Würschmidt family

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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. 65edo is the point where it is combined with schismic/nestoria and gravity/larry, and that suggests an extension for prime 11 through 243/242 and prime 19 through 513/512 or 1216/1215. 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.

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.

Würschmidt

Subgroup: 2.3.5

Comma list: 393216/390625

Mapping[1 7 3], 0 -8 -1]]

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

Optimal ET sequence3, 28, 31, 34, 65, 99, 164, 721c, 885c

Badness: 0.040603

Music

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 7 3 15], 0 -8 -1 -18]]

Wedgie⟨⟨8 1 18 -17 6 39]]

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

Optimal ET sequence31, 96, 127, 285bd, 412bbdd

Badness: 0.050776

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [1 7 3 15 17], 0 -8 -1 -18 -20]]

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

Optimal ET sequence31, 65d, 96, 127, 223d

Badness: 0.024413

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 7 3 15 17 1], 0 -8 -1 -18 -20 4]]

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

Optimal ET sequence31, 65d, 161df

Badness: 0.023593

Worseschmidt

Subgroup: 2.3.5.7.11.13

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

Mapping: [1 7 3 15 17 22], 0 -8 -1 -18 -20 -27]]

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

Optimal ET sequence3def, 28def, 31

Badness: 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 7 3 -6], 0 -8 -1 13]]

Wedgie⟨⟨8 1 -13 -17 -43 -33]]

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

Optimal ET sequence31, 65, 96d, 127d

Badness: 0.064614

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 385/384

Mapping: [1 7 3 -6 17], 0 -8 -1 13 -20]]

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

Optimal ET sequence31, 65, 96d, 127d

Badness: 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 7 3 38], 0 -8 -1 -52]]

Wedgie⟨⟨8 1 52 -17 60 118]]

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

Optimal ET sequence34d, 65, 99

Badness: 0.086334

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [1 7 3 38 17], 0 -8 -1 -52 -20]]

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

Optimal ET sequence34d, 65, 99e

Badness: 0.058325