# Schismatic family

(Redirected from Bischismic)

The 5-limit parent comma for the schismatic (or schismic) family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its monzo is [-15 8 1, and flipping that yields ⟨⟨1 -8 -15]] for the wedgie. This tells us the generator is a fifth and 5/4 is represented by a diminished fourth.

This defies the tradition of tertian harmony, as the just major triad on C is C-Fb-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C-vE-G.

## Schismatic aka helmholtz

The 5-limit version of the temperament is a microtemperament, sometimes called helmholtz, schismic or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismatic, but you need 118edo if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.

Subgroup: 2.3.5

Comma list: 32805/32768

Mapping[1 0 15], 0 1 -8]]

mapping generators: ~2, ~3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.736

• 5-odd-limit diamond monotone: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
• 5-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)
• 5-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.955]

### Overview to extensions

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

Those all have a fifth as generator.

• Bischismic adds [-69 40 0 2 and has a fifth generator with a half-octave period.
• Hemischis adds [-34 25 0 -2 and has a hemififth generator.
• Guiron adds [-10 1 0 3, with an ~8/7 generator, three of which give the fifth.
• Term adds [-94 54 0 3 with a 1/3 octave period.
• Sesquiquartififths adds [-35 15 0 4 and slices the fifth in four.

Temperaments discussed elsewhere include

The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in Subgroup extensions.

## Garibaldi

Garibaldi tempers out the garischisma, equating the septimal comma with both the syntonic comma and the Pythagorean comma. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its S-expression-based comma list is {S8/S9, S15}.

Subgroup: 2.3.5.7

Comma list: 225/224, 3125/3087

Mapping[1 0 15 25], 0 1 -8 -14]]

mapping generators: ~2, ~3

Wedgie⟨⟨1 -8 -14 -15 -25 -10]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.085

[[1 0 0 0, [5/3 1/15 0 -1/15, [5/3 -8/15 0 8/15, [5/3 -14/15 0 14/15]
eigenmonzo (unchanged-interval) basis: 2.7/3
[[1 0 0 0, [25/16 1/8 0 -1/16, [5/2 -1 0 1/2, [25/8 -7/4 0 7/8]
eigenmonzo (unchanged-interval) basis: 2.9/7

### Cassandra

Cassandra is one of the best extension of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2200/2187

Mapping: [1 0 15 25 -33], 0 1 -8 -14 23]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157

Minimax tuning:

• 11-odd-limit: ~3/2 = [9/16 1/8 0 -1/16
eigenmonzo (unchanged-interval) basis: 2.9/7

Tuning ranges:

• 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
• 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 325/324, 385/384

Mapping: [1 0 15 25 -33 -28], 0 1 -8 -14 23 20]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.113

Minimax tuning:

• 13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34
eigenmonzo (unchanged-interval) basis: 2.13/7

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
• 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
• 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]

##### Cassie

Subgroup: 2.3.5.7.11.13.17

Comma list: 120/119, 154/153, 225/224, 273/272, 325/324

Mapping: [1 0 15 25 -33 -28 -7], 0 1 -8 -14 23 20 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.092

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272

Mapping: [1 0 15 25 -33 -28 -7 9], 0 1 -8 -14 23 20 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.079

##### Cassandric

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 275/273, 325/324, 375/374, 385/384

Mapping: [1 0 15 25 -33 -28 77], 0 1 -8 -14 23 20 -46]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.097

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374

Mapping: [1 0 15 25 -33 -28 77 9], 0 1 -8 -14 23 20 -46 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.098

##### Cassander

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 225/224, 275/273, 325/324, 385/384

Mapping: [1 0 15 25 -33 -28 -72], 0 1 -8 -14 23 20 48]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.144

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324

Mapping: [1 0 15 25 -33 -28 -72 9], 0 1 -8 -14 23 20 48 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.135

### Andromeda

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 245/242

Mapping: [1 0 15 25 32], 0 1 -8 -14 -18]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321

Minimax tuning:

• 11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20
eigenmonzo (unchanged-interval) basis: 2.11/9

Tuning ranges:

• 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
• 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 196/195, 245/242

Mapping: [1 0 15 25 32 37], 0 1 -8 -14 -18 -21]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.559

Minimax tuning:

• 13- and 15-odd-limit: ~3/2 = [14/23 2/23 0 0 0 -1/23
eigenmonzo (unchanged-interval) basis: 2.13/9

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
• 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
• 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 189/187, 196/195

Mapping: [1 0 15 25 32 37 -7], 0 1 -8 -14 -18 -21 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.312

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195

Mapping: [1 0 15 25 32 37 -7 9], 0 1 -8 -14 -18 -21 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.357

