Subgroup temperaments
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; [91/90|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
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; [126|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
RMS error: 1.321 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; [2401|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
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; [1275|833/832 1275]/1274 [2205|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
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: [729|625/624 729]/728 4096/4095
Gencom: [2 4/3; [729|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
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; [729|625/624 729]/728 [2880|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
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; [4375|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
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; [1225|833/832 1225]/1224 [4096|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
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; [2401|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
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; [833|625/624 833]/832 [4096|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
RMS error: 0.1367 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
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; [121|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
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; [243|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
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; [243|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
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; [4000|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
RMS error: 0.3025 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, [[1]]
Gencom: [2 11/8; [9453125|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 = [[2]]
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 = [[3]]
RMS error: 0.1250 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
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
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
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
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; [864|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; [99/98|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; [78/77|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; [243|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; [78/77|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; [243|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
Leapfrog
Subgroup: 2.3.7
Comma list: [[4]]
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; [1331|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; [352|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
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; [1029|169/168 1029]/1024]
Sval mapping: [⟨2 2 6 7], ⟨0 3 -1 1]]
POL2 generator: ~8/7 = [[6]]
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; [273|169/168 273]/272 289/288]
Sval mapping: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]
POL2 generator: ~8/7 = [[7]]
RMS error: 0.5073 cents
Hemif
Related temperament: hemififths, namo
Subgroup: 2.3.7
Gencom: [2 2187/1792; [[9]]/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; [896|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; [243|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; [364|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; [243|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
No-threes subgroup
Llywelyn
Subgroup: 2.5.7
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
RMS error: 0.5391 cents
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
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
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
RMS error: 0.0586 cents
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
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: ~2744/2560 = 116.291
2.5.7.13
Subgroup: 2.5.7.13
Comma list: 343/338, 640/637
Gencom: [2 14/13; [640|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
2.5.7.13.17
Subgroup: 2.5.7.13.17
Comma list: 170/169, 224/221, 640/637
Gencom: [2 14/13; [224|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
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; [343|170/169 343]/338 [16384|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
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
RMS error: 1.977 cents
Mechanism
Subgroup: 2.9.7.11
Comma list: 896/891, 26411/26244
Gencom: [2 9/7; [26411|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
RMS error: 0.4262 cents
Apparatus
Subgroup: 2.9.7.11
Comma list: 41503/41472, [[12]]
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
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; [245|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
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; [225|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: Template:Val list
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; [105|100/99 105]/104 [196|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: Template:Val list
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
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 = [[13]]
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
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
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
RMS error: 2.063 cents