No-fives subgroup temperaments

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This is a collection of subgroup temperaments which omit the prime harmonic of 5.

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

Optimal GPV sequence5, 14, 19, 24, 67d, 91d

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, 60d

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, 60d

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

Optimal GPV sequence17, 43, 60d

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

Optimal GPV sequence5, 12, 17, 22, 27, 137bd

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

Optimal GPV sequence5, 12, 17, 39d, 56d

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

Optimal GPV sequence5f, 12f, 17, 22, 39d, 56d

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

Optimal GPV sequence7, 10, 17, 44e, 61de, 78de

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

Optimal GPV sequence7, 10, 17, 44e, 61de, 78de

RMS error: 1.953 cents

Skwares

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

Optimal GPV sequence14, 17, 31, 48, 79, 189b, 268bd, 347bd

RMS error: 1.149 cents

Related temperament: squares

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

Optimal GPV sequence14, 17, 31, 48, 79, 127, 206bde

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

Optimal GPV sequence17, 48f, 65ef, 82d, 147df

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

Optimal GPV sequence14, 17, 31

RMS error: 1.290 cents

Harrison

Subgroup: 2.3.7

Comma: 59049/57344

Gencom: [2 3/2; 59049/57344]

Gencom mapping: [1 1 0 -3], 0 1 0 10]]

Sval mapping: [1 1 -3], 0 1 10]]

POL2 generator: ~3/2 = 696.544

Optimal GPV sequence12, 19, 31, 112b, 143b, 174b

RMS error: 1.226 cents

Related temperament: meantone

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

Optimal GPV sequence17, 46, 63

RMS error: 0.6202 cents

Related temperaments: leapday, leapweek, srutal

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

Optimal GPV sequence17, 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

Optimal GPV sequence17, 46, 63

RMS error: 0.7541 cents

Skidoo

Subgroup: 2.3.7.11.13.23

Comma list: 169/168, 208/207, 352/351, 364/363

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

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

Sval mapping: [1 0 -21 -14 -9 -5], 0 1 15 11 8 6]]

POL2 generator: ~3/2 = 704.729 cents

Optimal GPV sequence17, 46, 63

RMS error: 0.6265 cents

2.3.7.11.13.23.29

Subgroup: 2.3.7.11.13.23.29

Comma list: 169/168, 208/207, 232/231, 352/351, 364/363

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

Gencom mapping: [1 1 0 -6 -3 -1 0 0 1 -11], 0 1 0 15 11 8 0 0 6 27]]

Sval mapping: [1 0 -21 -14 -9 -5 -38], 0 1 15 11 8 6 27]]

POL2 generator: ~3/2 = 704.729 cents

Optimal GPV sequence17, 46, 63

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

Optimal GPV sequence17, 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

Optimal GPV sequence5, 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

Optimal GPV sequence10, 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

Optimal GPV sequence10, 26, 36, 154…, 190…, 226

RMS error: 0.5073 cents

Hemif

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

Optimal GPV sequence7, 17, 41, 58, 99

RMS error: 0.2344 cents

Related temperaments: hemififths, namo

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

Optimal GPV sequence7, 17, 41, 58, 99e

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

Optimal GPV sequence7, 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

Optimal GPV sequence54, 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

Optimal GPV sequence17, 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

Optimal GPV sequence17, 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

Optimal GPV sequence7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

Scales

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

Optimal GPV sequence7, 10, 17, 24, 41

RMS error: 0.7038 cents

Reversed meantone

Main article: Reversed meantone

Subgroup: 2.3.41

Comma list: 82/81

Gencom: [2 4/3; 82/81]

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

POL2 generator: ~4/3 = 494.509

Optimal GPV sequence5, 12, 17

Magi

Subgroup: 2.3.7.11

Comma list: 896/891, 537824/531441

Gencom: [2 96/77; 896/891, 537824/531441]

Balthazar

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 169/168

Caspar

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 144/143

Melchior

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 364/363

Hogwarts

Subgroup: 2.3.7.29

Comma list: 537824/531441, 784/783

Gencom: [2 36/29; 537824/531441, 784/783]

Twenothology

Subgroup: 2.3.7.11.13.29

Comma list: 537824/531441, 896/891, 144/143, 784/783