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-sevens subgroup
Porkypine
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
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
RMS error: 1.392 cents
Music
Mohaha10ping2 by Billy Stiltner
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
RMS error: 0.3025 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
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 suhajira
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
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
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
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
2.3.7.11 skwares
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 skwares
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
2.3.7.11.13 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
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 bleu
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 bleu
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
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 hemif
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 hemif
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
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 leapfrog
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 leapfrog
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
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 parapyth
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
No-threes subgroup
Didacus
Subgroup: 2.5.7
Comma: 3136/3125
Related temperaments: roulette, hemithirds
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
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
RMS error: 0.5391 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: 0.058596 cents
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
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
RMS error: 0.0673 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
RMS error: 0.4262 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
RMS error: 1.074 cents
2.9.15.7.11 stacks
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: Template:Val list
RMS error: 1.226 cents
2.9.15.7.11.13 stacks
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: 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
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
Subgroup: 2.3.7.11/5.13
Comma list: 169/168, 540/539, 729/728
Related temperament: hypnos, tricot
Sval mapping: [⟨1 0 -3 8 0], ⟨0 3 11 -13 7]]
POL2 generator: ~13/9 = 633.518
RMS error: 0.5379 cents