##### Schisicosiennic

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 154/153, 170/169, 196/195

Mapping: [1 0 15 25 32 37 58], 0 1 -8 -14 -18 -21 -34]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.725

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189

Mapping: [1 0 15 25 32 37 58 9], 0 1 -8 -14 -18 -21 -34 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.753

##### Schisicosiennoid

Subgroup: 2.3.5.7.11.13.17

Comma list: 85/84, 100/99, 105/104, 119/117, 221/220

Mapping: [1 0 15 25 32 37 12], 0 1 -8 -14 -18 -21 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.717

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152

Mapping: [1 0 15 25 32 37 12 9], 0 1 -8 -14 -18 -21 -5 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.716

### Helenus

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 3125/3087

Mapping: [1 0 15 25 51], 0 1 -8 -14 -30]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725

Minimax tuning:

• 11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32
eigenmonzo (unchanged-interval) basis: 2.11/9

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 275/273, 847/845

Mapping: [1 0 15 25 51 56], 0 1 -8 -14 -30 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.747

Minimax tuning:

• 13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32
eigenmonzo (unchanged-interval) basis: 2.11/9

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 99/98, 120/119, 176/175, 275/273, 442/441

Mapping: [1 0 15 25 51 56 -7], 0 1 -8 -14 -30 -33 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.680

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245

Mapping: [1 0 15 25 51 56 -7 9], 0 1 -8 -14 -30 -33 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.705

### Hemigari

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 3125/3087

Mapping: [1 0 15 25 9], 0 2 -16 -28 -7]]

mapping generators: ~2, ~110/63

Optimal tuning (POTE): ~2 = 1\1, ~110/63 = 951.082 (~63/55 = 248.918)

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 225/224, 275/273

Mapping: [1 0 15 25 9 14], 0 2 -16 -28 -7 -13]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 3125/3087

Mapping: [1 1 7 11 2], 0 2 -16 -28 5]]

mapping generators: ~2, ~11/9

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 640/637

Mapping: [1 1 7 11 2 -8], 0 2 -16 -28 5 40]]

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014

### Sanjaab

Subgroup: 2.3.5.7.11

Comma list: 225/224, 1331/1323, 3125/3087

Mapping: [1 2 -1 -3 0], 0 -3 24 42 25]]

mapping generators: ~2, ~11/10

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.974

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 847/845, 1331/1323

Mapping: [1 2 -1 -3 0 -1], 0 -3 24 42 25 34]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963

## Schism

Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.

Subgroup: 2.3.5.7

Comma list: 64/63, 360/343

Mapping[1 0 15 6], 0 1 -8 -2]]

• CTE: ~2 = 1\1, ~3/2 = 702.2696
• POTE: ~2 = 1\1, ~3/2 = 701.556

Wedgie⟨⟨1 -8 -2 -15 -6 18]]

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 64/63, 99/98

Mapping: [1 0 15 6 13], 0 1 -8 -2 -6]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 703.3833
• POTE: ~2 = 1\1, ~3/2 = 702.136

## Pontiac

Pontiac tempers out the ragisma, rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^3A).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 32805/32768

Mapping[1 0 15 -59], 0 1 -8 39]]

Wedgie⟨⟨1 -8 39 -15 59 113]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.757

[[1 0 0 0, [74/47 0 -1/47 1/47, [113/47 0 8/47 -8/47, [113/47 0 -39/47 39/47]
eigenmonzo (unchanged-interval) basis: 2.7/5
[[1 0 0 0, [3/2 1/5 -1/10 0, [3 -8/5 4/5 0, [-1/2 39/5 -39/10 0]
eigenmonzo (unchanged-interval) basis: 2.9/5
• 7- and 9-odd-limit diamond monotone: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
• 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.711, 701.886]

### Helenoid

The helenoid temperament (53 & 118) is closely related to the helenus temperament, but with the ragisma rather than the marvel comma tempered out.

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 4375/4374

Mapping: [1 0 15 -59 51], 0 1 -8 39 -30]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.722

Minimax tuning:

• 11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69
eigenmonzo (unchanged-interval) basis: 2.11/7

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 625/624, 729/728

Mapping: [1 0 15 -59 51 56], 0 1 -8 39 -30 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.745

Minimax tuning:

• 13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72
eigenmonzo (unchanged-interval) basis: 2.13/7

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 625/624, 729/728

Mapping: [1 0 15 -59 51 56 -91], 0 1 -8 39 -30 -33 60]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.742

Minimax tuning:

• 17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93
eigenmonzo (unchanged-interval) basis: 2.17/13

#### Helena

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 3146/3125

Mapping: [1 0 15 -59 51 -28], 0 1 -8 39 -30 20]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125

