# Subgroup temperaments

(Redirected from Machine)

A subgroup temperament is a regular temperament defined on a just intonation subgroup that is not a full p-limit group.

For temperaments that omit various prime harmonics, see:

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on Chromatic pairs.

# Integer subgroup temperaments

## 2.3.35 subgroup

### Shaka

Two commas that split 2/1 in half, corresponding to convergents to sqrt(2), are the shaftesburisma S29/S41 and the kalisma S99, prompting to temper out {S29, S41, S99}, approximating /29 and /41 primodal chords well.

Subgroup: 2.3.35.11.29.41

Comma list: 841/840, 1189/1188, 1681/1680

Sval mapping[2 2 6 5 7 8], 0 1 1 -1 1 1], 0 0 2 2 1 1]]

Optimal tuning (CTE): ~41/29 = 1\2, ~3/2 = 702.031, ~41/24 = 926.693

Supporting ETs: 22, 26, 36, 48, 70, 96, 106, 118, 140, 154, 176, 188, 224, 272, 294, 342

Scale: Shaka10

## 2.9.5.7 subgroup

### Commatose

Commatose is a dual-fifth temperament which uses the Pythagorean comma as a generator. It was developed by Eliora to highlight the near-perfect expression of 9/8 by 1789edo, while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to ~13/9.

Subgroup: 2.9.5.7

Comma list: [28 -2 -19 8, [9 -25 23 6

Sval mapping[1 9 6 13], 0 -298 -188 -521]]

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4765

#### 2.9.5.7.11

Subgroup: 2.9.5.7.11

Comma list: [-7 7 -3 2 -4, [17 0 -13 1 3, [11 -2 -6 7 -3

Sval mapping: [1 9 6 13 16], 0 -298 -188 -521 -641]]

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

#### 2.9.5.7.11.13

Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: [0 9 6 13 16 10], -298 -188 -521 -641 -322]]

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

### Daemotertiaschis

Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a daemotonic 7L 4s scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

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

Sval mapping[1 1 11 -16 13 -18 20], 0 3 -12 26 -11 30 -22]]

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

### Baldy

Baldy results from taking every other generator of the garibaldi temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

Subgroup: 2.9.5.7

Comma list: 225/224, 3125/3087

Sval mapping[1 3 3 4], 0 1 -4 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.170

Related temperament: Garibaldi

#### 2.9.5.7.13

Subgroup: 2.9.5.7.13

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

Sval mapping[1 3 3 4 2], 0 1 -4 -7 10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.090

Related temperament: Cassandra

#### Baldanders

Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

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

Sval mapping[1 3 3 4 5], 0 1 -4 -7 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

Related temperament: Andromeda

##### 2.9.5.7.11.13

Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

Sval mapping[1 3 3 4 5 2], 0 1 -4 -7 -9 10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

## 2.9.7 subgroup

### Mabon

Derived from a calendar leap cycle built for the autumn equinox, hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

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

Optimal tuning (CTE): ~729/448 = 870.792

Optimal ET sequence7d, 11, 18d, 29, 40, 62, ...

#### 2.9.7.11 subgroup

Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

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

Optimal tuning (CTE): ~16/11 = 870.966

## 2.9.7.11 subgroup

### Machine

Subgroup: 2.9.7.11

Comma list: 64/63, 99/98

Sval mapping[1 0 6 13], 0 1 -1 -3]]

sval mapping generators: ~2, ~9

Gencom mapping[1 3/2 0 3 4], 0 1/2 0 -1 -3]]

gencom: [2 8/7; 64/63 99/98]
• CTE: ~2 = 1\1, ~9/8 = 216.9128
• POTE: ~2 = 1\1, ~9/8 = 214.3843

### Mechanism

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

Sval mapping[1 0 -1 6], 0 5 6 -4]]

sval mapping generators: ~2, ~14/9

Gencom mapping[1 5/2 0 5 2], 0 -5/2 0 -6 4]]

gencom: [2 9/7; 896/891 26411/26244]

Optimal tuning (POTE): ~2 = 1\1, ~14/9 = 761.3782

### Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Sval mapping[1 5 3 5], 0 -19 -2 -16]]

mapping generators: ~2, ~77/72

Gencom mapping[1 5/2 0 3 5], 0 -19/2 0 -2 -16]]

gencom: [2 77/72; 41503/41472 322102/321489]

Optimal tuning (CTE): ~77/72 = 115.5685

## 2.9.11 subgroup

### Demon

Demon is a temperament which equates 3 11/9 with 16/9, or equivalently 3 18/11 with 9/8, tempering out 1331/1296. This results in 11/9 being tuned flat to a supraminor third, and 27/22 being tuned sharp to a submajor third. It was discovered by CompactStar while searching for temperaments assosciated with the 7L 4s ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed 18edo supports demon temperament.

