Subgroup temperaments
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular 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-thirteens subgroup temperaments
- No-elevens subgroup temperaments
- No-sevens subgroup temperaments
- No-fives subgroup temperaments
- No-threes subgroup temperaments
- No-twos subgroup temperaments (additionally, 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.
Composite subgroup temperaments
2.9.5.7 subgroup
See also antikythera and isra.
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⟩
Sval 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
Daemotertiaschis
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a daemotonic 7L 4s scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.
Subgroup: 2.9.5.7.33.13.17
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
Sval mapping: [⟨1 1 11 -16 13 -18 20], ⟨0 3 -12 26 -11 30 -22]]
Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982
Supporting ETs: 47, 65f, 112, 159, 206, 253
Baldy
Baldy results from taking every other generator of the garibaldi temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.
Subgroup: 2.9.5.7
Comma list: 225/224, 3125/3087
Sval mapping: [⟨1 3 3 4], ⟨0 1 -4 -7]]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.170
Optimal ET sequence: 6, 29, 35, 41, 47
Related temperament: Garibaldi
2.9.5.7.13
Baldy is every other step of garibaldi, without the mapping of prime 11. It can be described as the 6 & 35 temperament.
Subgroup: 2.9.5.7.13
Comma list: 225/224, 325/324, 640/637
Sval mapping: [⟨1 0 15 25 -28], ⟨0 1 -4 -7 10]]
Gencom mapping: [⟨1 3/2 3 4 0 2], ⟨0 1/2 -4 -7 0 10]]
- gencom: [2 9/8; 225/224 325/324 640/637]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.090
Optimal ET sequence: 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd
RMS error: 0.5999 cents
Related temperament: Cassandra
Baldanders
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.
Subgroup: 2.9.5.7.11
Comma list: 100/99, 225/224, 245/242
Sval mapping: [⟨1 3 3 4 5], ⟨0 1 -4 -7 -9]]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743
Optimal ET sequence: 6, 23de, 29, 35, 41
Related temperament: Andromeda
2.9.5.7.11.13
Subgroup: 2.9.5.7.11.13
Comma list: 100/99, 144/143, 225/224, 245/242
Sval mapping: [⟨1 3 3 4 5 2], ⟨0 1 -4 -7 -9 10]]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414
Optimal ET sequence: 6, 23def, 29f, 35, 41, 47
2.9.5.11 subgroup
Glacial
Subgroup: 2.9.5.11.13
Comma list: 45/44, 65/64, 81/80
Sval mapping: [⟨1 0 -4 -6 10], ⟨0 1 2 3 -2]]
Gencom mapping: [⟨1 3/2 2 0 3 4], ⟨0 1/2 2 0 3 -2]]
- gencom: [2 9/8; 45/44 65/64 81/80]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 186.151
Optimal ET sequence: 6, 13, 45be, 58bce, 71bce, 84bce
RMS error: 2.887 cents
Music:
- Thundersnow - Sevish (2021)
2.9.7 subgroup
Mabon
Derived from a calendar leap cycle built for the autumn equinox, hence the name. Defined as the 11 & 62 temperament.
Subgroup: 2.9.7
Comma basis: 44957696/43046721
Sval mapping: [⟨1 1 -3], ⟨0 3 8]]
Optimal tuning (CTE): ~729/448 = 870.792
Optimal ET sequence: 7d, 11, 18d, 29, 40, 62, ...
2.9.7.11 subgroup
Subgroup: 2.9.7.11
Comma basis: 896/891, 1331/1296
Sval mapping: [⟨1 1 -3 2], ⟨0 3 8 2]]
Optimal tuning (CTE): ~16/11 = 870.966
Optimal ET sequence: 7d, 11, 40, 51, 62
2.9.7.11 subgroup
Apparatus
Subgroup: 2.9.7.11
Comma list: 41503/41472, 322102/321489
Sval mapping: [⟨1 5 3 5], ⟨0 -19 -2 -16]]
- mapping generators: ~2, ~77/72
Gencom mapping: [⟨1 5/2 0 3 5], ⟨0 -19/2 0 -2 -16]]
- gencom: [2 77/72; 41503/41472 322102/321489]
Optimal tuning (CTE): ~77/72 = 115.5685
Optimal ET sequence: 10e, 21, 31, 52, 83, 135, 353, 488, 623
Badness: 0.00263
Joan
Joan is related to casablanca as well as to orwell.
Subgroup: 2.9.7.11
Comma list: 99/98, 9317/9216
Sval mapping: [⟨1 0 1 3], ⟨0 7 4 1]]
Gencom mapping: [⟨1 0 0 1 3], ⟨0 7/2 0 4 1]]
- gencom: [2 11/8; 99/98 9317/9216]
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 542.672 cents
Optimal ET sequence: 11, 20, 31, 42, 115bd, 157bd
RMS error: 1.424 cents
Machine
Machine is every other step of supra, most interesting for its scale patterns.
