Subgroup temperaments

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

Below are some subgroups and temperaments for them. 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.

No-elevens subgroup

Ultrapyth

For full 13-limit extensions, see Archytas clan #Ultrapyth.

Subgroup: 2.3.5.7.13

Comma list: 64/63, 91/90, 4394/4375

Gencom: [2 4/3; 64/63 91/90 4394/4375]

Gencom mapping: [1 2 8 2 0 11], 0 -1 -14 2 0 -18]]

Sval mapping: [1 2 8 2 11], 0 -1 -14 2 -18]]

POL2 generator: ~4/3 = 486.255

Vals5, 32, 37

RMS error: 2.318 cents

Sensi (aka Sensation)

For full 13-limit extensions, see Sensipent family or Sensi extensions.

Subgroup: 2.3.5.7.13

Comma list: 91/90, 126/125, 169/168

Gencom: [2 9/7; 91/90 126/125 169/168]

Gencom mapping: [1 6 8 11 0 10], 0 -7 -9 -13 0 -10]]

Sval mapping: [1 6 8 11 10], 0 -7 -9 -13 -10]]

POL2 generator: ~9/7 = 443.322

Vals19, 27, 46, 111de, 157de

RMS error: 1.321 cents

Quartonic

For full 13-limit extensions, see Orwellismic temperaments #Quartonic.

Subgroup: 2.3.5.7.13

Comma list: 169/168, 325/324, 640/637

Gencom: [2 36/35; 169/168 325/324 640/637]

Gencom mapping: [1 2 3 3 0 4], 0 -11 -18 -5 0 -8]]

Sval mapping: [1 2 3 3 4], 0 -11 -18 -5 -8]]

POL2 generator: ~36/35 = 45.1821

Vals26, 27, 53

RMS error: 0.7439 cents

Septidiasemi

For full 13- and 17-limit extensions, see Breedsmic temperaments #Septidiasemi.

Subgroup: 2.3.5.7.13

Comma list: 2205/2197, 2401/2400, 4096/4095

Gencom: [2 15/14; 2205/2197 2401/2400 4096/4095]

Gencom mapping: [1 -1 6 4 0 4], 0 26 -37 -12 0 -3]]

Sval mapping: [1 -1 6 4 4], 0 26 -37 -12 -3]]

POL2 generator: ~15/14 = 119.297

Vals10, 151, 161, 171

RMS error: 0.2002 cents

2.3.5.7.13.17

Subgroup: 2.3.5.7.13.17

Comma list: 833/832, 1275/1274, 2025/2023, 2205/2197

Gencom: [2 15/14; 833/832 1275/1274 2025/2023 2205/2197]

Gencom mapping: [1 -1 6 4 0 4 2], 0 26 -37 -12 0 -3 21]]

Sval mapping: [1 -1 6 4 4 2], 0 26 -37 -12 -3 21]]

POL2 generator: ~15/14 = 119.297

Vals10, 151, 161, 171

RMS error: 0.1867 cents

Pontiac

For full 13- and 17-limit extensions, see Schismatic family #Pontiac.

Subgroup: 2.3.5.7.13

Comma list: 625/624 729/728 4096/4095

Gencom: [2 4/3; 625/624 729/728 4096/4095]

Gencom mapping: [1 2 -1 19 0 -10], 0 -1 8 -39 0 33]]

Sval mapping: [1 2 -1 19 -10], 0 -1 8 -39 33]]

POL2 generator: ~3/2 = 701.773

Vals53, 118, 171, 224, 395

RMS error: 0.1525 cents

2.3.5.7.13.17

Subgroup: 2.3.5.7.13.17

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

Gencom: [2 4/3; 625/624 729/728 1225/1224 2880/2873]

Gencom mapping: [1 2 -1 19 0 -10 29], 0 -1 8 -39 0 33 -60]]

Sval mapping: [1 2 -1 19 -10 29], 0 -1 8 -39 33 -60]]

POL2 generator: ~3/2 = 701.764

Vals53, 118, 171, 395, 566

RMS error: 0.1696 cents

Mitonic

For full 13- and 17-limit extensions, see Minortonic family #Mitonic.

