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.
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.
MOS scales of würschmidt are even more extreme than those of magic. Proper scales does not appear until 28, 31 or even 34 notes.
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 hemiwürschmidt adds 6144/6125 = [11 1 -3 -2⟩.
Würschmidt
Subgroup: 2.3.5
Comma list: 393216/390625
Mapping: [⟨1 7 3], ⟨0 -8 -1]]
POTE generator: ~5/4 = 387.799
Optimal ET sequence: 31, 34, 65, 99, 164, 721c, 885c
Badness: 0.040603
- Music
- Ancient Stardust, play by Chris Vaisvil; Würschmidt[13] in 5-odd-limit minimax tuning
- Extrospection by Jake Freivald; Würschmidt[16] tuned in 31edo.
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.
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]]
POTE generator: ~5/4 = 387.383
Optimal ET sequence: 31, 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]]
POTE generator: ~5/4 = 387.447
Optimal ET sequence: 31, 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]]
POTE generator: ~5/4 = 387.626
Optimal ET sequence: 31, 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]]
POTE generator: ~5/4 = 387.099
Optimal ET sequence: 3def, 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]]
POTE generator: ~5/4 = 387.392
Optimal ET sequence: 31, 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]]
POTE generator: ~5/4 = 387.407
Optimal ET sequence: 31, 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]]
POTE generator: ~5/4 = 387.881
Optimal ET sequence: 34d, 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]]
POTE generator: ~5/4 = 387.882
Optimal ET sequence: 34d, 65, 99e
Badness: 0.058325
Hemiwürschmidt
- See also: Hemimean clan #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 2401/2400, 3136/3125, and 6144/6125. 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 …]].
Subgroup: 2.3.5.7
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
Optimal ET sequence: 31, 68, 99, 229, 328, 557c, 885cc
Badness: 0.020307
11-limit
Subgroup: 2.3.5.7.11
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
Optimal ET sequence: 31, 99e, 130, 650ce, 811ce
Badness: 0.021069
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal ET sequence: 31, 99e, 130, 291, 421e, 551ce
Badness: 0.023074
Hemithir
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 196/195, 243/242, 625/624
Mapping: [⟨1 15 4 7 37 -3], ⟨0 -16 -2 -5 -40 8]]
POTE generator: ~28/25 = 193.918
Optimal ET sequence: 31, 68e, 99ef
Badness: 0.031199
Hemiwur
Subgroup: 2.3.5.7.11
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
Optimal ET sequence: 31, 68, 99, 130e, 229e
Badness: 0.029270
13-limit
Subgroup: 2.3.5.7.11.13
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
Optimal ET sequence: 31, 68, 99f, 167ef
Badness: 0.028432
Hemiwar
Subgroup: 2.3.5.7.11.13
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
Badness: 0.044886
Quadrawürschmidt
This has been documented in Graham Breed's temperament finder as semihemiwürschmidt, but quadrawürschmidt arguably makes more sense.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 3136/3125
Mapping: [⟨1 15 4 7 24], ⟨0 -32 -4 -10 -49]]
Mapping generators: ~2, ~147/110
POTE generator: ~147/110 = 503.0404
Optimal ET sequence: 31, 105be, 136e, 167, 198, 427c
Badness: 0.034814
Semihemiwür
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3136/3125, 9801/9800
Mapping: [⟨2 14 6 9 -10], ⟨0 -16 -2 -5 25]]
Mapping generators: ~99/70, ~495/392
POTE generator: ~28/25 = 193.9021
Optimal ET sequence: 62e, 68, 130, 198, 328
Badness: 0.044848
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
Mapping: [⟨2 14 6 9 -10 25], ⟨0 -16 -2 -5 25 -26]]
POTE generator: ~28/25 = 193.9035
Optimal ET sequence: 62e, 68, 130, 198, 328
Badness: 0.023388
Semihemiwürat
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
Mapping: [⟨2 14 6 9 -10 25 19], ⟨0 -16 -2 -5 25 -26 -16]]
POTE generator: ~28/25 = 193.9112
Optimal ET sequence: 62e, 68, 130, 198, 328g, 526cfgg
Badness: 0.028987
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
Mapping: [⟨2 14 6 9 -10 25 19 20], ⟨0 -16 -2 -5 25 -26 -16 -17]]
POTE generator: ~19/17 = 193.9145
Optimal ET sequence: 62e, 68, 130, 198, 328g, 526cfgg
Badness: 0.021707
Semihemiwürand
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
Mapping: [⟨2 14 6 9 -10 25 -4], ⟨0 -16 -2 -5 25 -26 18]]
POTE generator: ~28/25 = 193.9112
Optimal ET sequence: 62eg, 68, 130g, 198g
Badness: 0.029718
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
Mapping: [⟨2 14 6 9 -10 25 -4 -3], ⟨0 -16 -2 -5 25 -26 18 17]]
POTE generator: ~19/17 = 193.9428
Optimal ET sequence: 62egh, 68, 130gh, 198gh
Badness: 0.029545
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.