Subgroup: 2.9.7.11
Comma list: 64/63, 99/98
Sval mapping: [⟨1 0 6 13], ⟨0 1 -1 -3]]
- sval mapping generators: ~2, ~9
Gencom mapping: [⟨1 3/2 0 3 4], ⟨0 1/2 0 -1 -3]]
- gencom: [2 8/7; 64/63 99/98]
Optimal ET sequence: 5, 6, 11, 17, 28
Badness: 0.00233
Penta a.k.a. mechanism
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.
Subgroup: 2.9.7.11
Comma list: 896/891, 26411/26244
Sval mapping: [⟨1 0 -1 6], ⟨0 5 6 -4]]
- sval mapping generators: ~2, ~14/9
Gencom mapping: [⟨1 5/2 0 5 2], ⟨0 -5/2 0 -6 4]]
- gencom: [2 9/7; 896/891 26411/26244]
Optimal tuning (POTE): ~2 = 1\1, ~14/9 = 761.3782
Optimal ET sequence: 8, 11, 30, 41, 52
RMS error: 0.4262 cents
Badness: 0.00439
Scales: penta5, penta8, penta11, penta19
2.9.11 subgroup
Demon
Demon is a temperament which equates 3 11/9 with 16/9, or equivalently 3 18/11 with 9/8, tempering out 1331/1296. This results in 11/9 being tuned flat to a supraminor third, and 27/22 being tuned sharp to a submajor third. It was discovered by CompactStar while searching for temperaments assosciated with the 7L 4s ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed 18edo supports demon temperament.
Subgroup: 2.9.11
Sval mapping: [⟨1 1 2], ⟨0 3 2]]
Optimal tuning (CTE): ~18/11 = 870.060
Optimal ET sequence: 4, 7, 11, 18, 29, 76e
Genius
Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.
Subgroup: 2.9.11
Sval mapping: [⟨1 1 4], ⟨0 4 -1]]
Optimal tuning (CTE): ~16/11 = 650.863
Optimal ET sequence: 9, 11, 24, 59, 83, 142, 225, 367[-11], 592[-11], 959[-9, --11], 1326[-9, --11]
2.9.15.7 subgroup
Stacks (a.k.a. 2magic)
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of magic.
Subgroup: 2.9.15.7
Comma list: 225/224, 245/243
Sval mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]
- sval mapping generators: ~2, ~14/9
Gencom mapping: [⟨1 5/2 5/2 5], ⟨0 -5/2 -1/2 -6]]
- gencom: [2 9/7; 225/224 245/243]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 760.704
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 mapping: [⟨1 5/2 5/2 5 2], ⟨0 -5/2 -1/2 -6 4]]
- gencom: [2 9/7; 100/99 225/224 245/243]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393
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 mapping: [⟨1 5/2 5/2 5 2 7], ⟨0 -5/2 -1/2 -6 4 -9]]
- gencom: [2 9/7; 100/99 105/104 144/143 196/195]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023
Optimal ET sequence: 11, 30, 41, 153cdef, 194cdef, 235cdef
RMS error: 1.540 cents
2.9.21 subgroup
A-team
A-team is every other step of slendric; the 2.9.5.21.11 extension below specifically restricts mothra.
Subgroup: 2.9.21
Comma list: 1029/1024
Sval mapping: [⟨1 2 4], ⟨0 3 1]]
- sval mapping generators: ~2, ~21/16
Gencom mapping: [⟨1 1 0 3], ⟨0 3/2 0 -1/2]]
- gencom: [2 21/16; 1029/1024]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 467.375
Optimal ET sequence: 5, 13, 18, 41, 59, 77, 95
RMS error: 0.3202 cents
2.9.5.21
Lookalike temperament: Dual-3 A-Team
Subgroup: 2.9.5.21
Comma list: 81/80, 1029/1024
Sval mapping: [⟨1 2 0 4], ⟨0 3 6 1]]
Mapping generators: ~2, ~21/16
Optimal (POL2) generator: 464.3865
Optimal ET sequence: 13, 18, 31, 44
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 mapping: [⟨1 1 0 3 5], ⟨0 3/2 6 -1/2 -4]]
- gencom: [2 21/16; 81/80 99/98 385/384]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956
Optimal ET sequence: 5, 13, 31
B-team
B-team (23 & 41) is every other step of rodan.
