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
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
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
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
RMS error: 1.226 cents
Related temperament: meantone
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
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
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
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
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
- 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
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
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
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
- 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
RMS error: 0.7038 cents
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
2.3.7.41 subgroup
Subgroup: 2.3.7.41
Comma list: 64/63, 82/81
Gencom: [2 4/3; 64/63 82/81]
Sval mapping: [⟨1 2 2 7], ⟨0 -1 2 -4]]
POTE generator: ~4/3 = 490.0323
TOP generators: ~2 = 1197.2342, ~4/3 = 488.9029
Optimal GPV sequence: Template:Val list
2.3.7.11.41 subgroup
Subgroup: 2.3.7.11.41
Comma list: 64/63, 82/81, 99/98
Gencom: [2 4/3; 64/63 82/81 99/98]
Sval mapping: [⟨1 2 2 1 7], ⟨0 -1 2 6 -4]]
POTE generator: ~4/3 = 492.1787
TOP generators: ~2 = 1197.9683, ~4/3 = 491.3454
Optimal GPV sequence: Template:Val list
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
Hectosaros leap week
Defined as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.
Subgroup: 2.3.7
Comma list: [-50 -746 439⟩
Mapping: [⟨1 313 532], ⟨0 -439 -746]]
Optimal tuning (CTE): ~[17 343 143⟩ = 851.248
2.3.7.13 subgroup
Subgroup: 2.3.7.13
Comma list: [-42 -2 -5 16⟩, [10 -46 29 -5⟩
Mapping: [⟨1 313 532 208], ⟨0 -439 -746 -288]]
Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248
2.3.7.13.17 subgroup
Subgroup: 2.3.7.13.17
Comma list: 39337984/39328497, [0 -14 7 4 -3⟩, [-18 -24 14 -1 5⟩
Mapping: [⟨1 313 532 208 58], ⟨0 -439 -746 -288 -76]]
Optimal tuning (CTE): ~6144/3757 = 851.248
2.3.7.13.17.19 subgroup
Subgroup: 2.3.7.13.17
Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256
Mapping: [⟨1 313 532 208 58 432], ⟨0 -439 -746 -288 -76 -603]]
Optimal tuning (CTE): ~6144/3757 = 851.248
Sematology
This temperament tempers out 4107/4096 and thus equates 2 37/32's with 4/3.
Subgroup: 2.3.37
Comma list: 4107/4096
Gencom: [2 37/32; 4107/4096]
Mapping: [⟨1 1 5], ⟨0 -2 1]]
POTE generator: ~37/32 = 249.075
2.3.7.37 subgroup
Subgroup: 2.3.7.37
Comma list: 4107/4096, 259/256
Gencom: [2 37/32; 4107/4096 259/256]
Mapping: [⟨1 1 1 5], ⟨0 -2 -1 1]]
POTE generator: ~37/32 = 247.782
2.3.5.37 subgroup
It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.
Subgroup: 2.3.5.37
Comma list: 4107/4096, 17592186044416/17562397269605
Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605]
Mapping: [⟨1 1 4 5], ⟨0 -2 -8 1]]
POTE generator: ~37/32 = 251.393
2.3.5.7.37 subgroup
Subgroup: 2.3.5.7.37
Comma list: 4107/4096, 17592186044416/17562397269605, 259/256
Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]
Mapping: [⟨1 1 4 1 5], ⟨0 -2 -8 -1 1]]
POTE generator: ~37/32 = 251.204
Reversed mavila
Subgroup: 2.3.37
Comma list: 81/74
Gencom: [2 4/3; 81/74]
Mapping: [⟨1 1 0], ⟨0 -1 12]]
POTE generator: ~4/3 = 521.397
Aerophore
Subgroup: 2.3.11.19
Comma list: 363/361, 729/704
Mapping: [⟨1 0 -6 -6], ⟨0 2 12 13]]
POTE generator: ~19/11 = 945.4
Vals: 14, 5[+11, +19], 19, 33, 9[-11, -19], 47[-11, -19], 24[+11, +19], 52, 23[--11, --19], 61[-3, -11, -19], 43[+11, +19], 71[-3], 37[-3, --11, --19], 80[-3, -11, -19]
Semaerophore
Subgroup: 2.3.7.11.19
Comma list: 49/48, 77/76, 729/704
Mapping: [⟨1 0 2 -6 -6], ⟨0 2 1 12 13]]
POTE generator: ~7/4 = 944.667
Vals: 14, 5[+11, +19], 19, 9[-11, -19], 33[-7], 47[-7, -11, -19], 23[-7, --11, --19], 61[-3, -7, -11, -19], 37[-3, -7, --11, --19], 75[-3, --7, -11, --19], 89[-3, --7, --11, --19]