Würschmidt family
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 [17 1 -8⟩, and flipping that yields ⟨⟨ 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 11\34 and especially 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 scales 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
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; 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
Sval 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: 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: [⟨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: 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 ⟨⟨ 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.
Subgroup: 2.3.5.7
Comma list: 225/224, 8748/8575
Mapping: [⟨1 -1 2 -3], ⟨0 8 1 18]]
Wedgie: ⟨⟨ 8 1 18 -17 6 39 ]]
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 = 1\1, ~5/4 = 387.441
- POTE: ~2 = 1\1, ~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 = 1\1, ~5/4 = 387.469
- POTE: ~2 = 1\1, ~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 = 1\1, ~5/4 = 387.179
- POTE: ~2 = 1\1, ~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]]
Wedgie: ⟨⟨ 8 1 -13 -17 -43 -33 ]]
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 = 1\1, ~5/4 = 387.472
- POTE: ~2 = 1\1, ~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]]
Wedgie: ⟨⟨ 8 1 52 -17 60 118 ]]
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 = 1\1, ~5/4 = 387.829
- POTE: ~2 = 1\1, ~5/4 = 387.882
Optimal ET sequence: 34d, 65, 99e
Badness (Smith): 0.058325