Subgroup: 2.9.11

Sval mapping[1 1 2], 0 3 2]]

Optimal tuning (CTE): ~18/11 = 870.060

### Genius

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

Subgroup: 2.9.11

Sval mapping[1 1 4], 0 4 -1]]

Optimal tuning (CTE): ~16/11 = 650.863

Optimal ET sequence9, 11, 24, 59, 83, 142, 225, 367[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

## 2.9.15.7 subgroup

### Stacks (aka 2magic)

Subgroup: 2.9.15.7

Comma list: 225/224, 245/243

Sval mapping[1 0 2 -1], 0 5 3 6]]

sval mapping generators: ~2, ~14/9

Gencom mapping[1 5/2 5/2 5], 0 -5/2 -1/2 -6]]

gencom: [2 9/7; 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 760.704

RMS error: 1.074 cents

#### 2.9.15.7.11

Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: [1 0 2 -1 6], 0 5 3 6 -4]]

Gencom mapping: [1 5/2 5/2 5 2], 0 -5/2 -1/2 -6 4]]

gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

RMS error: 1.226 cents

#### 2.9.15.7.11.13

Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: [1 0 2 -1 6 -2], 0 5 3 6 -4 9]]

Gencom mapping: [1 5/2 5/2 5 2 7], 0 -5/2 -1/2 -6 4 -9]]

gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

RMS error: 1.540 cents

## 2.9.21 subgroup

### A-team

A-team is every other step of mothra.

Subgroup: 2.9.21

Comma list: 1029/1024

Sval mapping[1 2 4], 0 3 1]]

sval mapping generators: ~2, ~21/16

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

gencom: [2 21/16; 1029/1024]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 467.375

RMS error: 0.3202 cents

#### 2.9.5.21.11

Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: [1 2 0 4 5], 0 3 6 1 -4]]

Gencom mapping: [1 1 0 3 5], 0 3/2 6 -1/2 -4]]

gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

## 2.15.55 subgroup

### Spog

This temperament produces superpelog-like semiquartal scales while being more accurate (see rational approximations to their intervals).

Subgroup: 2.15.55

Sval mapping[1 0 5], 0 5 1]]

Optimal tuning (subgroup CTE): ~55/32 = 937.655

#### 2.15.55.325

Subgroup: 2.15.55.325

Sval mapping[1 0 5 6], 0 5 1 3]]

Optimal tuning (subgroup CTE): ~55/32 = 937.647

Supporting ETs: 5, 9, 13[-15], 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311

#### 2.15.189.55.325

Related temperament: lux

Subgroup: 2.15.189.55.325

Sval mapping[1 0 6 5 6], 0 5 2 1 3]]

Optimal tuning (subgroup CTE): ~55/32 = 937.677

Supporting ETs: 5, 9, 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311

#### 2.15.189.55.325.725

Subgroup: 2.15.189.55.325.725

Sval mapping[1 0 6 5 6 -3], 0 5 2 1 3 16]]

Optimal tuning (subgroup CTE): ~55/32 = 937.649

Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 142[+725], 151, 183, 206[+725], 311

#### 2.15.189.55.325.725.279

Here are rational approximations to the intervals of the semiquartal scale.

Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.

Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.

Subgroup: 2.15.189.55.325.725.279

Sval mapping[1 0 6 5 6 -3 5], 0 5 2 1 3 16 4]]

Optimal tuning (subgroup CTE): ~55/32 = 937.638

Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311

## 4.3.5 subgroup

### Tetrahanson

Subgroup: 4.3.5

Comma list: 15625/15552

Sval mapping[1 3 3], 0 -6 -5]]

Mapping generators: ~4, ~5/3

Optimal tuning (CTE): ~4 = 2\1, ~5/3 = 882.941

Supporting ETs: 19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79

### Tetrameantone

Subgroup: 4.3.5

Comma list: 81/80

Sval mapping[1 1 2], 0 -1 -4]]

Mapping generators: ~4, ~4/3

Optimal tuning (POTE): 4 = 2400.0, ~4/3 = 503.761

Supporting ETs: 5, 9, 14, 19, 24, 43, 62, 81, 100

### Tetramagic

Subgroup: 4.3.5

Comma list: 3125/3072

Sval mapping[1 0 1], 0 5 1]]

Mapping generators: ~4, ~5/4

Optimal tuning (POTE): 4 = 2400.0, ~5/4 = 380.059

Supporting ETs: 6, 13, 19, 25, 38, 44, 63, 82

### Blacktetra

Subgroup: 4.3.5

Comma list: 256/243

Sval mapping[5 4 6], 0 0 -1]]