Subgroup: 2.3.5.7.13

Comma list: 4096/4095, 4375/4374, 13720/13689

Gencom: [2 10/9; 4096/4095 4375/4374 13720/13689]

Gencom mapping: [1 -1 -3 6 0 11], 0 17 35 -21 0 -48]]

Sval mapping: [1 -1 -3 6 11], 0 17 35 -21 -48]]

POL2 generator: ~10/9 = 182.471

Vals46, 125, 171, 388

RMS error: 0.1442 cents

2.3.5.7.13.17

Subgroup: 2.3.5.7.13.17

Comma list: 833/832, 1225/1224, 1701/1700, 4096/4095

Gencom: [2 10/9; 833/832 1225/1224 1701/1700 4096/4095]

Gencom mapping: [1 -1 -3 6 0 11 5], 0 17 35 -21 0 -48 -6]]

Sval mapping: [1 -1 -3 6 11 5], 0 17 35 -21 -48 -6]]

POL2 generator: ~10/9 = 182.471

Vals46, 125, 171, 388

RMS error: 0.1341 cents

Tertiaseptal

For full 13- and 17-limit extensions, see Breedsmic temperaments #Tertiaseptal.

Subgroup: 2.3.5.7.13

Comma list: 625/624, 2401/2400, 4096/4095

Gencom: [2 117/112; 625/624 2401/2400 4096/4095]

Gencom mapping: [1 3 2 3 0 1], 0 -22 5 -3 0 42]]

Sval mapping: [1 3 2 3 1], 0 -22 5 -3 42]]

POL2 generator: ~117/112 = 77.173

Vals31, 109, 140, 171, 311

RMS error: 0.1383 cents

2.3.5.7.13.17

Subgroup: 2.3.5.7.13.17

Comma list: 625/624, 833/832, 1225/1224, 4096/4095

Gencom: [2 68/65; 625/624 833/832 1225/1224 4096/4095]

Gencom mapping: [1 3 2 3 0 1 1], 0 -22 5 -3 0 42 48]]

Sval mapping: [1 3 2 3 1 1], 0 -22 5 -3 42 48]]

POL2 generator: ~68/65 = 77.177

Vals31, 109f, 140, 171, 311

RMS error: 0.1367 cents

Oquatonic

For full 13-limit extensions, see Horwell temperaments #Oquatonic.

Subgroup: 2.3.5.7.13

Comma list: 625/624, 4096/4095, 10985/10976

Gencom: [40/39 105/104; 625/624 4096/4095 10985/10976]

Gencom mapping: [28 44 65 79 0 104], 0 1 0 -1 0 -1]]

Sval mapping: [28 44 65 79 104], 0 1 0 -1 -1]]

POL2 generator: ~105/104 = 16.4240

Vals28, 56, 84, 140, 224, 364

RMS error: 0.1347 cents

No-sevens subgroup

Porkypine

Related temperament: Porcupine

Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

Gencom: [2 10/9; 55/54, 100/99]

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

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

POL2 generator: ~11/10 = 164.078

Vals7, 15, 22, 37, 73cd, 95cd, 117bcd

RMS error: 2.287 cents

Mohaha

Related temperament: mohajira, migration

Subgroup: 2.3.5.11

Comma list: 81/80, 121/120

Gencom: [2 11/9; 81/80 121/120]

Gencom mapping: [1 1 0 0 2], 0 2 8 0 5]]

Sval mapping: [1 1 0 2], 0 2 8 5]]

POL2 generator: ~11/9 = 348.094

Vals7, 10, 17, 24, 31, 55, 69d, 100d, 131bd

RMS error: 1.392 cents

Music

Mohaha10ping2 by Billy Stiltner

Tetracot

For full 11- and 13-limit extensions, see Tetracot family.

Subgroup: 2.3.5.11

Comma list: 100/99, 243/242

Gencom: [2 10/9; 100/99 243/242]

Gencom mapping: [1 1 1 0 2], 0 4 9 0 10]]

Sval mapping: [1 1 1 2], 0 4 9 10]]

POL2 generator: ~10/9 = 175.985

Vals7, 27d, 34, 41, 75d

RMS error: 1.182 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 100/99, 144/143, 243/242

Gencom: [2 10/9; 100/99 243/242]

Gencom mapping: [1 1 1 0 2 4], 0 4 9 0 10 -2]]

Sval mapping: [1 1 1 2 4], 0 4 9 10 -2]]

POL2 generator: ~10/9 = 176.196

Vals7, 27d, 34, 41, 75d

RMS error: 1.140 cents

Larry

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

Related temperaments: gravity, harry

Gencom: [2 40/27; 243/242 4000/3993]

Gencom mapping: [1 5 12 0 12], 0 -6 -17 0 -15]]

Sval mapping: [1 5 12 12], 0 -6 -17 -15]]

POL2 generator: ~40/27 = 683.166

Vals7, 58, 65, 137, 202

RMS error: 0.3025 cents

Twentcufo

For full 11- and 13-limit extensions, see Hemimean clan #Undetrita.

