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

Comma: 393216/390625

POTE generator: ~5/4 = 387.799

Map: [<1 7 3|, <0 -8 -1|]

EDOs: 31, 34, 65, 99, 164, 721c, 885c

Music

Ancient Stardust play by Chris Vaisvil; Würschmidt[13] in 5-limit minimax tuning

Extrospection by Jake Freivald; Würschmidt[16] tuned in 31et.

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Wurschmidt 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>.

7-limit

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 wurschmidt, as well as for minerva, the 11-limit rank three temperament tempering out 99/98 and 176/175.

Commas: 225/224, 8748/8575

POTE generator: ~5/4 = 387.383

Map: [<1 7 3 15|, <0 -8 -1 -18|]

EDOs: 31, 96, 127, 285bd, 412bbdd

Badness: 0.0508

11-limit

Commas: 99/98, 176/175, 243/242

POTE generator: ~5/4 = 387.447

Map: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]

EDOs: 31, 65d, 96, 127, 223d

Badness: 0.0244

13-limit

Commas: 99/98, 144/143, 176/175, 275/273

POTE generator: ~5/4 = 387.626

Map: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]

EDOs: 31, 65d, 161df

Badness: 0.0236

Worseschmidt

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

POTE generator: ~5/4 = 387.099

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

EDOs: 31

Badness: 0.0344

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| and not <127 201 295 357| as with wurschmidt. 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.

Commas: 126/125, 33075/32768

POTE generator: ~5/4 = 387.392

Map: [<1 7 3 -6|, <0 -8 -1 13|]

EDOs: 31, 65, 96d, 127d

Badness: 0.0646

11-limit

Commas: 126/125, 243/242, 385/384

POTE generator: ~5/4 = 387.407

Map: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]

EDOs: 31, 65, 96d, 127d

Badness: 0.0334

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.

Commas: 4375/4374, 393216/390625

POTE generator: ~5/4 = 387.881

Map: [<1 7 3 38|, <0 -8 -1 -52|]

EDOs: 31dd, 34d, 65, 99

Hemiwürschmidt

Hemiwürschmidt, 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...

Commas: 2401/2400, 3136/3125

POTE generator: ~28/25 = 193.898

Map: [<1 15 4 7|, <0 -16 -2 -5|]

<<16 2 5 -34 -37 6||

EDOs: 31, 68, 99, 229, 328, 557c, 885cc

Badness: 0.0203

11-limit

Commas: 243/242, 441/440, 3136/3125

POTE generator: ~28/25 = 193.840

Map: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]

EDOs: 31, 99e, 130, 650ce, 811ce

Badness: 0.0211

13-limit

Commas: 243/242, 351/350, 441/440, 3584/3575

POTE generator: ~28/25 = 193.840

Map: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]

EDOs: 31, 99e, 130, 291, 421e, 551ce

Badness: 0.0231

Hemithir

Commas: 121/120, 176/175, 196/195, 275/273

POTE generator: ~28/25 = 193.918

Map: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]

EDOs: 31, 68e, 99ef

Badness: 0.0312

Hemiwur

Commas: 121/120, 176/175, 1375/1372

POTE generator: ~28/25 = 193.884

Map: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]

EDOs: 31, 68, 99, 130e, 229e

Badness: 0.0293

13-limit

Commas: 121/120, 176/175, 196/195, 275/273

POTE generator: ~28/25 = 194.004

Map: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]

EDOs: 31, 68, 99f, 167ef

Badness: 0.0284

Hemiwar

Commas: 66/65, 105/104, 121/120, 1375/1372

POTE generator: ~28/25 = 193.698

Map: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]

EDOs: 31

Badness: 0.0449

Relationships to other temperaments

around 775.489 which is approximately

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.