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

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⟩

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

2.9.11 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

Comma list: 1331/1296

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

2.9.25 subgroup

Squared meantone

Subgroup: 2.9.25

Comma list: 6561/6400

Subgroup-val mapping: [⟨1 3 4], ⟨0 1 4]]

Optimal tuning (CTE): ~9/8 = 194.429

Supporting ETs: 6, 7[-25], 13, 19, 5[+25], 25, 31, 11[+25], 8[-25], 37, 20[-25], 32[-25], 17[+25], 43

Squared dominant

Subgroup: 2.9.25.49

Comma list: 1296/1225, 4096/3969

Subgroup-val mapping: [⟨1 3 4 6], ⟨0 1 4 -2]]

Optimal tuning (CTE): ~9/8 = 199.244

Supporting ETs: 6, 5[+25], 11[+25], 17[+25, +49], 23[+25, +49], 29[+25, +49]

Squared septimal meantone

Subgroup: 2.9.25.49

Comma list: 6561/6400, 50625/50176

Subgroup-val mapping: [⟨1 3 4 4], ⟨0 1 4 10]]

Optimal tuning (CTE): ~9/8 = 193.904

Supporting ETs: 6, 25, 19[-49], 31, 13[-49], 7[-25, -49], 37, 56[-9], 44[-49], 43[+49], 32[-25, --49], 68[-9], 81[-9, -49], 69[-9, -49]

Squared flattone

Subgroup: 2.9.25.49

Comma list: 6561/6400, 765625/746496

Subgroup-val mapping: [⟨1 3 4 7], ⟨0 1 4 -9]]

Optimal tuning (CTE): ~9/8 = 187.104

Supporting ETs: 13, 6[-49], 19[-49], 32[-25, -49], 45[-9, -25, -49], 51[-9, -25, -49], 58[-9, -25, -49], 70[-9, -25, --49], 77[-9, --25, -49], 83[-9, -25, --49], 109[--9, --25, --49], 115[--9, --25, ---49], 122[--9, ---25, --49], 141[--9, ---25, ---49]

2.9.49 subgroup

Squared archy

Subgroup: 2.9.49

Comma list: 5, 11, 6, 16, 17[+49], 7[+49], 27, 21, 9[-49], 28[+49], 23[+49], 13[+49], 26[+9], 38[+9, +49]

Subgroup-val mapping: [⟨1 3 4], ⟨0 1 4]]

Optimal tuning (CTE): ~9/8 = 219.190

Supporting ETs: 5, 11, 6, 16, 17[+49], 7[+49], 27, 21, 9[-49], 28[+49], 23[+49], 13[+49], 26[+9], 38[+9, +49]

8.9.7 subgroup

Sixscared

Subgroup: 8.9.7

Comma list: 64/63

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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⟩

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Pythium

Subgroup: 3/2.4.5

Comma list: 81/80

Subgroup-val mapping: [⟨1 0 4], ⟨0 1 0]]

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~6/5 = 294.135

Supporting ETs: *7, *5, *9, *12, *11, *8, *16, *13[+4], *19[+5], *17[+5], *15[+4], *23[+5], *20[+4, +5], *25[+4, +5]

* wart for 3/2

3/2.4.5.7

Subgroup: 3/2.4.5.7

Comma list: 36/35, 64/63

Subgroup-val mapping: [⟨1 0 4 4], ⟨0 1 0 2]]

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~6/5 = 286.062

Supporting ETs: *5, *7, *12, *9[+7], *8, *17[+5], *11[+7], *19[+5, +7], *13, *16[+7], *22[+5], *26[+5, +7], *29[+5, +7], *23[+5, ++7]

* wart for 3/2