No-fives subgroup temperaments
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
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
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
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
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
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
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
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
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
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
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
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
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
RMS error: 1.290 cents
Harrison
Subgroup: 2.3.7
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
Leapfrog
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
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
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
RMS error: 0.7541 cents
- Music
- Suite for Harpsichord in A Locrian, tuning: Eb-G# in 46EDO by IlL (in progress):
- I. Prelude
- II. Allemande
- III. Courante
- IV. Sarabande (score, 17EDO version)
- V. Menuet and Trio
- VI. Gavotte I and II
- VII. Gigue
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
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
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
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
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
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
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
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
RMS error: 0.0383 cents
Parapyth (rank 3)
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
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
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
RMS error: 0.3021 cents
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
RMS error: 0.7038 cents