Mapping: [1 0 15 -59 51 -28 -91], 0 1 -8 39 -30 20 60]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.730

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625

Mapping: [1 0 15 -59 51 -28 -91 9], 0 1 -8 39 -30 20 60 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.729

### Ponta

The ponta temperament (53 & 171) tempers out the swetisma and the ragisma.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 32805/32768

Mapping: [1 0 15 -59 135], 0 1 -8 39 -83]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.783

Minimax tuning:

• 11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122
eigenmonzo (unchanged-interval) basis: 2.11/7

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2200/2197

Mapping: [1 0 15 -59 135 56], 0 1 -8 39 -83 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.784

Minimax tuning:

• 13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122
Eigenmonzo (unchanged-interval) basis: 2.11/7

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197

Mapping: [1 0 15 -59 135 56 -91], 0 1 -8 39 -83 -33 60]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.777

Minimax tuning:

• 17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143
Eigenmonzo (unchanged-interval) basis: 2.17/11

### Pontic

The pontic temperament (118 & 171) tempers out the werckisma and the ragisma.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 32805/32768

Mapping: [1 0 15 -59 -136], 0 1 -8 39 88]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.724

Minimax tuning:

• 11-odd-limit: ~3/2 = [6/11 0 0 0 1/88
eigenmonzo (unchanged-interval) basis: 2.11

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 625/624, 729/728, 3584/3575

Mapping: [1 0 15 -59 -136 56], 0 1 -8 39 88 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.738

Minimax tuning:

• 13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121
eigenmonzo (unchanged-interval) basis: 2.13/11

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873

Mapping: [1 0 15 -59 -136 56 -91], 0 1 -8 39 88 -33 60]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.740

Minimax tuning:

• 17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121
Eigenmonzo (unchanged-interval) basis: 2.13/11

#### Pontoid

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 4375/4374, 32805/32768

Mapping: [1 0 15 -59 -136 -215], 0 1 -8 39 88 138]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768

Mapping: [1 0 15 -59 -136 -215 -91], 0 1 -8 39 88 138 60]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.735

### Bipont

The bipont temperament (118 & 224) has a period of half octave and tempers out the lehmerisma (3025/3024) and the kalisma (9801/9800).

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 32805/32768

Mapping: [2 0 30 -118 -85], 0 1 -8 39 29]]

mapping generators: ~99/70, ~3

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.757

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 4096/4095

Mapping: [2 0 30 -118 -85 112], 0 1 -8 39 29 -33]]

Mapping generators: ~99/70, ~3

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.773

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873

Mapping: [2 0 30 -118 -85 112 -182], 0 1 -8 39 29 -33 60]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.765

#### Counterbipont

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768

Mapping: [2 0 30 -118 -85 -243], 0 1 -8 39 29 79]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.769

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768

Mapping: [2 0 30 -118 -85 -243 -182], 0 1 -8 39 29 79 60]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.764

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864

Mapping: [2 0 30 -118 -85 -243 -182 -169], 0 1 -8 39 29 79 60 56]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 701.761

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768

Mapping: [4 0 60 -236 -170 -131], 0 1 -8 39 29 23]]

mapping generators: ~208/175, ~3

Optimal tuning (POTE): ~208/175 = 1\4, ~3/2 = 701.756

## Grackle

Grackle tempers out [-44 26 0 1. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.

Subgroup: 2.3.5.7

Comma list: 126/125, 32805/32768

Mapping[1 0 15 44], 0 1 -8 -26]]

mapping generators: ~2, ~3

Wedgie⟨⟨1 -8 -26 -15 -44 -38]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.239

• 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/3
• 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 32805/32768

Mapping: [1 0 15 44 70], 0 1 -8 -26 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.172

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 5445/5408

Mapping: [1 0 15 44 70 75], 0 1 -8 -26 -42 -45]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.226

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873

Mapping: [1 0 15 44 70 75 -7], 0 1 -8 -26 -42 -45 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.206

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323

Mapping: [1 0 15 44 70 75 -7 9], 0 1 -8 -26 -42 -45 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217

#### Grackloid

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 729/728, 1287/1280

Mapping: [1 0 15 44 70 -47], 0 1 -8 -26 -42 32]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.217

### Grack

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 896/891

Mapping: [1 0 15 44 51], 0 1 -8 -26 -30]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.401

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 245/242, 832/825

Mapping: [1 0 15 44 51 75], 0 1 -8 -26 -30 -45]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.348

#### Catahelenic

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 126/125, 245/242, 352/351

Mapping: [1 0 15 44 51 56], 0 1 -8 -26 -30 -33]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.529

