# 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.

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.