Subgroup: 2.3.5.11

Comma list: 8019/8000, 14641/14580

Gencom: [2 400/363; 8019/8000 14641/14580]

Gencom mapping: [1 0 -2 0 0], 0 11 30 0 24]]

Sval mapping: [1 0 -2 0], 0 11 30 24]]

POL2 generator: ~400/363 = 172.8796

Vals7, 111, 118

RMS error: 0.2393 cents

Emka

For full 11- and 13-limit extensions, see Hemimean clan #Emka or Horwell temperaments #Emkay.

Subgroup: 2.3.5.11

Comma list: 4000/3993, 9453125/9437184

Gencom: [2 11/8; 4000/3993 9453125/9437184]

Gencom mapping: [1 14 6 0 3], 0 -27 -8 0 1]]

Sval mapping: [1 14 6 3], 0 -27 -8 1]]

POL2 generator: ~11/8 = 551.7778

Vals37, 50, 87, 137, 224

RMS error: 0.1188 cents

2.3.5.11.13

Subgroup: 2.3.5.11.13

Comma list: 625/624, 2200/2197, 4000/3993

Gencom: [2 11/8; 625/624 2200/2197 4000/3993]

Gencom mapping: [1 14 6 0 3 6], 0 -27 -8 0 1 -5]]

Sval mapping: [1 14 6 3 6], 0 -27 -8 1 -5]]

POL2 generator: ~11/8 = 551.7753

Vals37, 50, 87, 137, 224

RMS error: 0.1250 cents

Majvam

For full 13- and 17-limit extensions, see Parkleiness temperaments #Majvamic or Cataharry temperaments #Majvamoid.

Subgroup: 2.3.5.13

Comma list: 676/675, 127401984/126953125

Gencom: [2 125/96; 676/675 127401984/126953125]

Gencom mapping: [1 10 5 0 0 19], 0 -22 -7 0 0 -40]]

Sval mapping: [1 10 5 19], 0 -22 -7 -40]]

POL2 generator: ~125/96 = 458.988

Vals34, 149, 183, 217, 400

RMS error: 0.1171 cents

2.3.5.13.17

Subgroup: 2.3.5.13.17

Comma list: 676/675, 2601/2600, 24576/24565

Gencom: [2 125/96; 676/675 2601/2600 24576/24565]

Gencom mapping: [1 10 5 0 0 19 6], 0 -22 -7 0 0 -40 -5]]

Sval mapping: [1 10 5 19 6], 0 -22 -7 -40 -5]]

POL2 generator: ~125/96 = 458.991

Vals34, 149, 183, 217, 400

RMS error: 0.1129 cents

Photia

Related temperament: Schismic

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

Vals12, 41, 53, 65

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

Vals12, 41, 53, 65

RMS error: 0.5374 cents

Nestoria

Related temperament: Schismic

Subgroup: 2.3.5.19

Comma list: 361/360, 513/512

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

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

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

POL2 generator: ~3/2 = 701.746

Vals12, 29, 41, 53, 118, 171

RMS error: 0.1763 cents

No-fives subgroup

Semaphore

Subgroup: 2.3.7

Comma: 49/48

Gencom: [2 8/7; 49/48]

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

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

POL2 generator: ~7/6 = 250.385

Vals5, 14, 19, 24, 67c, 91c

RMS error: 2.523 cents

Bleu

Subgroup: 2.3.7

Comma: 17496/16807

Gencom: [2 54/49; 17496/16807]

Gencom mapping: [1 1 0 2], 0 5 0 7]]

Sval mapping: [1 1 2], 0 5 7]]

POL2 generator: ~54/49 = 139.848

9, 17, 43, 60c

RMS error: 1.917 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Gencom: [2 12/11; 99/98 864/847]

Gencom mapping: [1 1 0 2 3], 0 5 0 7 4]]

Sval mapping: [1 1 2 3], 0 5 7 4]]

POL2 generator: ~12/11 = 140.005

9, 17, 43, 60c

RMS error: 1.829 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Gencom: [2 12/11; 78/77 99/98 144/143]

Gencom mapping: [1 1 0 2 3 3], 0 5 0 7 4 6]]

Sval mapping: [1 1 2 3 3], 0 5 7 4 6]]

POL2 generator: ~12/11 = 139.990

Vals17, 43, 60c

RMS error: 1.752 cents

Archy

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives dominant" or "no-fives superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Subgroup: 2.3.7

