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.
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
Optimal GPV sequence: 460, 1329, 1789
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
Subgroup: 2.9.5.7.11.13
Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156
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 GPV 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 GPV 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 GPV 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 GPV 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
Vals: 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 GPV 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 GPV 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 GPV sequence: 5, 13, 31
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 GPV 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 GPV 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 GPV 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 mapping: [⟨1 2 1], ⟨0 1 -2]]
Optimal tuning (subgroup POTE): ~5/4 = 383.638
Optimal GPV sequence: 3, 13, 16, 19, 22, 25
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 GPV 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