Subgroup temperaments
A subgroup temperament is a regular temperament defined on a just intonation subgroup that is not a full p-limit group.
For temperaments that omit various prime harmonics, see:
- No-elevens subgroup temperaments
- No-sevens subgroup temperaments
- No-fives subgroup temperaments
- No-threes subgroup temperaments
- For no-twos, see Catalog of 3.5.7 subgroup rank two temperaments and Catalog of 3.5.11 subgroup rank two temperaments .
Below are some temperaments for composite subgroups and fractional subgroups. 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.
2.9.5.7 subgroup
Commatose
Commatose is a fifthless temperament which uses the Pythagorean comma as a generator. It was developed by Eliora to highlight the near-perfect expression of 9/8 by 1789edo, while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1789 temperament.
Subgroup: 2.9.5.7
Comma list: [28 -2 -19 8⟩, [9 -25 23 6⟩
Mapping: [⟨1 9 6 13], ⟨0 -298 -188 -521]]
Optimal tuning (POTE): ~531441/524288 = 23.4763
2.9.5.7.11
Subgroup: 2.9.5.7.11
Comma list: [-7 7 -3 2 -4⟩, [17 0 -13 1 3⟩, [11 -2 -6 7 -3⟩
Mapping: ⟨1 9 6 13 16], ⟨0 -298 -188 -521 -641]
Optimal tuning (POTE): ~531441/524288 = 23.4766
2.9.5.7.11.13
24 generators are equal to 13/9.
Subgroup: 2.9.5.7.11.13
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
Sval mapping: [⟨0 9 6 13 16 10], ⟨-298 -188 -521 -641 -322]]
Optimal tuning (CTE): ~3575/3528 = 23.477
Antikythera
Subgroup: 2.9.5.7
Commas: 50/49, 64/63
Related temperament: pajara
POT2 generator: ~8/7 = 214.095
Gencom: [7/5 8/7; 50/49 64/63]
Gencom mapping: [<2 3 5 6|, <0 1/2 -1 -1|]
Mapping: [<2 0 11 12|, <0 1 -1 -1|]
RMS error: 2.572 cents
2.9.5.7.11 subgroup
Tutone
Subgroup: 2.9.5.7.11
Commas: 81/80, 99/98, 126/125
Related temperaments: meantone, belobog, hemithirds, hemiwürschmidt
POT2 generator: ~9/8 = 193.937
Gencom: [2 9/8; 81/80 99/98 126/125]
Gencom mapping: [<1 3/2 2 2 2|, <0 1/2 2 5 9|]
Mapping: [<1 0 -4 -13 -25|, <0 1 2 5 9|]
RMS error: 1.439 cents
2.9.7.11 subgroup
Machine
Subgroup: 2.9.7.11
Comma list: 64/63, 99/98
Sval mapping: [⟨1 0 6 13], ⟨0 1 -1 -3]]
Gencom: [2 8/7; 64/63 99/98]
Gencom mapping: [⟨1 3/2 0 3 4], ⟨0 1/2 0 -1 -3]]
Optimal tuning (POTE): ~8/7 = 214.384
RMS error: 1.977 cents
Mechanism
Subgroup: 2.9.7.11
Comma list: 896/891, 26411/26244
Sval mapping: [⟨1 5 5 2], ⟨0 -5 -6 4]]
Gencom: [2 9/7; 896/891 26411/26244]
Gencom mapping: [⟨1 5/2 0 5 2], ⟨0 -5/2 0 -6 4]]
Optimal tuning (POTE): ~9/7 = 438.465
RMS error: 0.4262 cents
Apparatus
Subgroup: 2.9.7.11
Comma list: 41503/41472, 322102/321489
Sval mapping: [⟨1 5 3 5], ⟨0 -19 -2 -16]]
Gencom: [2 77/72; 41503/41472 322102/321489]
Gencom mapping: [⟨1 5/2 0 3 5], ⟨0 -19/2 0 -2 -16]]
Optimal tuning (POTE): ~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
Sval mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]
Gencom: [2 9/7; 225/224 245/243]
Gencom mapping: [⟨1 5/2 5/2 5], ⟨0 -5/2 -1/2 -6]]
Optimal tuning (subgroup POTE): ~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
Sval mapping: [⟨1 0 2 -1 6], ⟨0 5 3 6 -4]]
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]]
Optimal tuning (subgroup POTE): ~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
Sval mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 3 6 -4 9]]
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]]
Optimal tuning (subgroup POTE): ~9/7 = 438.977
Optimal GPV sequence: Template:Val list
RMS error: 1.540 cents
2.9.21 subgroup
A-team
Subgroup: 2.9.21
Comma list: 1029/1024
Sval mapping: [⟨1 2 4], ⟨0 3 1]]
Gencom: [2 21/16; 1029/1024]
Gencom mapping: [⟨1 1 0 3], ⟨0 3/2 0 -1/2]]