Subgroup: 2.9.15.21.33
Comma list: 245/243, 385/384, 441/440
Sval mapping: [⟨1 2 0 4 7], ⟨0 3 10 1 -5]]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918
Optimal ET sequence: 5, 13c, 18, 23, 41, 64, 87, 151
4.3.5 subgroup
Tetrahanson
Subgroup: 4.3.5
Comma list: 15625/15552
Sval mapping: [⟨1 3 3], ⟨0 -6 -5]]
- Mapping generators: ~4, ~5/3
Optimal tuning (CTE): ~4 = 2\1, ~5/3 = 882.941
Supporting ETs: 19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79
Tetrameantone
Subgroup: 4.3.5
Comma list: 81/80
Sval mapping: [⟨1 1 2], ⟨0 -1 -4]]
- Mapping generators: ~4, ~4/3
Optimal tuning (POTE): 4 = 2400.0, ~4/3 = 503.761
Supporting ETs: 5, 9, 14, 19, 24, 43, 62, 81, 100
Tetramagic
Subgroup: 4.3.5
Comma list: 3125/3072
Sval mapping: [⟨1 0 1], ⟨0 5 1]]
- Mapping generators: ~4, ~5/4
Optimal tuning (POTE): 4 = 2400.0, ~5/4 = 380.059
Supporting ETs: 6, 13, 19, 25, 38, 44, 63, 82
Blacktetra
Subgroup: 4.3.5
Comma list: 256/243
Sval mapping: [⟨5 4 6], ⟨0 0 -1]]
- Mapping generators: ~4, ~16/15
Optimal tuning (POTE): 1\5ed4 = 480.0, ~16/15 = 80.4062
Supporting ETs: 5, 10, 15, 20, 25, 30, 55, 85, 115
4.6.5 subgroup
Meanquad
Subgroup: 4.6.5
Comma list: 81/80 = [-4 4 -1⟩
Sval mapping: [⟨1 0 -4], ⟨0 1 4]]
- mapping generators: ~4, ~6
Optimal tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 697.214
Supporting ETs: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69
* Wart for 4
4.6.5.7 subgroup (tetrominant)
Subgroup: 4.6.5.7
Comma list: 36/35 = [0 2 -1 -1⟩, 64/63 = [4 -2 0 -1⟩
Sval mapping: [⟨1 0 -4 4], ⟨0 1 4 -2]]
Optimal tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 699.622
Supporting ETs: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]
* Wart for 4
Fourwar
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.
Fourwar is named after the closely related hemiwar temperament.
Reduced Mapping 4 6 5 [ ⟨ 1 0 1 ] ⟨ 0 16 2 ] ⟩ TE Generator Tunings (cents) ⟨2399.3973, 193.8643] TE Step Tunings (cents) ⟨25.21211, 47.81337] TE Tuning Map (cents) ⟨2399.397, 3101.829, 2787.126] TE Mistunings (cents) ⟨-0.603, -0.126, 0.812] Complexity 1.369085 Adjusted Error 0.692892 cents TE Error 0.268047 cents/octave Unison Vector [8, 1, -8⟩ (393216:390625) Subsets q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
4.6.5.7
Reduced Mapping 4 6 5 7 [ ⟨ 1 0 1 1 ] ⟨ 0 16 2 5 ] ⟩ TE Generator Tunings (cents) ⟨2399.4195, 193.8654] TE Step Tunings (cents) ⟨25.23883, 47.79592] TE Tuning Map (cents) ⟨2399.420, 3101.846, 2787.150, 3368.747] TE Mistunings (cents) ⟨-0.580, -0.109, 0.837, -0.079] Complexity 1.192044 Adjusted Error 0.653313 cents TE Error 0.232715 cents/octave Unison Vectors [-2, -1, -2, 4⟩ (2401:2400) [3, 0, -5, 2⟩ (3136:3125) [5, 1, -3, -2⟩ (6144:6125) [8, 1, -8, 0⟩ (393216:390625) Subsets q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
4.6.5.7.11
Reduced Mapping 4 6 5 7 11 [ ⟨ 1 0 1 1 1 ] ⟨ 0 16 2 5 9 ] ⟩ TE Generator Tunings (cents) ⟨2400.1097, 193.9498] TE Step Tunings (cents) ⟨24.18752, 48.52491] TE Tuning Map (cents) ⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658] TE Mistunings (cents) ⟨0.110, 1.241, 1.696, 1.033, -5.660] Complexity 1.068792 Adjusted Error 2.926965 cents TE Error 0.846083 cents/octave Unison Vectors [-1, -1, -1, 0, 2⟩ (121:120) [2, 0, -2, -1, 1⟩ (176:175) [-3, -1, 1, 1, 1⟩ (385:384) [-1, 0, 3, -3, 1⟩ (1375:1372) [-2, -1, -2, 4, 0⟩ (2401:2400) [1, 0, 1, -4, 2⟩ (2420:2401) Subsets q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
4.6.5.7.11.13
Reduced Mapping 4 6 5 7 11 13 [ ⟨ 1 0 1 1 1 0 ] ⟨ 0 16 2 5 9 23 ] ⟩ TE Generator Tunings (cents) ⟨2401.2305, 193.5378] TE Step Tunings (cents) ⟨42.79107, 35.98524] TE Tuning Map (cents) ⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371] TE Mistunings (cents) ⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843] Complexity 1.219191 Adjusted Error 6.699599 cents TE Error 1.810487 cents/octave Unison Vectors [0, 1, -1, 0, 1, -1⟩ (66:65) [-1, -1, -1, 0, 2, 0⟩ (121:120) [1, 2, 0, 0, -1, -1⟩ (144:143) [2, 0, -2, -1, 1, 0⟩ (176:175) [-2, 1, 1, 1, 0, -1⟩ (105:104) [-3, -1, 1, 1, 1, 0⟩ (385:384) [-3, 0, 0, 1, 2, -1⟩ (847:832) [1, 3, -1, 0, 0, -2⟩ (864:845) [-1, 0, 3, -3, 1, 0⟩ (1375:1372) Subsets q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
4.6.5.7.11.13.17
