# Diaschismic family

(Redirected from Echidna)

The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is [11 -4 -2, and flipping that yields ⟨⟨2 -4 -11]] for the wedgie for 5-limit diaschismic, or srutal, temperament. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. 34edo is a good tuning choice, with 46edo, 56edo, 58edo or 80edo being other possibilities. Both 12edo and 22edo support it, and retuning them to a MOS of diaschismic gives two scale possibilities.

## Srutal aka diaschismic

Subgroup: 2.3.5

Comma list: 2048/2025

Mapping[2 0 11], 0 1 -2]]

mapping generators: ~45/32, ~3

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.898

• 5-odd-limit diamond monotone: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
• 5-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
• 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 706.843]

### Overview to extensions

#### 7-limit extensions

To get the 7-limit extensions, we add another comma:

• Pajara derives from 64/63 and is a popular and well-known choice.
• Diaschismic adds 126/125, the starling comma, to obtain 7-limit harmony by more complex methods, but with greater accuracy.
• Srutal adds 4375/4374, the ragisma. It does no significant tuning damage, so we keep the 5-limit label srutal.
• Bidia adds 3136/3125, the hemimean comma.
• Echidna adds 1728/1715, the orwellisma.
• Shrutar adds 245/243, the sensamagic comma.

Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as 36/35, the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator.

#### Subgroup extensions

Since the diaschisma factors into (256/255)2(289/288) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup, resulting in srutal archagall.

### Srutal archagall

The S-expression-based comma list of this temperament is {S16, S17}.

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

Sval mapping: [2 0 11 5], 0 1 -2 1]]

mapping generators: ~17/12, ~3

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272

Optimal ET sequence: 10, 12, 22, 34, 80, 114, 194bc

## Srutal

Subgroup: 2.3.5.7

Comma list: 2048/2025, 4375/4374

Mapping[2 0 11 -42], 0 1 -2 15]]

Wedgie⟨⟨2 -4 30 -11 42 81]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.814

• 7- and 9-odd-limit diamond monotone: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
• 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 705.882]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891, 1331/1323

Mapping: [2 0 11 -42 -28], 0 1 -2 15 11]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856

Tuning ranges:

• 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
• 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
• 11-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 364/363

Mapping: [2 0 11 -42 -28 -18], 0 1 -2 15 11 8]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
• 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
• 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 169/168, 176/175, 221/220, 256/255

Mapping: [2 0 11 -42 -28 -18 5], 0 1 -2 15 11 8 1]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840

Tuning ranges:

• 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
• 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
• 17-odd-limit diamond monotone and tradeoff: ~3/2 = [704.348, 705.882]

### 19-limit

Srutal, shrutar and bidia have similar 19-limit properties, tempering 190/189, related rank-3 julius.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255

Mapping: [2 0 11 -42 -28 -18 5 -55], 0 1 -2 15 11 8 1 20]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905

#### Srutaloo

Srutaloo adds 576/575, 736/729 or 208/207, rhymes with Skidoo.

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255

Mapping: [2 0 11 -42 -28 -18 5 -55 -10], 0 1 -2 15 11 8 1 20 6]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899

##### 29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255

Mapping: [2 0 11 -42 -28 -18 5 -55 -10 -76], 0 1 -2 15 11 8 1 20 6 27]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906

## Pajara

Pajara is closely associated with 22edo (not to mention Paul Erlich) but other tunings are possible. The 1/2-octave period serves as both a 10/7 and a 7/5. Aside from 22edo, 34 with the val 34 54 79 96] and 56 with the val 56 89 130 158] are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.