## Bischismic

Subgroup: 2.3.5.7

Comma list: 3136/3125, 32805/32768

Mapping[2 0 30 69], 0 1 -8 -20]]

mapping generators: ~567/400, ~3

Wedgie⟨⟨2 -16 -40 -30 -69 -48]]

Optimal tuning (CTE): ~567/400 = 1\2, ~3/2 = 701.5899

• 7-odd-limit eigenmonzo (unchanged-interval) basis: 2.7/3
• 9-odd-limit eigenmonzo (unchanged-interval) basis: 2.9/7

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 8019/8000

Mapping: [2 0 30 69 102], 0 1 -8 -20 -30]]

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.6077

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 729/728, 1001/1000, 3136/3125

Mapping: [2 0 30 69 102 -75], 0 1 -8 -20 -30 26]]

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.5949

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125

Mapping: [2 0 30 69 102 -75 5], 0 1 -8 -20 -30 26 1]]

Optimal tuning (CTE): ~99/70 = 1\2, ~3/2 = 701.5959

#### Bischis

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 3136/3125

Mapping: [2 0 30 69 102 131], 0 1 -8 -20 -30 -39]]

Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 701.5708

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125

Mapping: [2 0 30 69 102 131 5], 0 1 -8 -20 -30 -39 1]]

Optimal tuning (CTE): ~55/39 = 1\2, ~3/2 = 701.5717

## Kleischismic

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1500625/1492992

Mapping[2 1 22 -15], 0 2 -16 19]]

mapping generators: ~1225/864, ~35/24

Wedgie⟨⟨4 -32 38 -60 49 178]]

Optimal tuning (POTE): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 9801/9800, 14641/14580

Mapping: [2 1 22 -15 8], 0 2 -16 19 -1]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1575/1573

Mapping: [2 1 22 -15 8 15], 0 2 -16 19 -1 -7]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.938 (~36/35 = 50.938)

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 289/288, 352/351, 385/384, 561/560

Mapping: [2 1 22 -15 8 15 6], 0 2 -16 19 -1 -7 2]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)

#### Kleischis

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1573/1568, 14641/14580

Mapping: [2 1 22 -15 8 -36], 0 2 -16 19 -1 40]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580

Mapping: [2 1 22 -15 8 -36 6], 0 2 -16 19 -1 40 2]]

Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)

## Hemischis

Subgroup: 2.3.5.7

Comma list: 6144/6125, 19683/19600

Mapping[1 0 15 -17], 0 2 -16 25]]

mapping generators: ~2, ~140/81

Wedgie⟨⟨2 -16 25 -30 34 103]]

Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.797

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625, 8019/8000

Mapping: [1 0 15 -17 51], 0 2 -16 25 -60]]

Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.801

### 13-limit

Its S-expression-based comma list is {S12/S14, S13/S15 = S26, S27, S64(, S65)}. Tempering S13, S15 or S25 leads to 53edo (through Catakleismic) while tempering S12/S13, S13/S14, S14/S15 or S49 (thus leading to S12 = S13 = S14 = S15) leads to 130edo.

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 4096/4095

Mapping: [1 0 15 -17 51 14], 0 2 -16 25 -60 -13]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095

Mapping: [1 0 15 -17 51 14 -49], 0 2 -16 25 -60 -13 67]]

Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810

Music

## Squirrel

The squirrel temperament (29 & 36) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.

Subgroup: 2.3.5.7

Comma list: 686/675, 32805/32768

Mapping[1 2 -1 1], 0 -3 24 13]]

Wedgie⟨⟨3 -24 -13 -45 -29 37]]

Optimal tuning (POTE): ~2 = 1\1, ~160/147 = 166.140

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 686/675, 896/891

Mapping: [1 2 -1 1 0], 0 -3 24 13 25]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.097

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 169/168, 245/242, 896/891

Mapping: [1 2 -1 1 0 3], 0 -3 24 13 25 5]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.054

## Tertiaschis

The tertiaschis temperament (94 & 159) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with #Squirrel, but tempers out 1071785/1062882 for prime 7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 1071875/1062882

Mapping[1 2 -1 10], 0 -3 24 -52]]

Wedgie⟨⟨3 -24 52 -45 74 188]]

Optimal tuning (POTE): ~2 = 1\1, ~192/175 = 166.019

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 19712/19683

Mapping: [1 2 -1 10 0], 0 -3 24 -52 25]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.017

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 10985/10976

Mapping: [1 2 -1 10 0 12], 0 -3 24 -52 25 -60]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.016

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

Mapping: [1 2 -1 10 0 12 -2], 0 -3 24 -52 25 -60 44]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.012

## Countertertiaschis

The countertertiaschis temperament (159 & 224) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with #Squirrel, but tempers out 244140625/243045684 for prime 7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 244140625/243045684