Comma: 64/63

Gencom: [2 3/2; 64/63]

Gencom mapping: [1 1 0 4], 0 1 0 -2]]

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

POL2 generator: ~3/2 = 709.321

Vals5, 12, 17, 22, 27, 137bc

RMS error: 1.856 cents

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

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

Sval mapping: [1 0 6 13], 0 1 -2 -6]]

POL2 generator: ~3/2 = 707.192

Vals5, 12, 17, 39c, 56c

RMS error: 1.977 cents

Supraphon

Subgroup: 2.3.7.11.13

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

Gencom: [2 3/2; 64/63 78/77 99/98]

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

Sval mapping: [1 0 6 13 18], 0 1 -2 -6 -9]]

POL2 generator: ~3/2 = 706.137

Vals5e, 12e, 17, 22, 39c, 56c

RMS error: 2.095 cents

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Gencom: [2 11/9; 64/63 243/242]

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

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

POL2 generator: ~11/9 = 353.958

Vals7, 10, 17, 44d, 61cd, 78cd

RMS error: 1.968 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

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

Gencom: [2 11/9; 64/63 78/77 144/143]

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

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

POL2 generator: ~11/9 = 353.775

Vals7, 10, 17, 44d, 61cd, 78cd

RMS error: 1.953 cents

Skwares

Related temperament: squares

Subgroup: 2.3.7

Comma: 19683/19208

Gencom: [2 9/7; 19683/19208]

Gencom mapping: [1 3 6], 0 -4 -9]]

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

POL2 generator: ~9/7 = 425.365

Vals14, 17, 31, 48, 79, 189b, 268bc, 347bc

RMS error: 1.149 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Gencom: [2 9/7; 99/98 243/242]

Gencom mapping: [1 3 0 6 7], 0 -4 0 -9 -10]]

Sval mapping: [1 3 6 7], 0 -4 -9 -10]]

POL2 generator: ~9/7 = 425.244

Vals14, 17, 31, 48, 79, 127, 206bcd

RMS error: 1.099 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 243/242

Gencom: [2 9/7; 78/77, 99/98, 243/242]

Gencom mapping: [1 3 0 6 7 9], 0 -4 0 -9 -10 -15]]

Sval mapping: [1 3 6 7 9], 0 -4 -9 -10 -15]]

POL2 generator: ~9/7 = 424.457

Vals17, 48e, 65de, 82c, 147ce

RMS error: 1.769 cents

Skwairs

Subgroup: 2.3.7.11.13

Comma list: 99/98, 144/143, 243/242

Gencom: [2 9/7; 99/98, 144/143, 243/242]

Gencom mapping: [1 3 0 6 7 3], 0 -4 0 -9 -10 2]]

Sval mapping: [1 3 6 7 3], 0 -4 -9 -10 2]]

POL2 generator: ~9/7 = 424.702

Vals14, 17, 31

RMS error: 1.290 cents

Leapfrog

See also: Gentle region

Subgroup: 2.3.7

Comma list: 14680064/14348907

Gencom: [2 3/2; 14680064/14348907]

Gencom mapping: [1 1 0 -6], 0 1 0 15]]

Sval mapping: [1 0 -21], 0 1 15]]

POL2 generator: ~3/2 = 704.721 cents

Vals17, 46, 63

RMS error: 0.6202 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 1331/1323

Gencom: [2 3/2; 896/891 1331/1323]

Gencom mapping: [1 1 0 -6 -3], 0 1 0 15 11]]

Sval mapping: [1 0 -21 -14], 0 1 15 11]]

POL2 generator: ~3/2 = 704.753 cents

Vals17, 46, 63

RMS error: 0.6047 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [1 1 0 -6 -3 -1], 0 1 0 15 11 8]]

Sval mapping: [1 0 -21 -14 -9], 0 1 15 11 8]]

POL2 generator: ~3/2 = 704.745 cents

Vals17, 46, 63

RMS error: 0.7541 cents

Music

Lee

Subgroup: 2.3.7

Comma: 177147/175616

Gencom: [2 81/56; 177147/175616]

Gencom mapping: [1 0 0 -3], 0 3 0 11]]

Sval mapping: [1 0 -3], 0 3 11]]

POL2 generator: ~81/56 = 633.525

Vals17, 36, 89, 125, 161, 358, 519b

RMS error: 0.3519 cents

Slendric

Subgroup: 2.3.7

Comma: 1029/1024

Gencom: [2 8/7; 1029/1024]