Mapping generators: ~4, ~16/15

Optimal tuning (POTE): 1\5ed4 = 480.0, ~16/15 = 80.4062

Supporting ETs: 5, 10, 15, 20, 25, 30, 55, 85, 115

## 4.6.5 subgroup

Subgroup: 4.6.5

Comma list: 81/80 = [-4 4 -1

Sval mapping[1 0 -4], 0 1 4]]

mapping generators: ~4, ~6

Optimal tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 697.214

Supporting ETs: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

* wart for 4

#### 4.6.5.7 subgroup (tetrominant)

Subgroup: 4.6.5.7

Comma list: 36/35 = [0 2 -1 -1, 64/63 = [4 -2 0 -1

Sval mapping[1 0 -4 4], 0 1 4 -2]]

Optimal tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 699.622

Supporting ETs: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

* wart for 4

## 4.9.25 subgroup

### Meansquared

Subgroup: 4.9.25

Sval mapping[1 3 4], 0 1 4]]

Mapping generators: ~4, ~9/64

Optimal tuning (CTE): ~4 = 2\1, ~9/4 = 1394.429

Supporting ETs: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

## 4.9.49 subgroup

### Archsquared

Subgroup: 4.9.49

Comma list: 4096/3969

Sval mapping[1 3 0], 0 1 -2]]

Mapping generators: ~4, ~9/64

Optimal tuning (CTE): ~9/8 = 219.190

Supporting ETs: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

## 8.9.7 subgroup

### Sixscared

Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

Subgroup: 8.9.7

Comma list: 64/63

Sval mapping[1 0 2], 0 1 -1]]

sval mapping generators: ~8, ~9
gencom: [8 9/8; 64/63]

Optimal tuning (CTE): 1\3ed8 = 1600.0, ~9/8 = 219.1898

Optimal ET sequence: 16 17 15], 33 35 31], 148 …], 181 …], 214 …], 247 …]

Badness: 0.0215 × 10-3

# Fractional subgroup temperaments

## 3/2.5/2… subgroup

### Hemihemi

Subgroup: 3/2.5/2.7/2

Sval mapping[1 2 3], 0 3 1]]

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/27 = 60.909

Supporting ETs: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

### Halftone

Subgroup: 3/2.5/2.7/2

Comma list: 9604/9375

Sval mapping[1 3 4], 0 -4 -5]]

sval mapping generators: ~3/2, ~15/14

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49

* wart for 3/2

#### 3/2.5/2.7/2.11/2

Subgroup: 3/2.5/2.7/2.11/2

Comma list: 1232/1215, 27783/27500

Sval mapping[1 3 4 4], 0 -4 -5 1]]

sval mapping generators: ~3/2, ~15/14

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 129.186

Supporting ETs: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]

* wart for 3/2

#### 3/2.5/2.7/2.11/2.13/2

Subgroup: 3/2.5/2.7/2.11/2.13/2

Comma list: 275/273, 1232/1215, 1323/1300

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

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 129.381

Supporting ETs: *11, *5, *16, *6, *27[-11/2]

* wart for 3/2

### Semiwolf

Subgroup: 3/2.5/2.7/4

Comma list: 245/243

Sval mapping[1 1 2], 0 2 -1]]

sval mapping generators: ~3/2, ~9/7

Optimal tuning (subgroup POTE): ~7/6 = 262.1728

#### Semilupine

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 100/99, 245/243

Sval mapping[1 1 2 0], 0 2 -1 4]]

Optimal tuning (subgroup POTE): ~7/6 = 264.3771

#### Hemilycan

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 245/243, 441/440

Sval mapping[1 1 2 5], 0 2 -1 -4]]

Optimal tuning (subgroup POTE): ~7/6 = 261.5939

## 5/3.7/3… subgroup

### Greeley

Related temperaments: Opossum, Nusecond

Subgroup: 2.5/3.7/3.11/3

Comma list: 121/120, 126/125

Sval mapping[1 1 2 2], 0 -2 -6 -1]]

Gencom mapping[1 -5/4 -1/4 3/4 3/4], 0 9/4 1/4 -15/4 5/4]]

gencom: [2 11/10; 121/120 126/125]

Optimal tuning (subgroup POTE): ~11/10 = 155.776

Optimal ET sequence8, 15, 23, 54, 77, 100, 131†, 208*†

* wart for 5/3

† wart for 11/3

RMS error: 1.034 cents

## 7/5.11/5… subgroup

### Historical

Not to be confused with Historical temperaments.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 364/363, 441/440, 1001/1000