Reduced Mapping 4 6 5 7 11 13 17 [ ⟨ 1 0 1 1 1 0 1 ] ⟨ 0 16 2 5 9 23 13 ] ⟩ TE Generator Tunings (cents) ⟨2400.4701, 193.4599] TE Step Tunings (cents) ⟨43.39350, 35.55764] TE Tuning Map (cents) ⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449] TE Mistunings (cents) ⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494] Complexity 1.129881 Adjusted Error 8.082725 cents TE Error 1.977443 cents/octave Unison Vectors [0, 1, -1, 0, 1, -1, 0⟩ (66:65) [1, 1, 1, -1, 0, 0, -1⟩ (120:119) [1, 2, 0, 0, -1, -1, 0⟩ (144:143) [-2, 1, 1, 1, 0, -1, 0⟩ (105:104) [-1, 2, 2, 0, 0, -1, -1⟩ (225:221) [-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274) Subsets q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
4.6.5.7.11.13.17.19
Reduced Mapping 4 6 5 7 11 13 17 19 [ ⟨ 1 0 1 1 1 0 1 1 ] ⟨ 0 16 2 5 9 23 13 14 ] ⟩ TE Generator Tunings (cents) ⟨2399.9219, 193.3952] TE Step Tunings (cents) ⟨44.14256, 35.03670] TE Tuning Map (cents) ⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455] TE Mistunings (cents) ⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942] Complexity 1.058472 Adjusted Error 8.712222 cents TE Error 2.050935 cents/octave Unison Vectors [0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65) [-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76) [2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95) [1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119) [0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209) [0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931) [2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125) Subsets q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
4.6.5.7.11.13.17.19.23
Reduced Mapping 4 6 5 7 11 13 17 19 23 [ ⟨ 1 0 1 1 1 0 1 1 0 ] ⟨ 0 16 2 5 9 23 13 14 28 ] ⟩ TE Generator Tunings (cents) ⟨2399.3286, 193.5316] TE Step Tunings (cents) ⟨37.31613, 39.63311] TE Tuning Map (cents) ⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885] TE Mistunings (cents) ⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389] Complexity 1.115920 Adjusted Error 9.502017 cents TE Error 2.100561 cents/octave Unison Vectors [0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65) [1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91) [0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114) [1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119) [2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175) [-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384) [1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475) [1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833) [0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931) [1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867) Subsets q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
4.9.25 subgroup
Meansquared
Subgroup: 4.9.25
Sval mapping: [⟨1 3 4], ⟨0 1 4]]
Mapping generators: ~4, ~9/64
Optimal tuning (CTE): ~4 = 2\1, ~9/4 = 1394.429
Supporting ETs: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]
4.9.49 subgroup
Archsquared
Subgroup: 4.9.49
Comma list: 4096/3969
Sval mapping: [⟨1 3 0], ⟨0 1 -2]]
Mapping generators: ~4, ~9/64
Optimal tuning (CTE): ~9/8 = 219.190
Supporting ETs: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
8.9.7 subgroup
Sixscared
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."
Subgroup: 8.9.7
Comma list: 64/63
Sval mapping: [⟨1 0 2], ⟨0 1 -1]]
- sval mapping generators: ~8, ~9
- gencom: [8 9/8; 64/63]
Optimal tuning (CTE): 1\3ed8 = 1600.0, ~9/8 = 219.1898
Optimal ET sequence: ⟨16 17 15], ⟨33 35 31], ⟨148 …], ⟨181 …], ⟨214 …], ⟨247 …]
Badness: 0.0215 × 10-3
Fractional subgroup temperaments
2.5/3… subgroups
Magicaltet
Magicaltet is related to supermagic, superkleismic, and magic. The tonic and the first three generator steps make a magical seventh chord, hence the name.
Subgroup: 2.5/3.7.11
Comma list: 100/99 ([2 2 0 -1⟩), 385/384 ([-7 1 1 1⟩)
Sval mapping: [⟨1 0 5 2], ⟨0 1 -3 2]]
- mapping generators: ~2, ~5/3
Gencom mapping: [⟨1 -1/2 1/2 2 4], ⟨0 1/2 -1/2 3 -2]]
- gencom: [2 6/5; 100/99 385/384]
Optimal ET sequence: 4, 7, 11, 15, 26, 67, 93*
* Wart for 5/3
RMS error: 1.206 cents
Starlingtet
Starlingtet, the 4 & 15 temperament in the 2.5/3.7/3 subgroup, is related to starling as well as to myna. The tonic and the first three generator steps make a starling tetrad, hence the name.
Subgroup: 2.5/3.7/3
Comma list: 126/125 ([1 -3 1⟩)
Sval mapping: [⟨1 0 -1], ⟨0 1 3]]
- mapping generators: ~2, ~5/3
Gencom mapping: [⟨1 -1 0 1], ⟨0 4/3 1/3 -5/3]]
- gencom: [2 6/5; 126/125]
Optimal ET sequence: 4, 15, 19, 23, 27
RMS error: 0.8398 cents
Greeley
Greeley is related to opossum as well as to nusecond.