Mapping[1 2 -1 -12], 0 -3 24 107]]

Optimal tuning (POTE): ~2 = 1\1, ~625/567 = 166.0621

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 32805/32768

Mapping: [1 2 -1 -12 0], 0 -3 24 107 25]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976

Mapping: [1 2 -1 -12 0 -10], 0 -3 24 107 25 99]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 166.0628

## Pogo

The pogo temperament (94 & 130) splits the period in two to address the difference between #Tertiaschis and #Countertertiaschis. The schismic tempering of the fifth is just about right for tempering out the stearnsma.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 118098/117649

Mapping[2 1 22 2], 0 3 -24 5]]

mapping generators: ~343/243, ~9/7

Wedgie⟨⟨6 -48 10 -90 -1 158]]

Optimal tuning (POTE): ~343/243 = 1\2, ~9/7 = 433.901

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4000/3993, 32805/32768

Mapping: [2 1 22 2 25], 0 3 -24 5 -25]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1575/1573, 4096/4095

Mapping: [2 1 22 2 25 -2], 0 3 -24 5 -25 13]]

Optimal tuning (POTE): ~99/70 = 1\2, ~9/7 = 433.911

## Term

Subgroup: 2.3.5.7

Comma list: 32805/32768, 250047/250000

Mapping[3 0 45 94], 0 1 -8 -18]]

mapping generators: ~63/50, ~3

Wedgie⟨⟨3 -24 -54 -45 -94 -58]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.742

### Terminal

The terminal temperament (12 & 159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 32805/32768

Mapping: [3 0 45 94 134], 0 1 -8 -18 -26]]

Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.824

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 13720/13689

Mapping: [3 0 45 94 134 168], 0 1 -8 -18 -26 -33]]

Optimal tuning (POTE): ~44/35 = 1\3, ~3/2 = 701.821

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619

Mapping: [3 0 45 94 134 168 -2], 0 1 -8 -18 -26 -33 3]]

Optimal tuning (POTE): ~34/27 = 1\3, ~3/2 = 701.810

### Terminator

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 137781/137500

Mapping: [3 0 45 94 -137], 0 1 -8 -18 31]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.685

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 31250/31213

Mapping: [3 0 45 94 -137 -103], 0 1 -8 -18 31 24]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.689

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095

Mapping: [3 0 45 94 -137 -103 -2], 0 1 -8 -18 31 24 3]]

Optimal tuning (POTE): ~63/50 = 1\3, ~3/2 = 701.688

### Semiterm

The semiterm temperament (12 & 342) has a period of 1/6 octave and tempers out 9801/9800 (kalisma) and 151263/151250 (odiheim comma).

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 151263/151250

Mapping: [6 0 90 188 287], 0 1 -8 -18 -28]]

mapping generators: ~55/49, ~3

Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7460

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375

Mapping: [6 0 90 188 287 355], 0 1 -8 -18 -28 -35]]

Optimal tuning (POTE): ~55/49 = 1\6, ~3/2 = 701.7256

* optimal patent val: 354

### Hemiterm

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 32805/32768, 102487/102400

Mapping: [3 0 45 94 8], 0 2 -16 -36 1]]

mapping generators: ~63/50, ~693/400

Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712

Mapping: [3 0 45 94 8 42], 0 2 -16 -36 1 -13]]

Optimal tuning (POTE): ~63/50 = 1\3, ~26/15 = 950.873 (~12/11 = 150.873)

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264

Mapping: [3 0 45 94 8 42 -2], 0 2 -16 -36 1 -13 6]]

Optimal tuning (POTE): ~34/27 = 1\3, ~26/15 = 950.867 (~12/11 = 150.867)

## Altinex

Subgroup: 2.3.5.7

Comma list: 32805/32768, 367653125/362797056

Mapping[3 0 45 -32], 0 2 -16 17]]

mapping generators: ~1536/1225, ~34300/19683

Optimal tuning (CTE): ~1536/1225 = 1\3, ~34300/19683 = 950.9654

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 14700/14641, 19712/19683

Mapping: [3 0 45 -32 8], 0 2 -16 17 1]]

Optimal tuning (CTE): ~44/35 = 1\3, ~121/70 = 950.9658

Optimal ET sequence: 24, …, 111c, 135, 159, 612ccdd, 771ccdd

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384, 676/675, 19712/19683

Mapping: [3 0 45 -32 8 42], 0 2 -16 17 1 -13]]

Optimal tuning (CTE): ~44/35 = 1\3, ~26/15 = 950.9360

Optimal ET sequence: 24, …, 111cf, 135f, 159

## Sesquiquartififths

Subgroup: 2.3.5.7

Comma list: 2401/2400, 32805/32768

Mapping[1 1 7 5], 0 4 -32 -15]]

mapping generators: ~2, ~448/405

Wedgie⟨⟨4 -32 -15 -60 -35 55]]

