Subgroup temperaments

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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:

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

See also: Jubilismic clan

Antikythera is every other step of pajara.

Subgroup: 2.9.5.7

Comma list: 50/49, 64/63

Sval 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⟩

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

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

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 (CTE): ~26/21 = 1\2edf, ~28/9 = 1971.772[clarification needed]


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 (CTE): ~3/2 = 1\1edf, ~28/9 = 1950.859[clarification needed]


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

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 (CTE): ~3/2 = 1\1edf, ~15/14 = 128.783[clarification needed]


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 (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

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