Subgroup: 2.5/3.7/3.11/3
Comma list: 121/120 ([-3 -1 0 2⟩), 126/125 ([1 -3 1⟩)
Sval mapping: [⟨1 1 2 2], ⟨0 -2 -6 -1]]
Gencom mapping: [⟨1 -5/4 -1/4 3/4 3/4], ⟨0 9/4 1/4 -15/4 5/4]]
- gencom: [2 11/10; 121/120 126/125]
Optimal ET sequence: 8, 15, 23, 54, 77, 100, 131*
* Wart for 11/3
RMS error: 1.034 cents
Skateboard
Skateboard is related to thrasher.
Subgroup: 2.5/3.7/3.11.13/9
Comma list: 56/55 ([3 -1 1 -1⟩), 91/90 ([-1 -1 1 0 1⟩), 100/99 ([2 2 0 -1⟩)
Sval mapping: [⟨1 0 -1 2 2], ⟨0 1 3 2 -2]]
Gencom mapping: [⟨1 -3/7 4/7 11/7 4 -6/7], ⟨0 0 -1 -3 -2 2]]
- gencom: [2 6/5; 56/55 91/90 100/99]
Optimal ET sequence: 11, 15, 19, 23, 42d, 65d
RMS error: 2.396 cents
Gariberttet
Gariberttet is the 2.5/3.7/3 altergene of sirius.
Gariberttet (2.5/3.7/3.13/11 subgroup)
Gariberttet can be described as the 4 & 29 temperament in the 2.5/3.7/3.13/11 subgroup.
Subgroup: 2.5/3.7/3.13/11
Comma list: 275/273 ([0 2 -1 -1⟩), 847/845 ([0 -1 1 -2⟩)
Sval mapping: [⟨1 0 0 0], ⟨0 3 5 1]]
Gencom mapping: [⟨1 0 0 0 0 0], ⟨0 -8/3 1/3 7/3 -1/2 1/2]]
- gencom: [2 13/11; 275/273 847/845]
Optimal ET sequence: 29, 33, 37, 41, 45, 49, 78, 94, 143*
* Wart for 13/11
RMS error: 0.6914 cents
Indium
Indium can be described as the 8 & 33 temperament in the 2.5/3.7/3.11/3 subgroup.
Subgroup: 2.5/3.7/3.11/3
Comma list: 3025/3024 ([-4 2 -1 2⟩), 3125/3087 ([0 5 -3⟩)
Sval mapping: [⟨1 0 0 2], ⟨0 6 10 -1]]
Gencom mapping: [⟨1 -1/2 -1/2 -1/2 3/2], ⟨0 -15/4 9/4 25/4 -19/4]]
- gencom: [2 12/11; 3025/3024 3125/3087]
Optimal ET sequence: 8, 33, 41, 49, 204*†
* Wart for 7/3
† Wart for 11/3
RMS error: 0.7788 cents
Ammon
Ammon can be described as the 8 & 29 temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends tridec, and is related to ammonite. It is generated by a semidiminished fourth, hence the old name semidim, which has been rejected since 2025 to avoid confusion with another temperament of the same name.
Subgroup: 2.5/3.7/3.11/3.13/3
Comma list: 121/120 ([-3 -1 0 2⟩), 169/168 ([-3 0 -1 0 2⟩), 275/273 ([0 2 -1 1 -1⟩)
Sval mapping: [⟨1 3 5 3 4], ⟨0 -6 -10 -3 -5]]
Gencom mapping: [⟨1 -3 0 2 0 1], ⟨0 24/5 -6/5 -26/5 9/5 -1/5]]
- gencom: [2 13/10; 121/120 169/168 275/273]
Optimal ET sequence: 8, 29, 37, 45
RMS error: 1.052 cents
Sentry
Sentry, the 3 & 5 temperament in the 2.5/3.9/7 subgroup, is related to sensi.
Subgroup: 2.5/3.9/7
Comma list: 245/243 ([0 1 -2⟩)
Sval mapping: [⟨1 0 0], ⟨0 2 1]]
Gencom mapping: [⟨1 0 0 0], ⟨0 0 2 -1]]
- gencom: [2 9/7; 245/243]
Optimal ET sequence: 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*†
* Wart for 5/3
† Wart for 9/7
RMS error: 0.7105 cents
Marveltwintri
Marveltwintri can be described as the 3 & 4 temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a marveltwin triad, hence the name.
Subgroup: 2.5/3.13/9
Comma list: 325/324 ([-2 2 1⟩)
Sval mapping: [⟨1 0 2], ⟨0 1 -2]]
Gencom mapping: [⟨1 -1/6 5/6 0 0 -1/3], ⟨0 -1/2 -3/2 0 0 1]]
- gencom: [2 6/5; 325/324]
Optimal ET sequence: 3, 4, 11, 15, 19, 34, 53, 87, 140
RMS error: 0.2444 cents
2.….7/3… subgroups
Guanyintet
Guanyintet, the 4 & 9 temperament in the 2.5.7/3.11/3 subgroup, is related to guanyin as well as to orwell. The tonic and the first three generator steps make a guanyin tetrad, hence the name.
