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
Antikythera
Antikythera is every other step of pajara.
Subgroup: 2.9.5.7
Comma list: 50/49, 64/63
Subgroup-val mapping: [⟨2 0 11 12], ⟨0 1 -1 -1]]
- mapping generators: ~7/5, ~9
Gencom mapping: [⟨2 3 5 6], ⟨0 1/2 -1 -1]]
- gencom: [7/5 8/7; 50/49 64/63]
Optimal tuning (POTE): ~7/5 = 1\2, ~8/7 = 214.095
Optimal ET sequence: 4, 6, 16, 22, 28
Badness: 0.00501
Commatose
Commatose is a dual-fifth 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 & 1329 temperament. In the 13-limit extension 24 generators are equal to ~13/9.
Subgroup: 2.9.5.7
Comma list: [28 -2 -19 8⟩, [9 -25 23 6⟩
Subgroup-val mapping: [⟨1 9 6 13], ⟨0 -298 -188 -521]]
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4765
Optimal ET sequence: 460, 869, 1329
Badness: 0.611
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⟩
Sval mapping: [⟨1 9 6 13 16], ⟨0 -298 -188 -521 -641]]
Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767
Optimal ET sequence: 460, 869e, 1329, 1789, 3118
Badness: 0.165
2.9.5.7.11.13
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): ~2 = 1\1, ~3575/3528 = 23.4767
Optimal ET sequence: 460, 869e, 1329, 1789, 3118
Badness: 0.0564
2.9.5.7.11 subgroup
Tutone
Related temperaments: meantone, belobog, hemithirds, hemiwürschmidt
Subgroup: 2.9.5.7.11
Comma list: 81/80, 99/98, 126/125
Subgroup-val mapping: [⟨1 0 -4 -13 -25], ⟨0 1 2 5 9]]
- mapping generators: ~2, ~9
Gencom mapping: [⟨1 3/2 2 2 2], ⟨0 1/2 2 5 9]]
- gencom: [2 9/8; 81/80 99/98 126/125]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 193.937
Optimal ET sequence: 25, 31, 68b, 99b
Badness: 0.00536
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
Optimal ET sequence: 5, 6, 11, 17, 28
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
Optimal ET sequence: 8, 11, 30, 41, 52
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
Optimal ET sequence: 10, 21, 31, 52, 83, 135, 353, 488, 623
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
Optimal ET sequence: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c
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
Optimal ET sequence: 8, 11, 30, 41, 52, 93, 145, 342bce
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 ET sequence: 11, 30, 41, 153cdef, 194cdef, 235cdef
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
Optimal ET sequence: 5, 13, 18, 41, 59, 77, 95
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 ET sequence: 5, 13, 31
7.8.9 subgroup
Sixscared
Subgroup: 7.8.9
Comma list: 64/63
Gencom: [7/1 9/8; 64/63]
Mapping: [⟨1 1 1], ⟨0 1 2]]
TE generator: ~9/8 = 218.0899
Ed7s: 15, 16
Ed8s: 16, 17
Ed9s: 17, 18
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
Optimal ET sequence: 14, 29, 72, 101, 130, 159
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
Optimal ET sequence: 17, 36, 118e, 125e, 161e, 197e
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
Optimal ET sequence: 5, 22, 27, 32, 37
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]]
Optimal ET sequence: 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269bc, 660bc
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
Optimal ET sequence: 3, …, 22, 25, 28, 31, 59
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 ET 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 ET 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 ET 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]]
Optimal ET sequence: 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]]
Optimal ET sequence: 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]]
Optimal ET sequence: 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]