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. Its [[monzo]] is {{monzo| 17 1 -8 }}, and flipping that yields {{multival| 8 1 17 }} for the wedgie. This tells us the [[generator]] is a classic major third, and that to get to the interval class of fifths will require 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.  

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]].  

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.799

{{Optimal ET sequence|legend=1| 3, 28, 31, 34, 65, 99, 164, 721c, 885c }}

[[Badness]]: 0.040603

=== 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 S46² × 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

{{Mapping|legend=1| 1 -1 2 0 | 0 8 1 14 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.805

[[Subgroup]]: 2.3.5.11.23

[[Comma list]]: 243/242, 276/275, 529/528

{{Mapping|legend=1| 1 -1 2 -3 0 | 0 8 1 20 14 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.690

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 {{Multival| 8 1 18 20 … }} 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.

{{Multival|legend=1| 8 1 18 -17 6 39 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.383

 

[[Badness]]: 0.050776

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.447

{{Optimal ET sequence|legend=1| 31, 65d, 96, 127, 223d }}

Badness: 0.024413

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.626

{{Optimal ET sequence|legend=1| 31, 65d, 161df }}

Badness: 0.023593

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.099

{{Optimal ET sequence|legend=1| 3def, 28def, 31 }}

Badness: 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.

{{Multival|legend=1| 8 1 -13 -17 -43 -33 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.392

 

[[Badness]]: 0.064614

Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.407

{{Optimal ET sequence|legend=1| 31, 65, 96d, 127d }}

Badness: 0.033436

{{Mapping|legend=1| 1 7 3 38 | 0 -8 -1 -52 }}

{{Multival|legend=1| 8 1 52 -17 60 118 }}

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 387.881

[[Badness]]: 0.086334

Mapping: {{mapping| 1 7 3 38 17 | 0 -8 -1 -52 -20 }}

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 387.882

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

Badness: 0.058325

[[Category:Rank 2]]