Gencom mapping: [1 1 0 3], 0 3 0 -1]]

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

POL2 generator: ~8/7 = 233.688

Vals5, 21, 26, 31, 36, 77, 113, 190

RMS error: 0.3202 cents

Baladic

Subgroup: 2.3.7.13

Comma list: 169/168, 1029/1024

Gencom: [91/64 8/7; 169/168 1029/1024]

Sval mapping: [2 2 6 7], 0 3 -1 1]]

POL2 generator: ~8/7 = 233.6044

Vals10, 26, 36, 154…, 190…, 226…, 262

RMS error: 0.5452 cents

2.3.7.13.17

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Gencom: [17/12 8/7; 169/168 273/272 289/288]

Sval mapping: [2 2 6 7 7], 0 3 -1 1 3]]

POL2 generator: ~8/7 = 233.6155

Vals10, 26, 36, 154…, 190…, 226

RMS error: 0.5073 cents

Hemif

Related temperament: hemififths, namo

Subgroup: 2.3.7

Comma: 1605632/1594323

Gencom: [2 2187/1792; 1605632/1594323]

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

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

POL2 generator: ~2187/1792 = 351.485

Vals7, 17, 41, 58, 99

RMS error: 0.2344 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Gencom: [2 11/9; 243/242 896/891]

Gencom mapping: [1 1 0 -1 2], 0 2 0 13 5]]

Sval mapping: [1 1 -1 2], 0 2 13 5]]

POL2 generator: ~11/9 = 351.535

Vals7, 17, 41, 58, 99d

RMS error: 0.6108 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 144/143, 243/242, 364/363

Gencom: [2 11/9; 144/143 243/242 364/363]

Gencom mapping: [1 1 0 -1 2 4], 0 2 0 13 5 -1]]

Sval mapping: [1 1 -1 2 4], 0 2 13 5 -1]]

POL2 generator: ~11/9 = 351.691

Vals7, 10, 17, 24, 41, 58

RMS error: 0.7167 cents

Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Gencom: [2 99/98; 41503/41472, 43923/43904]

Gencom mapping: [1 14/9 0 25/9 31/9], 0 2 0 2 1]]

Sval mapping: [9 0 11 24], 0 2 2 1]]

POL2 generator: ~99/98 = 17.6258

Vals54, 63, 72, 135, 342, 477, 1089, 1566

RMS error: 0.0383 cents

Parapyth (Rank 3)

See also: Pentacircle temperaments #Parapyth

Subgroup: 2.3.7.11

Comma list: 896/891

Gencom: [2 3/2 28/27; 896/891]

Gencom mapping: [1 1 0 1 4], 0 1 0 3 -1], 0 0 0 1 1]]

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

POL2 tuning: ~3 = 1903.834, ~7 = 3369.872

Vals17, 36, 41, 58, 63, 104

RMS error: 0.4149 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

The gencom below gives Margo Schulter's favored basis

Gencom: [2 3/2 28/27; 352/351 364/363]

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

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

POL2 tuning: ~3 = 1903.856, ~7 = 3369.907

Vals17, 41, 46, 58, 87, 104

RMS error: 0.3789 cents

Neutral

Neutral can be thought of as the 2.3.11 version of either mohajira or maqamic, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.

Subgroup: 2.3.11

Comma: 243/242

Gencom: [2 11/9; 243/242]

Gencom mapping: [1 1 0 0 2], 0 2 0 0 5]]

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

POL2 generator: ~11/9 = 350.525

Vals7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

Seven note albitonic scale

Ten note chromatic scale

Seventeen note mega chromatic scale

Namo

Subgroup: 2.3.11.13

Comma list: 144/143, 243/242

Gencom: [2 11/9; 144/143 243/242]

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

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

POL2 generator: ~11/9 = 351.488

Vals7, 10, 17, 24, 41

RMS error: 0.7038 cents

No-threes subgroup

Llywelyn

Subgroup: 2.5.7

Comma: 4194304/4117715

Gencom: [2 8/7; 4194304/4117715]

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

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

POL2 generator: ~8/7 = 226.910

Vals16, 37

RMS error: 0.5391 cents

Didacus

See also: Hemimean clan #Didacus

Related temperaments: roulette, hemithirds

Subgroup: 2.5.7

Comma: 3136/3125

Gencom: [2 28/25; 3136/3125]

Gencom map: [1 0 2 2], 0 0 2 5]]

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

POL2 generator: ~28/25 = 93.772

Vals6, 19, 25, 31, 37, 99, 130, 161, 353

RMS error: 0.2138 cents

Rainy

Three generators make an 8/7; five generators make a 5/4. This is the no-threes version of tertiaseptal.