Subgroup: 2.5.7/3.11/3
Comma list: 176/175 ([4 -2 -1 1⟩), 540/539 ([2 1 -2 -1⟩)
Sval mapping: [⟨1 0 2 -2], ⟨0 3 -1 5]]
- mapping generators: ~2, ~12/7
Gencom mapping: [⟨1 -4/3 3 -1/3 5/3], ⟨0 4/3 -3 7/3 -11/3]]
- gencom: [2 7/6; 176/175 540/539]
Optimal ET sequence: 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c*
* wart for 7/3
RMS error: 0.6028 cents
Laz
Laz is related to avalokita as well as to winston.
Subgroup: 2.5.7/3.11/3.13/3
Comma list: 144/143 ([4 0 0 -1 -1⟩), 176/175 ([4 -2 -1 1⟩), 196/195 ([2 -1 2 0 -1⟩
Sval mapping: [⟨1 0 2 -2 6], ⟨0 3 -1 5 -5]]
Gencom mapping: [⟨1 -5/4 3 -1/4 7/4 -1/4], ⟨0 -1/4 -3 3/4 -21/4 19/4]]
- gencom: [2 7/6; 144/143 176/175 196/195]
Optimal ET sequence: 9, 31, 40, 49, 156c*†, 205c*†
* wart for 7/3
† wart for 11/3
RMS error: 0.8790 cents
Kryptonite
Kryptonite is related to krypton.
Subgroup: 2.5.7/3.11/3.13/3
Comma list: 56/55 ([3 -1 1 -1⟩), 78/77 ([1 0 -1 -1 1⟩), 91/90 ([-1 -2 1 0 1⟩)
Sval mapping: [⟨1 2 1 2 2], ⟨0 3 2 -1 1]]
- mapping generators: ~2, ~13/12
Gencom mapping: [⟨1 -5/4 2 -1/4 3/4 3/4], ⟨0 -1/2 3 3/2 -3/2 1/2]]
- gencom: [2 13/12; 56/55 78/77 91/90]
Optimal ET sequence: 1, …, 8, 9
RMS error: 2.545 cents
Kiribati
Kiribati is related to nakika as well as to octacot.
Subgroup: 2.9/5.7/3.11/9
Comma list: 100/99 ([2 -2 0 -1⟩), 245/242 ([-1 -1 2 -2⟩)
Sval mapping: [⟨1 1 1 0], ⟨0 -2 3 4]]
- mapping generators: ~2, ~21/20
Gencom mapping: [⟨1 1/10 -4/5 11/10 1/5], ⟨0 -3/2 -1 3/2 1]]
- gencom: [2 21/20; 100/99 245/242]
Optimal ET sequence: 13, 14, 27, 41
RMS error: 1.245 cents
Mothwelltri
Mothwelltri, the 1 & 4 temperament in the 2.7/3.11 subgroup, is related to orwell. The tonic and the first two generator steps make a mothwellsmic triad, hence the name.
Subgroup: 2.7/3.11
Comma list: 99/98 ([-1 -2 1⟩)
Sval mapping: [⟨1 0 1], ⟨0 1 2]]
- mapping generators: ~2, ~7/3
Gencom mapping: [⟨1 -1/2 0 1/2 3], ⟨0 -1/2 0 1/2 2]]
- gencom: [2 7/6; 99/98]
Optimal ET sequence: 4, 9, 13, 22, 79
RMS error: 1.064 cents
2.….9/7… subgroups
Marveltri
Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament. The tonic and the first two generator steps make a marvel triad, hence the name.
Subgroup: 2.5.9/7
Comma list: 225/224 ([-5 2 1⟩)
Sval mapping: [⟨1 0 5], ⟨0 1 -2]]
- mapping generators: ~2, ~5
Gencom mapping: [⟨1 2 0 -1], ⟨0 -4/5 1 2/5]]
- gencom: [2 5; 225/224]
Optimal ET sequence: 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c*
* Wart for 9/7
RMS error: 0.4801 cents
Sulis
Sulis is related to minerva and würschmidt.
Subgroup: 2.5.9/7.11/9
Comma list: 99/98 ([-1 0 2 1⟩), 176/175 ([4 -2 1 1⟩)
Sval mapping: [⟨1 0 5 -9], ⟨0 1 -2 4]]]
Optimal ET sequence: 3, …, 22, 25, 28, 31, 59
RMS error: 1.074 cents
2.….7/5… subgroups
Hydrothermal
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
Subgroup: 2.3.7/5
Sval mapping: [⟨2 3 1], ⟨0 1 0]]
Optimal tuning (inharmonic TE): ~1\2 = 590.998, ~10/7-1\2 = 128.962
Supporting ETs: 4, 6, 8, 10, 18, 28, 46, 64, 110
Edson
Edson is the 2.3.7/5 subgroup temperament tempering out 5120/5103.