Sval mapping[1 2 0 1 2], 0 -6 7 2 -9]]

Optimal tuning (subgroup POTE): ~21/20 = 83.016

RMS error: 0.2562 cents

## 11/7.13/7… subgroup

### Pepperoni

Pepperoni is the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. The Pepper fifth, which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

Subgroup: 2.3.11/7.13/7

Comma list: 352/351, 364/363

Sval mapping[1 0 7 12], 0 1 -4 -7]]

Gencom mapping[1 1 0 -8/3 1/3 7/3], 0 1 0 11/3 -1/3 -10/3]]

gencom: [2 3/2; 352/351 364/363]

Optimal tuning (subgroup POTE): ~3/2 = 703.856

Optimal ET sequence5, 7, 12, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*†

* wart for 11/7

† wart for 13/7

RMS error: 0.3789 cents

## Other 3/2 subgroups

### Auk

Subgroup: 3/2.7.13

Comma list: 87808/85293

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

sval mapping generators: ~3/2, ~7

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]

* wart for 3/2

### Doubleton

Subgroup: 3/2.7.13

Comma list: 1352/1323

Sval mapping[2 0 3], 0 1 1]]

sval mapping generators: ~26/21, ~7

Optimal tuning (subgroup CTE): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]

* wart for 3/2

## Other 5/2 subgroups

### Hyperion

Subgroup: 5/2.7.11

Comma list: [11 1 -5

Sval mapping[1 4 3], 0 -5 -1]]

gencom: [5/2 125/88; 341796875/329832448]

Optimal tuning (subgroup POTE): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]

* wart for 5/2

## Other 7/5 subgroups

### Hydrothermal

A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

Subgroup: 2.3.7/5

Sval mapping[2 3 1], 0 1 0]]

Optimal tuning (inharmonic TE): ~1\2 = 590.998, ~10/7-1\2 = 128.962

Supporting ETs: 4, 6, 8, 10, 18, 28, 46, 64, 110

## Other 11/5 subgroups

### Hypnosis

Related temperaments: hypnos, tricot

Subgroup: 2.3.7.11/5.13

Comma list: 169/168, 540/539, 729/728

Sval mapping[1 0 -3 8 0], 0 3 11 -13 7]]

Optimal tuning (subgroup POTE): ~13/9 = 633.518

RMS error: 0.5379 cents

## Other 13/5 subgroups

Subgroup: 2.3.13/5

Comma list: 676/675 = [2 -3 2

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.621

Optimal ET sequence5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362

* wart for 3/2

### Oceanfront

Related temperaments: superpyth, ultrapyth

Subgroup: 2.3.7.13/5

Comma list: 64/63, 91/90

Sval mapping[1 0 6 -5], 0 1 -2 4]]

Optimal tuning (subgroup POTE): ~3/2 = 713.910

RMS error: 2.063 cents

### Pakkanian hemipyth

Subgroup: 2.3.11.13/5.17

Comma list: 221/220, 243/242, 289/288

Sval mapping[2 0 -1 -2 5], 0 2 5 3 2]]

• subgroup CTE: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
• subgroup CWE: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

Optimal ET sequence10, 14, 24, 106, 130, 154, 178*, 202*

* wart for 13/5

## Other 9/7 subgroups

### Marveltri

Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament.

Subgroup: 2.5.9/7

Comma list: 225/224

Sval mapping[1 2 1], 0 1 -2]]

Gencom mapping[1 2/5 2 -1/5], 0 -4/5 1 2/5]]

gencom: [2 5/4; 225/224]

Optimal tuning (subgroup POTE): ~5/4 = 383.638

Optimal ET sequence12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269c*, 660c*

* wart for 9/7

RMS error: 0.4801 cents

#### Sulis

Related temperament: minerva, würschmidt

Subgroup: 2.5.9/7.11/9

Comma list: 99/98, 176/175

Sval mapping[1 2 1 -1], 0 1 -2 4]]]

Optimal tuning (subgroup POTE): ~5/4 = 386.558

Optimal ET sequence3, …, 22, 25, 28, 31, 59

RMS error: 1.074 cents

## Other 15/11 subgroups

### Poggers

Related temperaments: pogo, supers

Subgroup: 2.9.7.15/11.13

Sval mapping[1 1 1 -1 -1], 0 6 5 4 13]]

Optimal tuning (subgroup CTE): ~9/7 = 433.888

Supporting ETs: 8[+9, +7, +13], 11, 14[-13], 19[+9, +7, ++13], 25[-13], 36, 47, 58, 61[-13], 69[+13], 80[+13], 83, 91[+9, +7, +13], 105