Optimal tuning (subgroup POTE): ~21/16 = 467.375
RMS error: 0.3202 cents
2.9.5.21.11
Subgroup: 2.9.5.21.11
Comma list: 81/80, 99/98, 385/384
Sval mapping: [⟨1 2 0 4 5], ⟨0 3 6 1 -4]]
Gencom: [2 21/16; 81/80 99/98 385/384]
Gencom mapping: [⟨1 1 0 3 5], ⟨0 3/2 6 -1/2 -4]]
Optimal tuning (subgroup POTE): ~21/16 = 463.956
Optimal GPV sequence: Template:Val list
Fractional 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]]
Optimal tuning (subgroup POTE): ~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
Sval mapping: [⟨1 0 -3 8 0], ⟨0 3 11 -13 7]]
Optimal tuning (subgroup POTE): ~13/9 = 633.518
RMS error: 0.5379 cents
Related temperament: hypnos, tricot
Oceanfront
Subgroup: 2.3.7.13/5
Comma list: 64/63, 91/90
Sval mapping: [⟨1 0 6 -5], ⟨0 1 -2 4]]
Optimal tuning (subgroup POTE): ~3/2 = 713.910
RMS error: 2.063 cents
Related temperament: superpyth, ultrapyth
Marveltri
Subgroup: 2.5.9/7
Comma list: 225/224
Sval name: 3&13
Related temperaments: marvel, 22&47, magic
Optimal tuning (subgroup POTE): ~5/4 = 383.638
Gencom: [2 5/4; 225/224]
Gencom mapping: [<1 2/5 2 -1/5|, <0 -4/5 1 2/5|]
Sval mapping: [⟨1 2 1], ⟨0 1 -2]]
RMS error: 0.4801 cents
Sulis
Subgroup: 2.5.9/7.11/9
Comma list: 99/98, 176/175
Sval mapping: [⟨1 2 1 -1], ⟨0 1 -2 4]]
Optimal tuning (subgroup POTE): ~5/4 = 386.558
RMS error: 1.074 cents
Related temperament: minerva, würschmidt
Breedsmic
Subgroup: 2.3.49/5
Comma list: 2401/2400
Sval mapping: [⟨1 1 3], ⟨0 2 1]]
Optimal tuning (subgroup POTE): ~49/40 = 350.966
RMS error: ?
Related temperament: hemithirds, newt
Semiwolf
Subgroup: 3/2.7/4.5/2
Comma list: 245/243
Sval mapping: [⟨1 1 3], ⟨0 1 -2]]
Optimal tuning (POL2): ~7/6 = 262.1728
Optimal GPV sequence: 3edf, 5edf, 8edf
Semilupine
Subgroup: 3/2.7/4.5/2.11/4
Comma list: 100/99, 245/243
Sval mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]
POL2 generator: ~7/6 = 264.3771
Optimal GPV sequence: 8edf, 13edf
Hemilycan
Subgroup: 3/2.7/4.5/2.11/4
Comma list: 245/243, 441/440
Sval mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]
Optimal tuning (POL2): ~7/6 = 261.5939
Optimal GPV sequence: 8edf, 11edf
Greeley
Subgroup: 2.5/3.7/3.11/3
Commas: 121/120, 126/125
Related temperament: Opossum, Nusecond
POT2 generator: ~11/10 = 155.776
Gencom: [2 11/10; 121/120 126/125]
Gencom mapping: <1 -5/4 -1/4 3/4 3/4|, <0 9/4 1/4 -15/4 5/4|]
Mapping: [<1 1 2 2|, <0 -2 -6 -1|]
EDOs: 8, 15, 23, 54, 77, 100, 131d, 208bd
RMS error: 1.034 cents
Pepperoni
Subgroup: 2.3.11/7.13/7
Commas: 352/351, 364/363
Sval name: 5&12
Related temperament: The Pepper fifth, which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
POT2 generator: ~3/2 = 703.856
Gencom: [2 3/2; 352/351 364/363]
Gencom mapping: [<1 1 0 -8/3 1/3 7/3|, <0 1 0 11/3 -1/3 -10/3|]
Mapping: [<1 0 7 12|, <0 1 -4 -7|]
EDOs: 5, 7, 12, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595bcd
RMS error: 0.3789 cents
Hyperion
Subgroup: 5/2.7.11
Comma list: 341796875/329832448
Gencom: [5/2 125/88; 341796875/329832448]
POTE generator: ~125/88 = 595.602
Mapping: [⟨1 4 3], ⟨0 -5 -1]]
Vals: 8, 5[-7], 11[+7], 13[-7], 19[+7], 21[-7], 27[+7], 29[-7], 35[+7], 43[+7], 37[-7], 51[+7, +11], 45[-7], 59[+7, +11]
Doubleton
Subgroup: 3/2.7.13
Comma list: 1352/1323
CTE generator: ~26/21 = 1\2edf, ~28/9 = 1971.772
Mapping: [⟨2 4 7], ⟨0 1 1]]
Vals: 6, 10, 16, 14[-13], 8[+7], 22, 18[-13], 26, 24[-13], 28[+7], 20[+7], 36[-13], 12[+7, +13], 34[-13]
Auk
Subgroup: 3/2.7.13
Comma list: 87808/85293
CTE generator: ~28/9 = 1950.859
Mapping: [⟨1 2 -2], ⟨0 1 3]]
Vals: 5, 9, 13, 14, 6[+13], 17[-7, -13], 7[-7, -13], 22[-7], 19[+13], 23[+13], 11[+13], 21[-7, -13], 31[-7], 25[-7, -13]