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 other miscellaneous subgroups and temperaments for them. 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.

2.9.7.11 subgroup

Machine

Subgroup: 2.9.7.11

Commas: 64/63, 99/98

Gencom: [2 8/7; 64/63 99/98]

Gencom mapping: [1 3/2 0 3 4], 0 1/2 0 -1 -3]]

Sval mapping: [1 0 6 13], 0 1 -1 -3]]

POL2 generator: ~8/7 = 214.384

Optimal GPV sequence5, 6, 11, 17, 28

RMS error: 1.977 cents

Mechanism

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

Gencom: [2 9/7; 896/891 26411/26244]

Gencom mapping: [1 5/2 0 5 2], 0 -5/2 0 -6 4]]

Sval mapping: [1 5 5 2], 0 -5 -6 4]]

POL2 generator: ~9/7 = 438.465

Optimal GPV sequence8, 11, 30, 41, 52

RMS error: 0.4262 cents

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Gencom: [2 77/72; 41503/41472 322102/321489]

Gencom mapping: [1 5/2 0 3 5], 0 -19/2 0 -2 -16]]

Sval mapping: [1 5 3 5], 0 -19 -2 -16]]

POL2 generator: ~77/72 = 115.570

Optimal GPV sequence10, 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

Gencom: [2 9/7; 225/224 245/243]

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

Sval mapping: [1 0 2 -1], 0 5 3 6]]

POL2 generator: ~9/7 = 439.296

Optimal GPV sequence8, 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

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]]

Sval mapping: [1 0 2 -1 6], 0 5 3 6 -4]]

POL2 generator: ~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

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]]

Sval mapping: [1 0 2 -1 6 -2], 0 5 3 6 -4 9]]

POL2 generator: ~9/7 = 438.977

Vals: 11, 30, 41, 153cdef, 194cdef, 235cdef

RMS error: 1.540 cents

2.9.21 subgroup

A-team

Subgroup: 2.9.21

Comma: 1029/1024

Gencom: [2 21/16; 1029/1024]

Gencom mapping: [1 1 0 3], 0 3/2 0 -1/2]]

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

POL2 generator: ~21/16 = 467.375

Optimal GPV sequence5, 13, 18, 41, 59, 77, 95

RMS error: 0.3202 cents

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]]

POL2 generator: ~21/20 = 83.016

Optimal GPV sequence14, 29, 72, 101, 130, 159

RMS error: 0.2562 cents

Hypnosis

Related temperament: 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]]

POL2 generator: ~13/9 = 633.518

Optimal GPV sequence17, 36, 118e, 125e, 161e, 197e

RMS error: 0.5379 cents

Oceanfront

Subgroup: 2.3.7.13/5

Related temperament: superpyth, ultrapyth

Comma list: 64/63, 91/90

Sval mapping: [1 0 6 -5], 0 1 -2 4]]

POL2 generator: ~3/2 = 713.910

Optimal GPV sequence5, 22, 27, 32, 37

RMS error: 2.063 cents