Optimal tuning (POTE): ~2 = 1\1, ~448/405 = 175.434

### Sesquart

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 16384/16335

Mapping: [1 1 7 5 2], 0 4 -32 -15 10]]

Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406

Optimal ET sequence: 41, 89, 130, 301e, 431e

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 3584/3575

Mapping: [1 1 7 5 2 -2], 0 4 -32 -15 10 39]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.409

Optimal ET sequence: 41, 89, 130, 301e, 431e

##### Sesquartia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575

Mapping: [1 1 7 5 2 -2 -6], 0 4 -32 -15 10 39 69]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.424

Optimal ET sequence: 41, 89g, 130, 171, 301e

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: [1 1 7 5 2 -2 -6 6], 0 4 -32 -15 10 39 69 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.419

Optimal ET sequence: 41, 89g, 130, 171, 301eh

###### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594

Mapping: [1 1 7 5 2 -2 -6 6 -6], 0 4 -32 -15 10 39 69 -12 72]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.412

Optimal ET sequence: 41i, 89gi, 130, 171, 301eh

##### Heartia

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 256/255, 273/272, 364/363, 441/440

Mapping: [1 1 7 5 2 -2 0], 0 4 -32 -15 10 39 28]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.386

Optimal ET sequence: 41, 89, 130g

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440

Mapping: [1 1 7 5 2 -2 0 6], 0 4 -32 -15 10 39 28 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.380

Optimal ET sequence: 41, 89, 130g

##### Hearty

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625

Mapping: [1 1 7 5 2 -2 13], 0 4 -32 -15 10 39 -61]]

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 175.377

Optimal ET sequence: 41g, 89, 130, 609ceefgg

###### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455

Mapping: [1 1 7 5 2 -2 13 6], 0 4 -32 -15 10 39 -61 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.377

Optimal ET sequence: 41g, 89, 130, 609ceefggh

###### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440

Mapping: [1 1 7 5 2 -2 13 6 13], 0 4 -32 -15 10 39 -61 -12 -58]]

Optimal tuning (POTE): ~2 = 1\1, ~21/19 = 175.376

Optimal ET sequence: 41g, 89, 130, 609ceefggh

### Bisesqui

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 32805/32768

Mapping: [2 2 14 10 23], 0 4 -32 -15 -55]]

mapping generators: ~99/70, ~448/405

Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435

## Quintilipyth

The quintilipyth temperament (12 & 253, formerly quintilischis) slices the pythagorean fourth (4/3) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 9765625/9680832

Mapping[1 2 -1 -4], 0 -5 40 82]]

Wedgie⟨⟨5 -40 -82 -75 -144 -78]]

Optimal tuning (POTE): ~2 = 1\1, ~625/588 = 99.625

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 32805/32768

Mapping: [1 2 -1 -4 -7], 0 -5 40 82 126]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.616

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647

Mapping: [1 2 -1 -4 -7 -9], 0 -5 40 82 126 153]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.612

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619

Mapping: [1 2 -1 -4 -7 -9 5], 0 -5 40 82 126 153 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.612

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971

Mapping: [1 2 -1 -4 -7 -9 5 4], 0 -5 40 82 126 153 -11 3]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.615

## Quintaschis

The quintaschis temperament (12 & 289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 49009212/48828125

Mapping[1 2 -1 -5], 0 -5 40 94]]

Wedgie⟨⟨5 -40 -94 -75 -163 -106]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.664

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 1953125/1951488

Mapping: [1 2 -1 -5 -8], 0 -5 40 94 138]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.653

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 32805/32768, 109512/109375

Mapping: [1 2 -1 -5 -8 -11], 0 -5 40 94 138 177]]

Optimal tuning (POTE): ~2 = 1\1, ~35/33 = 99.658

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768

Mapping: [1 2 -1 -5 -8 -11 5], 0 -5 40 94 138 177 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.656

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859

Mapping: [1 2 -1 -5 -8 -11 5 4], 0 -5 40 94 138 177 -11 3]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.659

### Quintahelenic

Subgroup: 2.3.5.7.11

Comma list: 5632/5625, 8019/8000, 151263/151250

Mapping: [1 2 -1 -5 -9], 0 -5 40 94 150]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.671

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000

Mapping: [1 2 -1 -5 -9 -11], 0 -5 40 94 150 177]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.661

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750

Mapping: [1 2 -1 -5 -9 -11 5], 0 -5 40 94 150 177 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.665

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700

Mapping: [1 2 -1 -5 -9 -11 5 4], 0 -5 40 94 150 177 -11 3]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.668

