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

Optimal ET sequence: 6f, 31

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.