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

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

24 generators are equal to 13/9.

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): ~3575/3528 = 23.477

Antikythera

Subgroup: 2.9.5.7

Commas: 50/49, 64/63

Related temperament: pajara

POT2 generator: ~8/7 = 214.095

Gencom: [7/5 8/7; 50/49 64/63]

Gencom mapping: [<2 3 5 6|, <0 1/2 -1 -1|]

Mapping: [<2 0 11 12|, <0 1 -1 -1|]

EDOS: 4, 6, 10, 16, 22, 28

RMS error: 2.572 cents

2.9.5.7.11 subgroup

Tutone

Subgroup: 2.9.5.7.11

Commas: 81/80, 99/98, 126/125

Related temperaments: meantone, belobog, hemithirds, hemiwürschmidt

POT2 generator: ~9/8 = 193.937

Gencom: [2 9/8; 81/80 99/98 126/125]

Gencom mapping: [<1 3/2 2 2 2|, <0 1/2 2 5 9|]

Mapping: [<1 0 -4 -13 -25|, <0 1 2 5 9|]

EDOs: 25, 31, 68b, 99b

RMS error: 1.439 cents

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 name: 3&13

Related temperaments: marvel, 22&47, magic

Optimal tuning (subgroup POTE): ~5/4 = 383.638

Gencom: [2 5/4; 225/224]

Gencom mapping: [<1 2/5 2 -1/5|, <0 -4/5 1 2/5|]

Sval mapping: [⟨1 2 1], ⟨0 1 -2]]

Optimal GPV sequence: 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269bc, 660bc

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

Optimal GPV sequence: ?

RMS error: ?

Related temperament: hemithirds, newt

Semiwolf

Subgroup: 3/2.7/4.5/2

Comma list: 245/243

Sval mapping: [⟨1 1 3], ⟨0 1 -2]]

Optimal tuning (POL2): ~7/6 = 262.1728

Optimal GPV sequence: 3edf, 5edf, 8edf

Semilupine

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 100/99, 245/243

Sval mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]

POL2 generator: ~7/6 = 264.3771

Optimal GPV sequence: 8edf, 13edf

Hemilycan

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 441/440

Sval mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]

Optimal tuning (POL2): ~7/6 = 261.5939

Optimal GPV sequence: 8edf, 11edf

Greeley

Subgroup: 2.5/3.7/3.11/3

Commas: 121/120, 126/125

Related temperament: Opossum, Nusecond

POT2 generator: ~11/10 = 155.776

Gencom: [2 11/10; 121/120 126/125]

Gencom mapping: <1 -5/4 -1/4 3/4 3/4|, <0 9/4 1/4 -15/4 5/4|]

Mapping: [<1 1 2 2|, <0 -2 -6 -1|]

EDOs: 8, 15, 23, 54, 77, 100, 131d, 208bd

RMS error: 1.034 cents

Pepperoni

Subgroup: 2.3.11/7.13/7

Commas: 352/351, 364/363

Sval name: 5&12

Related temperament: The Pepper fifth, which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

POT2 generator: ~3/2 = 703.856

Gencom: [2 3/2; 352/351 364/363]

Gencom mapping: [<1 1 0 -8/3 1/3 7/3|, <0 1 0 11/3 -1/3 -10/3|]

Mapping: [<1 0 7 12|, <0 1 -4 -7|]

EDOs: 5, 7, 12, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595bcd

RMS error: 0.3789 cents

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