#### Quintahelenoid

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436

Mapping: [1 2 -1 -5 -9 14], 0 -5 40 94 150 -124]]

Optimal tuning (POTE): ~2 = 1\1, ~200/189 = 99.672

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157

Mapping: [1 2 -1 -5 -9 14 5], 0 -5 40 94 150 -124 -11]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.671

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137

Mapping: [1 2 -1 -5 -9 14 5 4], 0 -5 40 94 150 -124 -11 3]]

Optimal tuning (POTE): ~2 = 1\1, ~18/17 = 99.672

## Sextilififths

The sextilififths (130 & 159, also known as sextilischis) slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 235298/234375

Mapping[1 2 -1 -1], 0 -6 48 55]]

mapping generators: ~2, ~21/20

Wedgie⟨⟨6 -48 -55 -90 -104 7]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.053

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 235298/234375

Mapping: [1 2 -1 -1 0], 0 -6 48 55 50]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 10985/10976

Mapping: [1 2 -1 -1 0 1], 0 -6 48 55 50 39]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.049

## Septiquarschis

The septiquarschis temperament (89 & 94) splits septimal minor seventh (7/4) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).

Subgroup: 2.3.5.7

Comma list: 32805/32768, 829440/823543

Mapping[1 3 -9 2], 0 -7 -56 4]]

Wedgie⟨⟨7 56 -4 231 -26 -76]]

Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.614

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 15488/15435, 32805/32768

Mapping: [1 3 -9 2 -2], 0 -7 -56 4 27]]

Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.616

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1573/1568, 4096/4095

Mapping: [1 3 -9 2 -2 13], 0 -7 -56 4 27 -46]]

Optimal tuning (POTE): ~2 = 1\1, ~147/128 = 242.610

## Tsaharuk

Subgroup: 2.3.5.7

Comma list: 32805/32768, 420175/419904

Mapping[1 1 7 0], 0 5 -40 24]]

mapping generators: ~2, ~243/224

Wedgie⟨⟨5 -40 24 -75 24 168]]

Optimal tuning (POTE): ~2 = 1\1, ~243/224 = 140.350

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 19712/19683

Mapping: [1 1 7 0 1], 0 5 -40 24 21]]

Optimal tuning (POTE): ~2 = 1\1, ~88/81 = 140.365

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 729/728, 1331/1323

Mapping: [1 1 7 0 1 3], 0 5 -40 24 21 6]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.363

## Quanharuk

Subgroup: 2.3.5.7

Comma list: 16875/16807, 32805/32768

Mapping[1 0 15 12], 0 5 -40 -29]]

mapping generators: ~2, ~56/45

Wedgie⟨⟨5 -40 -29 -75 -60 45]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.355

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 32805/32768

Mapping: [1 0 15 12 -7], 0 5 -40 -29 33]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 1375/1372, 4096/4095

Mapping: [1 0 15 12 -7 -15], 0 5 -40 -29 33 59]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351

The quadrant temperament (12 & 224) has a period of quarter octave and tempers out the dimcomp comma, 390625/388962. In this temperament, 25/21 is mapped into quarter octave.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 390625/388962

Mapping[4 0 60 119], 0 1 -8 -17]]

mapping generators: ~25/21, ~3

Wedgie⟨⟨4 -32 -68 -60 -119 -68]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8234

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 32805/32768

Mapping: [4 0 60 119 185], 0 1 -8 -17 -27]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8176

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647

Mapping: [4 0 60 119 185 224], 0 1 -8 -17 -27 -33]]

Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 701.8158

## Septant

The septant temperament (224 & 301) has a period of 1/7 octave and tempers out the akjaysma, [47 -7 -7 -7.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 516560652/514714375

Mapping[7 0 105 -56], 0 1 -8 7]]

mapping generators: ~8575/7776, ~3

Wedgie⟨⟨7 -56 49 -105 58 271]]

Optimal tuning (POTE): ~8575/7776 = 1\7, ~3/2 = 701.702

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 24057/24010, 32805/32768

Mapping: [7 0 105 -56 -120], 0 1 -8 7 13]]

Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.719

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024

Mapping: [7 0 105 -56 -120 37], 0 1 -8 7 13 -1]]

Optimal tuning (POTE): ~495/448 = 1\7, ~3/2 = 701.724

## Octant

The octant temperament (224 & 472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 2259436291848/2251875390625

Mapping[8 0 120 -117], 0 1 -8 11]]

mapping generators: ~42875/39366, ~3

Wedgie⟨⟨8 -64 88 -120 117 384]]

Optimal tuning (POTE): ~42875/39366 = 1\8, ~3/2 = 701.713

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 46656/46585

Mapping: [8 0 120 -117 15], 0 1 -8 11 1]]

Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.713

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655

Mapping: [8 0 120 -117 15 93], 0 1 -8 11 1 -5]]

Optimal tuning (POTE): ~12/11 = 1\8, ~3/2 = 701.725

## Nonant

The nonant temperament (36 & 135) has a period of 1/9 octave and tempers out the septimal ennealimma, [-11 -9 0 9.

