# Diaschismic family

(Redirected from Bidia)

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

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

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

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

## 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

Main article: 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]]

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].

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.

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