Subgroup: 2.3.5.7

Comma list: 50/49, 64/63

Mapping[2 0 11 12], 0 1 -2 -2]]

Wedgie⟨⟨2 -4 -4 -11 -12 2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.048

• 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
• 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 715.587]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 99/98

Mapping: [2 0 11 12 26], 0 1 -2 -2 -6]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.885

Tuning ranges:

• 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
• 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]
• 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 709.091]

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 65/63, 99/98

Mapping: [2 0 11 12 26 1], 0 1 -2 -2 -6 2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.919

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 64/63, 65/63, 99/98

Mapping: [2 0 11 12 26 1 5], 0 1 -2 -2 -6 2 1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.806

#### Pajarina

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 78/77, 99/98

Mapping: [2 0 11 12 26 36], 0 1 -2 -2 -6 -9]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.133

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 64/63, 78/77, 85/84, 99/98

Mapping: [2 0 11 12 26 36 5], 0 1 -2 -2 -6 -9 1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.410

#### Pajarita

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 64/63, 66/65

Mapping: [2 0 11 12 26 17], 0 1 -2 -2 -6 -3]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.450

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 64/63, 66/65, 85/84

Mapping: [2 0 11 12 26 17 5], 0 1 -2 -2 -6 -3 1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.947

### Pajarous

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 64/63

Mapping: [2 0 11 12 -9], 0 1 -2 -2 5]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 709.578

Tuning ranges:

• 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
• 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]
• 11-odd-limit diamond monotone and tradeoff: ~3/2 = 709.091

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 55/54, 64/63, 65/63

Mapping: [2 0 11 12 -9 1], 0 1 -2 -2 5 2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.240

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: [2 0 11 12 -9 1 5], 0 1 -2 -2 5 2 1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.221

#### Pajaro

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 55/54, 64/63

Mapping: [2 0 11 12 -9 17], 0 1 -2 -2 5 -3]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.818

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 55/54, 64/63, 85/84

Mapping: [2 0 11 12 -9 17 5], 0 1 -2 -2 5 -3 1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.866

### Pajaric

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 56/55

Mapping: [2 0 11 12 7], 0 1 -2 -2 0]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 705.524

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 45/44, 50/49, 56/55

Mapping: [2 0 11 12 7 17], 0 1 -2 -2 0 -3]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.442

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 34/33, 40/39, 45/44, 50/49, 56/55

Mapping: [2 0 11 12 7 17 5], 0 1 -2 -2 0 -3 1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.544

### Hemipaj

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 121/120

Mapping: [2 1 9 10 8], 0 2 -4 -4 -1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 546.383

### Hemifourths

Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 243/242

Mapping: [2 0 11 12 -1], 0 2 -4 -4 5]]

Optimal tuning (POTE): ~7/5 = 1\2, ~55/32 = 953.093

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 78/77, 144/143

Mapping: [2 0 11 12 -1 9], 0 2 -4 -4 5 -1]]

Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.074

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 64/63, 78/77, 85/84, 144/143

Mapping: [2 0 11 12 -1 9 5], 0 2 -4 -4 5 -1 2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.210

## Diaschismic

A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46 & 58. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. 58edo provides an excellent tuning, but an alternative is to make 7/4 just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.

Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; Mos of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.

Subgroup: 2.3.5.7

Comma list: 126/125, 2048/2025

Mapping[2 0 11 31], 0 1 -2 -8]]

Wedgie⟨⟨2 -4 -16 -11 -31 -26]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.681

• 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
• 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 705.882]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 896/891

Mapping: [2 0 11 31 45 0 1 -2 -8 -12]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714

Tuning ranges:

• 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
• 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
• 11-odd-limit diamond monotone and tradeoff: ~3/2 = [701.955, 704.348]

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 364/363

Mapping: [2 0 11 31 45 55], 0 1 -2 -8 -12 -15]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
• 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
• 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
• 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 136/135, 176/175, 196/195, 256/255

Mapping: [2 0 11 31 45 55 5], 0 1 -2 -8 -12 -15 1]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812

Tuning ranges:

• 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
• 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
• 17-odd-limit diamond monotone and tradeoff: ~3/2 = [703.448, 704.348]

#### Na"Naa'

Na"Naa' is a remarkable subgroup temperament of 46&58 with a prime harmonic of 23.

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255

Sval mapping: [2 0 11 31 45 55 5 63], 0 1 -2 -8 -12 -15 1 -17]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870

## Keen

Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22 & 56 temperament. 78edo is a good tuning choice, and remains a good one in the 11-limit, where keen, ⟨⟨2 -4 18 -12 …]], is really more interesting, adding 100/99 and 385/384 to the commas.