Subgroup: 2.3.5.7

Comma list: 32805/32768, 40353607/40310784

Mapping[9 0 135 11], 0 1 -8 1]]

mapping generators: ~2592/2401, ~3

Optimal tuning (CTE): ~2592/2401 = 1\9, ~3/2 = 701.7232

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 32805/32768, 42875/42592

Mapping: [9 0 135 11 131], 0 1 -8 1 -7]]

Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.8398

Optimal ET sequence: 36, 99c, 135, 171, 477ce, 648cee

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728, 4096/4095, 16807/16731

Mapping: [9 0 135 11 131 -38], 0 1 -8 1 -7 5]]

Optimal tuning (CTE): ~242/225 = 1\9, ~3/2 = 701.7998

Optimal ET sequence: 36, 99cf, 135, 171

## Tridecafifths

Tridecafifths divides the perfect 3/2 into 13 quartertones.

Subgroup: 2.3.5.7

Comma list: 32805/32768, [-14 -1 -9 13

Mapping[1 1 7 6], 0 13 -104 -71]]

mapping generators: ~2, ~1323/1280

Optimal tuning (CTE): ~2 = 1\1, ~1323/1280 = 53.9741

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 32805/32768, 55296000/55240493

Mapping: [1 1 7 6 4], 0 13 -104 -71 -12]]

Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 53.9744

Optimal ET sequence: 89, 200, 289

## Subgroup extensions

### Taylor (2.3.5.13)

This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.

Subgroup: 2.3.5.13

Comma list: 676/675, 32805/32768

Gencom: [2 15/13; 676/675 32805/32768]

Gencom mapping: [1 2 -1 0 0 1], 0 -2 16 0 0 13]]

Sval mapping: [1 2 -1 1], 0 -2 16 13]]

POL2 generator: ~15/13 = 249.145

RMS error: 0.1485 cents

### Photia (2.3.5.17)

Subgroup: 2.3.5.17

Comma list: 256/255, 1458/1445

Gencom: [2 4/3; 256/255 1458/1445]

Gencom mapping: [1 2 -1 0 0 0 7], 0 -1 8 0 0 0 -7]]

Sval mapping: [1 2 -1 7], 0 -1 8 -7]]

POL2 generator: ~3/2 = 701.491

RMS error: 0.4842 cents

#### 2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 171/170, 256/255, 324/323

Gencom: [2 4/3; 171/170 256/255 324/323]

Gencom mapping: [1 2 -1 0 0 0 7 3], 0 -1 8 0 0 0 -7 3]]

Sval mapping: [1 2 -1 7 3], 0 -1 8 -7 3]]

POL2 generator: ~3/2 = 701.470

RMS error: 0.5374 cents

### Quintilischis (2.3.5.17)

For full 17- and 19-limit extensions, see Quintilipyth or Quintaschis.

Subgroup: 2.3.5.17

Comma list: 32805/32768, 1419857/1417176

Gencom: [2 18/17; 32805/32768 1419857/1417176]

Gencom mapping: [1 2 -1 0 0 0 5], 0 -5 40 0 0 0 -11]]

Sval mapping: [1 2 -1 5], 0 -5 40 -11]]

POL2 generator: ~18/17 = 99.649

RMS error: 0.0719 cents

#### 2.3.5.17.19

Subgroup: 2.3.5.17.19

Comma list: 4624/4617, 6144/6137, 6885/6859

Gencom: [2 18/17; 4624/4617 6144/6137 6885/6859]

Gencom mapping: [1 2 -1 0 0 0 5 4], 0 -5 40 0 0 0 -11 3]]

Sval mapping: [1 2 -1 5 4], 0 -5 40 -11 3]]

POL2 generator: ~18/17 = 99.652

RMS error: 0.1636 cents

### Nestoria (2.3.5.19)

The S-expression-based comma list of this temperament is {S16/S18, S19 (, S15/S20)}.

Subgroup: 2.3.5.19

Comma list: 361/360, 513/512

Sval mapping[1 0 15 9], 0 1 -8 -3]]

mapping generators: ~2, ~3

Gencom mapping[1 2 -1 0 0 0 0 3], 0 -1 8 0 0 0 0 3]]

gencom: [2 4/3; 361/360 513/512]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.746

RMS error: 0.1763 cents