}

[[Badness]] (Smith): 0.050776

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 243/242

Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}

Optimal tunings:  

* POTE: ~2 = 1\1, ~5/4 = 387.447

Badness (Smith): 0.024413

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 176/175, 275/273

Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}

Optimal tunings:  

* POTE: ~2 = 1\1, ~5/4 = 387.626

Badness (Smith): 0.023593

Subgroup: 2.3.5.7.11.13

Commas: 66/65, 99/98, 105/104, 243/242

Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}

Optimal tunings:  

* CTE: ~2 = 1\1, ~5/4 = 387.179

* POTE: ~2 = 1\1, ~5/4 = 387.099

Badness (Smith): 0.034382

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

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 }}

[[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:  

* [[POTE]]: ~2 = 1\1, ~5/4 = 387.881

{{Optimal ET sequence|legend=1| 34d, 65, 99 }}

[[Badness]] (Smith): 0.086334

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891, 4375/4356

Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}

Optimal tunings:  

* POTE: ~2 = 1\1, ~5/4 = 387.882

Badness (Smith): 0.058325

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