Subgroup: 2.3.5.7

Comma list: 875/864, 2048/2025

Mapping[2 0 11 -23], 0 1 -2 9]]

Wedgie⟨⟨2 -4 18 -11 23 53]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.571

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1232/1215

Mapping: [2 0 11 -23 26], 0 1 -2 9 -6]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.609

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 1078/1053

Mapping: [2 0 11 -23 26 -18], 0 1 -2 9 -6 8]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.167

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 119/117, 144/143, 154/153

Mapping: [2 0 11 -23 26 -18 5], 0 1 -2 9 -6 8 1]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.155

#### Keenic

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 352/351, 385/384

Mapping: [2 0 11 -23 26 36], 0 1 -2 9 -6 -9]]

Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.257

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 100/99, 136/135, 154/153, 256/255

Mapping: [2 0 11 -23 26 36 5], 0 1 -2 9 -6 -9 1]]

Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.252

## Bidia

Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12 & 56 temperament.

Subgroup: 2.3.5.7

Comma list: 2048/2025, 3136/3125

Mapping[4 0 22 43], 0 1 -2 -5]]

Wedgie⟨⟨4 -8 -20 -22 -43 -24]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.364

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891, 1375/1372

Mapping: [4 0 22 43 71], 0 1 -2 -5 -9]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.087

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 325/324, 640/637, 896/891

Mapping: [4 0 22 43 71 -36], 0 1 -2 -5 -9 8]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.301

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 325/324, 640/637

Mapping: [4 0 22 43 71 -36 10], 0 1 -2 -5 -9 8 1]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.334

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637

Mapping: [4 0 22 43 71 -36 10 17], 0 1 -2 -5 -9 8 1 0]]

Optimal tuning (POTE): ~19/16 = 1\4, ~3/2 = 705.339

## Echidna

Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22 & 58 temperament. 58edo or 80edo make for good tunings, or their vals can be added to 138 219 321 388] (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with hedgehog; 58edo is the smallest tuning that is distinctly consistent in the 11-odd-limit and 80edo is the third smallest distinctly consistent in the 11-odd-limit.

The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes 99/70 which is extremely close to 600 ¢ and is equal to it if we temper out S99. Three 11/10's then make a 4/3 (tempering out S10/S11 thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.

Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out 176/175, tempering out 351/350 and 352/351 which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with srutal archagall, leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the 17-odd-limit.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 2048/2025

Mapping[2 1 9 2], 0 3 -6 5]]

Wedgie⟨⟨6 -12 10 -33 -1 57]]

Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.856

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 896/891

Mapping: [2 1 9 2 12], 0 3 -6 5 -7]]

Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.852

Minimax tuning:

• 11-odd-limit: ~9/7 = [5/12 0 0 1/12 -1/12
[[1 0 0 0 0, [7/4 0 0 1/4 -1/4, [2 0 0 -1/2 1/2, [37/12 0 0 5/12 -5/12, [37/12 0 0 -7/12 7/12]
Eigenmonzo (unchanged-interval) basis: 2.11/7

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 364/363, 540/539

Mapping: [2 1 9 2 12 19], 0 3 -6 5 -7 -16]]

Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.756

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

Mapping: [2 1 9 2 12 19 6], 0 3 -6 5 -7 -16 3]]

Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.816

## Echidnic

Subgroup: 2.3.5.7

Comma list: 686/675, 1029/1024

Mapping[2 2 7 6], 0 3 -6 -1]]

Wedgie⟨⟨6 -12 -2 -33 -20 29]]

Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 234.492

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 686/675

Mapping: [2 2 7 6 3], 0 3 -6 -1 10]]

Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.096

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 385/384, 441/440

Mapping: [2 2 7 6 3 7], 0 3 -6 -1 10 1]]

Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.088

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 136/135, 154/153, 169/168, 256/255

Mapping: [2 2 7 6 3 7 7], 0 3 -6 -1 10 1 3]]

Optimal tuning (POTE): ~17/12 = 1\2, ~8/7 = 235.088

Compositions

## Shrutar

Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. 68edo makes for a good tuning, but another excellent choice is a generator of 14(1/7), making 7's just.