Subgroup: 2.5.7

Comma: 2100875/2097152

Gencom: [2 256/245; 2100875/2097152]

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

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

POL2 generator: ~256/245 = 77.205

Vals31, 47, 78, 109, 140, 171, 202, 233

RMS error: 0.0586 cents

Mercy

See also: Quince clan #Mercy

Two generators make an 8/7; seven generators make an 8/5. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of 31edo, and is the no-threes or elevens version of miracle.

Subgroup: 2.5.7

Comma list: 823543/819200

Gencom: [2 2744/2560; 823543/819200]

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

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

POL2 generator: ~343/320 = 116.291

Vals10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd

2.5.7.13

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

Gencom: [2 14/13; 343/338 640/637]

Gencom mapping: [1 0 3 3 4], 0 0 -7 -2 -3]]

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

POL2 generator: ~14/13 = 116.094

Vals10, 21, 31

2.5.7.13.17

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 640/637

Gencom: [2 14/13; 170/169 224/221 640/637]

Gencom mapping: [1 0 3 3 4 4], 0 0 -7 -2 -3 1]]

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

POL2 generator: ~14/13 = 115.769

Vals10, 21, 31

2.5.7.13.17.19

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 343/338, 640/637, 16384/16055

Gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]

Gencom mapping: [1 0 3 3 4 4 3], 0 0 -7 -2 -3 1 13]]

Sval mapping: [1 3 3 4 4 3], 0 -7 -2 -3 1 13]]

POL2 generator: ~14/13 = 115.716

Vals10, 21, 31, 52f

2.9.7.11 subgroup

Machine

Subgroup: 2.9.7.11

Commas: 64/63, 99/98

Gencom: [2 8/7; 64/63 99/98]

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

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

POL2 generator: ~8/7 = 214.384

Vals5, 6, 11, 17, 28

RMS error: 1.977 cents

Mechanism

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

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

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

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

POL2 generator: ~9/7 = 438.465

Vals8, 11, 30, 41, 52

RMS error: 0.4262 cents

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

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

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

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

POL2 generator: ~77/72 = 115.570

Vals10, 21, 31, 52, 83, 135, 353, 488, 623

RMS error: 0.0673 cents

2.9.15.7 subgroup

Stacks (aka 2magic)

Subgroup: 2.9.15.7

Comma list: 225/224, 245/243

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

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

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

POL2 generator: ~9/7 = 439.296

Vals8, 11, 30, 41, 71, 93, 112c, 134c, 175c

RMS error: 1.074 cents

2.9.15.7.11

Subgroup: 2.9.15.7.11

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

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

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

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

POL2 generator: ~9/7 = 438.607

Vals: 8, 11, 30, 41, 52, 93, 145, 342bce

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

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

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

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

POL2 generator: ~9/7 = 438.977

Vals: 11, 30, 41, 153cdef, 194cdef, 235cdef

RMS error: 1.540 cents

2.9.21 subgroup

A-team

Subgroup: 2.9.21

Comma: 1029/1024

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

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

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

POL2 generator: ~21/16 = 467.375

Vals5, 13, 18, 41, 59, 77, 95

RMS error: 0.3202 cents

2.11.13.17.19 subgroup

Yamablu

Yamablu, with a generator of ~17/13, is named for it's tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The 13th Yamablu[13] scale is a linear-temperament version of Gjaeck.

Subgroup: 2.11.13.17.19

Comma list: 209/208, 2057/2048, 83521/83486

Sval mapping: [1 5 1 1 0], 0 -4 7 8 11]]

POL2 generator: ~17/13 = 462.9606

Vals13, 44, 57, 70

RMS error: 0.4898 cents

Miscellaneous subgroup temperaments

Historical

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]]

POL2 generator: ~21/20 = 83.016

Vals14, 29, 72, 101, 130, 159

RMS error: 0.2562 cents

Hypnosis

Related temperament: 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]]

POL2 generator: ~13/9 = 633.518

Vals17, 36, 118e, 125e, 161e, 197e

RMS error: 0.5379 cents

Oceanfront

Subgroup: 2.3.7.13/5

Related temperament: superpyth, ultrapyth

Comma list: 64/63, 91/90

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

POL2 generator: ~3/2 = 713.910

Vals5, 22, 27, 32, 37

RMS error: 2.063 cents