Edson (2.3.7/5.11/5.13/5 subgroup)
Edson is related to pele and andromeda.
Subgroup: 2.3.7/5.11/5.13/5
Comma list: 196/195 = [2 -1 2 0 -1⟩, 352/351 = [5 -3 0 1 -1⟩, 364/363 = [2 -1 1 -2 1⟩
Sval mapping: [⟨1 0 10 17 22], ⟨0 1 -6 -10 -13]]
- mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 -5 -1 2 4], ⟨0 1 29/4 5/4 -11/4 -23/4]]
- gencom: [2 3/2; 196/195, 352/351, 364/363]
Optimal ET sequence: 12, 17, 29
RMS error: 0.5102 cents
Haumea
Related temperaments include bridgetown, namaka, hemigari, barbados, and parizekmic.
Subgroup: 2.3.7/5.11/5.13/5
Comma list: 352/351, 676/675, 847/845
Sval mapping: [⟨1 0 10 -6 -1], ⟨0 2 -12 9 3]]
Gencom mapping: [⟨1 2 -3/4 -11/4 9/4 5/4], ⟨0 -2 0 12 -9 -3]]
- gencom: [2 15/13; 352/351 676/675 847/845]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.491
Optimal ET sequence: 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd
RMS error: 0.2668 cents
Historical
- Not to be confused with Historical temperaments.
- Not to be confused with History (temperament)., which is the rank-3 version of this temperament in the full 13-limit.
Historical is essentially an analogue of miracle that splits 4/3 in six rather than 3/2. It tempers out the comma S10/S11 = 4000/3993 to set 11/10 equal to one-third of 4/3, and S13/S15 = 676/675 to equate 15/13 to one-half of 4/3, and tempers out S21 = 441/440 to split 11/10 into two instances of 22/21~21/20. Sextilifourths adds the schismic mapping of prime 5 (reached by eight fourths) to complete the 13-limit.
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
Terrain
- "Terrain" redirects here. For the scale, see Terrain (scale).
Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to domain. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.
Subgroup: 2.7/5.9/5
Sval mapping: [⟨3 1 3], ⟨0 1 -1]]
Gencom mapping: [⟨3 10/9 -7/9 2/9], ⟨0 -2/3 -1/3 2/3]]
- gencom: [63/50 10/9; 250047/250000]
Optimal tuning (subgroup POTE): ~63/50 = 1\3, ~10/9 = 182.461
Optimal ET sequence: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558
RMS error: 0.00844 cents
Tridec
Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends #Petrtri.
Subgroup: 2.7/5.11/5.13/5
Comma list: 847/845, 1001/1000
Sval mapping: [⟨1 2 0 1], ⟨0 -4 3 1]]
Gencom mapping: [⟨1 0 -3/4 5/4 -3/4 1/4], ⟨0 0 0 -4 3 1]]
- gencom: [2 13/10; 847/845 1001/1000]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 454.556
Optimal ET sequence: 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc
RMS error: 0.1613 cents
2.….11/5… subgroups
Petrtri
Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.
Subgroup: 2.11/5.13/5
Sval mapping: [⟨1 0 1], ⟨0 3 1]]
Gencom mapping: [⟨1 0 -1/3 0 -1/3 2/3], ⟨0 0 -4/3 0 5/3 -1/3]]
- gencom: [2 13/10; 2200/2197]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 455.012
Optimal ET sequence: 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c
RMS error: 0.0749 cents
Bridgetown
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to haumea and barbados.
Subgroup: 2.3.11/5.13/5
Sval mapping: [⟨1 0 -6 -1], ⟨0 2 9 3]]
Gencom mapping: [⟨1 2 -5/3 0 4/3 1/3], ⟨0 -2 4 0 -5 1]]
- gencom: [2 15/13; 352/351 676/675]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.399
Optimal ET sequence: 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314
RMS error: 0.2513 cents
Hypnosis
Related temperaments: 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]]
Optimal tuning (subgroup POTE): ~13/9 = 633.518
Optimal ET sequence: 17, 36, 118f, 125f, 161f, 197f
RMS error: 0.5379 cents
2.….11/7… subgroups
Pepperoni
Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of parapyth. The Pepper fifth, which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
Subgroup: 2.3.11/7.13/7
Comma list: 352/351, 364/363
Sval mapping: [⟨1 0 7 12], ⟨0 1 -4 -7]]
Gencom mapping: [⟨1 1 0 -8/3 1/3 7/3], ⟨0 1 0 11/3 -1/3 -10/3]]
- gencom: [2 3/2; 352/351 364/363]
Optimal tuning (subgroup POTE): ~3/2 = 703.856
Optimal ET sequence: 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*†
* Wart for 11/7
† Wart for 13/7
RMS error: 0.3789 cents
2.….13/5… subgroups
Barbados
The minimax tuning for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are 24edo, 29edo, 53edo and 111edo, with mos scales of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
Subgroup: 2.3.13/5
Comma list: 676/675 = [2 -3 2⟩
Sval mapping: [⟨1 0 -1], ⟨0 2 3]]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.621
Optimal ET sequence: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362
Badness: 0.002335
- Music
- Desert Island Rain in 313edo tuned Barbados[9], by Sevish
Tobago
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends neutral and barbados.
