Würschmidt family
The 5-limit parent comma for the würschmidt family 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 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. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96EDO, 99EDO and 164EDO. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the minimax tuning. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note MOS all possibilities.
Würschmidt
(Würschmidt is sometimes spelled Wuerschmidt)
Comma: 393216/390625
Mapping: [⟨1 7 3], ⟨0 -8 -1]]
POTE generator: ~5/4 = 387.799
Badness: 0.040603
- Music
- Ancient Stardust, play by Chris Vaisvil; Würschmidt[13] in 5-limit minimax tuning
- Extrospection by Jake Freivald; Würschmidt[16] tuned in 31EDO.
Seven limit children
The second comma of the normal comma list defines which 7-limit family member we are looking at. Würschmidt adds [12 3 -6 -1⟩, worschmidt adds 65625/65536 = [-16 1 5 1⟩, whirrschmidt adds 4375/4374 = [-1 -7 4 1⟩ and hemiwuerschmidt adds 6144/6125 = [11 1 -3 -2⟩.
Septimal Würschmidt
Würschmidt, aside from the commas listed above, also tempers out 225/224. 31EDO or 127EDO can be used as tunings. Würschmidt has ⟨⟨ 8 1 18 -17 6 39 ]] for a wedgie. 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 three temperament tempering out 99/98 and 176/175.
Comma list: 225/224, 8748/8575
Mapping: [⟨1 7 3 15], ⟨0 -8 -1 -18]]
POTE generator: ~5/4 = 387.383
Badness: 0.050776
11-limit
Comma list: 99/98, 176/175, 243/242
Mapping: [⟨1 7 3 15 17], ⟨0 -8 -1 -18 -20]]
POTE generator: ~5/4 = 387.447
Vals: Template:Val list
Badness: 0.024413
13-limit
Comma list: 99/98, 144/143, 176/175, 275/273
Mapping: [⟨1 7 3 15 17 1], ⟨0 -8 -1 -18 -20 4]]
POTE generator: ~5/4 = 387.626
Vals: Template:Val list
Badness: 0.023593
Worseschmidt
Commas: 66/65, 99/98, 105/104, 243/242
Mapping: [⟨1 7 3 15 17 22], ⟨0 -8 -1 -18 -20 -27]]
POTE generator: ~5/4 = 387.099
Vals: Template:Val list
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. The wedgie now is ⟨⟨ 8 1 -13 -17 -43 -33 ]]. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.
Comma list: 126/125, 33075/32768
Mapping: [⟨1 7 3 -6], ⟨0 -8 -1 13]]
POTE generator: ~5/4 = 387.392
Badness: 0.064614
11-limit
Comma list: 126/125, 243/242, 385/384
Mapping: [⟨1 7 3 -6 17], ⟨0 -8 -1 13 -20]]
POTE generator: ~5/4 = 387.407
Vals: Template:Val list
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 ⟨⟨ 8 1 52 -17 60 118 ]] for a wedgie.
Comma list: 4375/4374, 393216/390625
Mapping: [⟨1 7 3 38], ⟨0 -8 -1 -52]]
POTE generator: ~5/4 = 387.881
Badness: 0.086334
Hemiwürschmidt
Hemiwürschmidt (sometimes spelled Hemiwuerschmidt), which splits the major third in two and uses that for a generator, is the most important of these temperaments even with the rather large complexity for the fifth. It tempers out 3136/3125, 6144/6125 and 2401/2400. 68EDO, 99EDO and 130EDO can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, ⟨⟨ 16 2 5 40 -39 -49 -48 28... ]]
Comma list: 2401/2400, 3136/3125
Mapping: [⟨1 15 4 7], ⟨0 -16 -2 -5]]
Wedgie: ⟨⟨ 16 2 5 -34 -37 6 ]]
POTE generator: ~28/25 = 193.898
Badness: 0.020307
11-limit
Comma list: 243/242, 441/440, 3136/3125
Mapping: [⟨1 15 4 7 37], ⟨0 -16 -2 -5 -40]]
POTE generator: ~28/25 = 193.840
Vals: Template:Val list
Badness: 0.021069
13-limit
Comma list: 243/242, 351/350, 441/440, 3584/3575
Mapping: [⟨1 15 4 7 37 -29], ⟨0 -16 -2 -5 -40 39]]
POTE generator: ~28/25 = 193.829
Vals: Template:Val list
Badness: 0.023074
Hemithir
Comma list: 121/120, 176/175, 196/195, 275/273
Mapping: [⟨1 15 4 7 37 -3], ⟨0 -16 -2 -5 -40 8]]
POTE generator: ~28/25 = 193.918
Vals: Template:Val list
Badness: 0.031199
Hemiwur
Comma list: 121/120, 176/175, 1375/1372
Mapping: [⟨1 15 4 7 11], ⟨0 -16 -2 -5 -9]]
POTE generator: ~28/25 = 193.884
Vals: Template:Val list
Badness: 0.029270
13-limit
Comma list: 121/120, 176/175, 196/195, 275/273
Mapping: [⟨1 15 4 7 11 -3], ⟨0 -16 -2 -5 -9 8]]
POTE generator: ~28/25 = 194.004
Vals: Template:Val list
Badness: 0.028432
Hemiwar
Comma list: 66/65, 105/104, 121/120, 1375/1372
Mapping: [⟨1 15 4 7 11 23], ⟨0 -16 -2 -5 -9 -23]]
POTE generator: ~28/25 = 193.698
Vals: Template:Val list
Badness: 0.044886
Relationships to other temperaments
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 temperament.