By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14(1/7) generator can again be used as tunings.

Subgroup: 2.3.5.7

Comma list: 245/243, 2048/2025

Mapping[2 1 9 -2], 0 2 -4 7]]

Wedgie⟨⟨4 -8 14 -22 11 55]]

Optimal tuning (POTE): ~45/32 = 1\2, ~35/24 = 652.811

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 245/243

Mapping: [2 1 9 -2 8], 0 2 -4 7 -1]]

Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.680

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 245/243

Mapping: [2 1 9 -2 8 -10], 0 2 -4 7 -1 16]]

Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.654

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195

Mapping: [2 1 9 -2 8 -10 6], 0 2 -4 7 -1 16 2]]

Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.647

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342

Mapping: [2 1 9 -2 8 -10 6 -10], 0 2 -4 7 -1 16 2 17]]

Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.730

## Sruti

Subgroup: 2.3.5.7

Comma list: 2048/2025, 19683/19600

Mapping[2 0 11 -15], 0 2 -4 13]]

Wedgie⟨⟨4 -8 26 -22 30 83]]

Optimal tuning (POTE): ~45/32 = 1\2, ~140/81 = 951.876

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242, 896/891

Mapping: [2 0 11 -15 -1], 0 2 -4 13 5]]

Optimal tuning (POTE): ~45/32 = 1\2, ~121/70 = 951.863

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 351/350, 676/675

Mapping: [2 0 11 -15 -1 9], 0 2 -4 13 5 -1]]

Optimal tuning (POTE): ~45/32 = 1\2, ~26/15 = 951.886

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 144/143, 170/169, 176/175, 221/220

Mapping: [2 0 11 -15 -1 9 5], 0 2 -4 13 5 -1 2]]

Optimal tuning (POTE): ~17/12 = 1\2, ~26/15 = 951.857

## Anguirus

Subgroup: 2.3.5.7

Comma list: 49/48, 2048/2025

Mapping[2 0 11 4], 0 2 -4 1]]

Wedgie⟨⟨4 -8 2 -22 -8 27]]

Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 953.021

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 243/242

Mapping: [2 0 11 4 -1], 0 2 -4 1 5]]

Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.184

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 243/242

Mapping: [2 0 11 4 -1 9], 0 2 -4 1 5 -1]]

Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.309

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 49/48, 56/55, 91/90, 119/117, 154/153

Mapping: [2 0 11 4 -1 9 5], 0 2 -4 1 5 -1 2]]

Optimal tuning (POTE): ~17/12 = 1\2, ~7/4 = 952.330

## Shru

Subgroup: 2.3.5.7

Comma list: 392/375, 1323/1280

Mapping[2 1 9 11], 0 2 -4 -5]]

Wedgie⟨⟨4 -8 -10 -22 -27 -1]]

Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.135

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 1323/1280

Mapping: [2 1 9 11 8], 0 2 -4 -5 -1]]

Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.130

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 77/75, 105/104, 507/500

Mapping: [2 1 9 11 8 15], 0 2 -4 -5 -1 -7]]

Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.535

Subgroup: 2.3.5.7

Comma list: 2048/2025, 2401/2400

Mapping[2 0 11 8], 0 4 -8 -3]]

Wedgie⟨⟨8 -16 -6 -44 -32 31]]

Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.216

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891, 2401/2400

Mapping: [2 0 11 8 22], 0 4 -8 -3 -19]]

Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.118

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 196/195, 512/507, 676/675

Mapping: [2 0 11 8 22 9], 0 4 -8 -3 -19 -2]]

Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.099

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 170/169, 176/175, 196/195, 256/255

Mapping: [2 0 11 8 22 9 5], 0 4 -8 -3 -19 -2 4]]

Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.162

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 2048/2025

Mapping: [2 0 11 8 -1], 0 4 -8 -3 10]]

Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.017

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 676/675

Mapping: [2 0 11 8 -1 9], 0 4 -8 -3 10 -2]]

Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.028