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.
2.3.35 subgroup
Shaka
- See also: Kalismic temperaments
Two commas that split 2/1 in half, corresponding to convergents to sqrt(2), are the shaftesburisma S29/S41 and the kalisma S99, prompting to temper out {S29, S41, S99}, approximating /29 and /41 primodal chords well.
Subgroup: 2.3.35.11.29.41
Comma list: 841/840, 1189/1188, 1681/1680
Sval mapping: [⟨2 2 6 5 7 8], ⟨0 1 1 -1 1 1], ⟨0 0 2 2 1 1]]
Optimal tuning (CTE): ~41/29 = 1\2, ~3/2 = 702.031, ~41/24 = 926.693
Supporting ETs: 22, 26, 36, 48, 70, 96, 106, 118, 140, 154, 176, 188, 224, 272, 294, 342
Scale: Shaka10
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
- See also: Schismatic family#Tertiaschis
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
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 ("daemonotonic") 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
Supporting ETs: 7, 11, 18, 15, 29, 25, 10[-9, -11], 5[+9, +11], 26[+9], 19[+9], 32[-11], 17[-9, -11], 9[+9, +11], 13[-9, -11]
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]]
- 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 tuning (POTE): ~2 = 1\1, ~9/8 = 216.9128
Optimal ET sequence: 5, 6, 11, 17, 28
Badness: 0.00233
Mechanism
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
Badness: 0.00439
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
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]]
- 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
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
2.9.5.21.11
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
2.15.55 subgroup
Spog
This temperament produces superpelog-like semiquartal scales while being more accurate (see rational approximations to their intervals).
Subgroup: 2.15.55
Comma list: 100663296/100656875
Sval mapping: [⟨1 0 5], ⟨0 5 1]]
Optimal tuning (subgroup CTE): ~55/32 = 937.655
Supporting ETs: 5, 9, 13[-15], 14, 19[+15], 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
2.15.55.325
Subgroup: 2.15.55.325
Comma list: 4225/4224, 6656/6655
Sval mapping: [⟨1 0 5 6], ⟨0 5 1 3]]
Optimal tuning (subgroup CTE): ~55/32 = 937.647
Supporting ETs: 5, 9, 13[-15], 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
2.15.189.55.325
Related temperament: lux
Subgroup: 2.15.189.55.325
Comma list: 2080/2079, 3025/3024, 4096/4095
Sval mapping: [⟨1 0 6 5 6], ⟨0 5 2 1 3]]
Optimal tuning (subgroup CTE): ~55/32 = 937.677
Supporting ETs: 5, 9, 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
2.15.189.55.325.725
Subgroup: 2.15.189.55.325.725
Comma list: 1625/1624, 2080/2079, 3025/3024, 4096/4095
Sval mapping: [⟨1 0 6 5 6 -3], ⟨0 5 2 1 3 16]]
Optimal tuning (subgroup CTE): ~55/32 = 937.649
Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 142[+725], 151, 183, 206[+725], 311
2.15.189.55.325.725.279
Here are rational approximations to the intervals of the semiquartal scale.
Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.
Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.
Subgroup: 2.15.189.55.325.725.279
Comma list: 1625/1624, 2016/2015, 2080/2079, 3025/3024, 4096/4095
Sval mapping: [⟨1 0 6 5 6 -3 5], ⟨0 5 2 1 3 16 4]]
Optimal tuning (subgroup CTE): ~55/32 = 937.638
Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311
4.3.5 subgroup
Tetrahanson
- Main article: 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
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
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): ~8 = 3\1, ~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
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
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
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
Marveltri
Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament.
Subgroup: 2.5.9/7
Comma list: 225/224
Sval mapping: [⟨1 2 1], ⟨0 1 -2]]
Gencom mapping: [⟨1 2/5 2 -1/5], ⟨0 -4/5 1 2/5]]
- gencom: [2 5/4; 225/224]
Optimal tuning (subgroup POTE): ~5/4 = 383.638
Optimal ET sequence: 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269c*, 660c*
* wart for 9/7
RMS error: 0.4801 cents
Sulis
Related temperament: minerva, würschmidt
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
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
Greeley
Related temperaments: Opossum, Nusecond
Subgroup: 2.5/3.7/3.11/3
Comma list: 121/120, 126/125
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 tuning (subgroup POTE): ~11/10 = 155.776
Optimal ET sequence: 8, 15, 23, 54, 77, 100, 131†, 208*†
* wart for 5/3
† wart for 11/3
RMS error: 1.034 cents
Pepperoni
- Main article: Parapyth
Pepperoni is the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. 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, 12, 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
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
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
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
Halftone
- Main article: 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
Meanquad aka pythium
- Main article: 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
Poggers
Related temperaments: pogo, supers
Subgroup: 2.9.7.15/11.13
Comma list: 540/539, 1716/1715, 2080/2079
Sval mapping: [⟨1 1 1 -1 -1], ⟨0 6 5 4 13]]
Optimal tuning (subgroup CTE): ~9/7 = 433.888
Supporting ETs: 8[+9, +7, +13], 11, 14[-13], 19[+9, +7, ++13], 25[-13], 36, 47, 58, 61[-13], 69[+13], 80[+13], 83, 91[+9, +7, +13], 105