Subgroup: 2.3.11.13/5
Sval mapping: [⟨2 0 -1 -2], ⟨0 2 5 3]]
Gencom mapping: [⟨2 4 -2 0 9 2], ⟨0 -2 3/2 0 -5 -3/2]]
- gencom: [55/39 15/13; 243/242 676/675]
Optimal tuning (subgroup POTE): ~55/39 = 1\2, ~15/13 = 249.312
Optimal ET sequence: 10, 14, 24, 58, 82, 130
RMS error: 0.3533 cents
Pakkanian hemipyth
Subgroup: 2.3.11.13/5.17
Comma list: 221/220, 243/242, 289/288
Sval mapping: [⟨2 0 -1 -2 5], ⟨0 2 5 3 2]]
- subgroup CTE: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
- subgroup CWE: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
Optimal ET sequence: 10, 14, 24, 106, 130, 154, 178*, 202*
* Wart for 13/5
Oceanfront
Related temperaments: superpyth, ultrapyth
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
Scales: Oceanfront scales
2.….49/5… subgroups
Direct breedsmic
Related temperament: hemithirds, newt
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: ?
3/2.5/2… subgroups
Hemihemi
Subgroup: 3/2.5/2.7/2
Sval mapping: [⟨1 2 3], ⟨0 3 1]]
Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/27 = 60.909
Supporting ETs: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]
Halftone
Subgroup: 3/2.5/2.7/2
Comma list: 9604/9375
Sval mapping: [⟨1 3 4], ⟨0 -4 -5]]
- sval mapping generators: ~3/2, ~15/14
Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 128.783
Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
* Wart for 3/2
3/2.5/2.7/2.11/2
Subgroup: 3/2.5/2.7/2.11/2
Comma list: 1232/1215, 27783/27500
Sval mapping: [⟨1 3 4 4], ⟨0 -4 -5 1]]
- sval mapping generators: ~3/2, ~15/14
Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 129.186
Supporting ETs: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
* Wart for 3/2
3/2.5/2.7/2.11/2.13/2
Subgroup: 3/2.5/2.7/2.11/2.13/2
Comma list: 275/273, 1232/1215, 1323/1300
Sval mapping: [⟨1 3 4 4 5], ⟨0 -4 -5 1 -2]]
Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 129.381
Supporting ETs: *11, *5, *16, *6, *27[-11/2]
* Wart for 3/2
Semiwolf
Subgroup: 3/2.5/2.7/4
Comma list: 245/243
Sval mapping: [⟨1 1 2], ⟨0 2 -1]]
- sval mapping generators: ~3/2, ~9/7
Optimal tuning (subgroup POTE): ~7/6 = 262.1728
Optimal ET sequence: 3edf, 5edf, 8edf
Semilupine
Subgroup: 3/2.5/2.7/4.11/4
Comma list: 100/99, 245/243
Sval mapping: [⟨1 1 2 0], ⟨0 2 -1 4]]
Optimal tuning (subgroup POTE): ~7/6 = 264.3771
Optimal ET sequence: 8edf, 13edf
Hemilycan
Subgroup: 3/2.5/2.7/4.11/4
Comma list: 245/243, 441/440
Sval mapping: [⟨1 1 2 5], ⟨0 2 -1 -4]]
Optimal tuning (subgroup POTE): ~7/6 = 261.5939
Optimal ET sequence: 8edf, 11edf
3/2.5/4… subgroups
Poseidon
This temperament will be subjected to renaming due to a conflict.
Subgroup: 3/2.5/4.11/8
Comma list: 121/120
Sval mapping: [⟨1 1 1], ⟨0 2 -1]]]
- gencom: [3/2 12/11; 121/120]
Optimal tuning (subgroup POTE): ~3/2, ~12/11 = 158.29
Optimal ET sequence: 9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49
Other 3/2-equave subgroups
Auk
Subgroup: 3/2.7.13
Comma list: 87808/85293
Sval mapping: [⟨1 0 -8], ⟨0 1 3]]
- sval mapping generators: ~3/2, ~7
Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/9 = 1950.859
Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
* Wart for 3/2
Doubleton
Subgroup: 3/2.7.13
Comma list: 1352/1323
Sval mapping: [⟨2 0 3], ⟨0 1 1]]
- sval mapping generators: ~26/21, ~7
Optimal tuning (subgroup CTE): ~26/21 = 1\2edf, ~28/9 = 1971.772
Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
* Wart for 3/2
5/2-equave subgroups
Hyperion
Subgroup: 5/2.7.11
Comma list: [11 1 -5⟩
Sval mapping: [⟨1 4 3], ⟨0 -5 -1]]
- gencom: [5/2 125/88; 341796875/329832448]
Optimal tuning (subgroup POTE): ~5/2 = 1586.3137, ~125/88 = 593.6668
Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
* Wart for 5/2
Related temperament collections
- Dual-fifth temperaments
- Equalizer subgroup temperaments
- Substitute harmonic temperaments