# 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.

Similar to meantone, würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives 31edo as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are 11\34 and especially 21\65, which notably is the point where it is combined with schismic/nestoria and gravity/larry. Other edo tunings include 96edo, 99edo and 164edo.

Another tuning solution is to sharpen the major third by 1/8 of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the minimax tuning.

Mos scales may not be the best approach for würschmidt since they are even more extreme than those of magic. Proper scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.

## Würschmidt

Subgroup: 2.3.5

Comma list: 393216/390625

Mapping: [⟨1 -1 2], ⟨0 8 1]]

- mapping generators: ~2, ~5/4

Optimal ET sequence: 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc

Badness (Smith): 0.040603

### Overview to extensions

#### 7-limit extensions

The 7-limit extensions can be obtained by adding another comma. Septimal würschmidt adds 225/224, worschmidt adds 126/125, whirrschmidt adds 4375/4374. These all use the same generator as 5-limit würschmidt.

Hemiwürschmidt adds 3136/3125 and splits the generator in two. This temperament is the best extension available for würschmidt despite its complexity. The details can be found in Hemimean clan.

#### Subgroup extensions

Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to 11/9, that is by tempering out 5632/5625 in addition to 243/242; furthermore, like practically any 5-limit temperament with this accuracy level of 3/2 available, extensions to prime 19 exist by tempering out either 513/512 or 1216/1215 (which meet at 65edo and nestoria).

However, as discussed in the main article, the "free" higher prime for würschmidt outside the 5-limit is in fact 23, via tempering out S24 = 576/575 and S46^{2} × S47 = 12167/12150. Therefore, the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions.

### 2.3.5.23 subgroup

Subgroup: 2.3.5.23

Comma list: 576/575, 12167/12150

Sval mapping: [⟨1 -1 2 0], ⟨0 8 1 14]]

Optimal tunings:

- CTE: ~2 = 1\1, ~5/4 = 387.734
- POTE: ~2 = 1\1, ~5/4 = 387.805

Optimal ET sequence: 3, …, 28i, 31, 34, 65, 99, 164

Badness (Smith): 0.00530

#### 2.3.5.11.23 subgroup

Subgroup: 2.3.5.11.23

Comma list: 243/242, 276/275, 529/528

Sval mapping: [⟨1 -1 2 -3 0], ⟨0 8 1 20 14]]

Optimal tuning:

- CTE: ~2 = 1\1, ~5/4 = 387.652
- POTE: ~2 = 1\1, ~5/4 = 387.690

Optimal ET sequence: 31, 34, 65

Badness (Smith): 0.00660

## 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.

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 subgroup temperament.

Subgroup: 2.3.5.7

Comma list: 225/224, 8748/8575

Mapping: [⟨1 -1 2 -3], ⟨0 8 1 18]]

Wedgie: ⟨⟨ 8 1 18 -17 6 39 ]]

Optimal ET sequence: 31, 96, 127

Badness (Smith): 0.050776

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 243/242

Mapping: [⟨1 -1 2 -3 -3], ⟨0 8 1 18 20]]

Optimal tunings:

- CTE: ~2 = 1\1, ~5/4 = 387.441
- POTE: ~2 = 1\1, ~5/4 = 387.447

Optimal ET sequence: 31, 65d, 96, 127

Badness (Smith): 0.024413

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 176/175, 275/273

Mapping: [⟨1 -1 2 -3 -3 5], ⟨0 8 1 18 20 -4]]

Optimal tunings:

- CTE: ~2 = 1\1, ~5/4 = 387.469
- POTE: ~2 = 1\1, ~5/4 = 387.626

Badness (Smith): 0.023593

#### Worseschmidt

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 2 -3 -3 -5], ⟨0 8 1 18 20 27]]

Optimal tunings:

- CTE: ~2 = 1\1, ~5/4 = 387.179
- POTE: ~2 = 1\1, ~5/4 = 387.099

Optimal ET sequence: 3def, 28def, 31

Badness (Smith): 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 -1 2 7], ⟨0 8 1 -13]]

Wedgie: ⟨⟨ 8 1 -13 -17 -43 -33 ]]

Optimal ET sequence: 31, 96d, 127d

Badness (Smith): 0.064614

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 385/384

Mapping: [⟨1 -1 2 7 -3], ⟨0 8 1 -13 20]]

Optimal tunings:

- CTE: ~2 = 1\1, ~5/4 = 387.472
- POTE: ~2 = 1\1, ~5/4 = 387.407

Optimal ET sequence: 31, 65, 96d, 127d

Badness (Smith): 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 -1 2 -14], ⟨0 8 1 52]]

Wedgie: ⟨⟨ 8 1 52 -17 60 118 ]]

Optimal ET sequence: 34d, 65, 99

Badness (Smith): 0.086334

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891, 4375/4356

Mapping: [⟨1 -1 2 -14 -3], ⟨0 8 1 52 20]]

Optimal tunings:

- CTE: ~2 = 1\1, ~5/4 = 387.829
- POTE: ~2 = 1\1, ~5/4 = 387.882

Optimal ET sequence: 34d, 65, 99e

Badness (Smith): 0.058325