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.

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

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

Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

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

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

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

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

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

2.5.11.13 subgroup temperaments

2.5.11.13 subgroup primarily contains temperaments developed for 1789edo, since it tempers out the jacobin comma 6656/6655, for which 2.5.11.13 is the subgroup, and the year 1789 is hallmark for the French revolution.

Estates general

Named so because it is defined as the 1789 & 3125 temperament due to 3125 providing optimal patent val for the jacobin comma, 3125 is 5 to the 5th power, and Estates General were called by Louis XVI on 5th May 1789 (05/05). Defined for the 2.5.11.13.19 subgroup.

Subgroup: 2.5.11.13

Comma list: 6656/6655, [314 -78 14 -49⟩

Mapping: [⟨1 118 -107 -212], ⟨0 -133 127 248]]

Optimal tuning (POTE): ~995765040283203125/544935554911830016 = 1043.712

2.5.11.13.19 subgroup

Todo: Weak extension, must use a different name

Subgroup: 2.5.11.13.19

Comma list: 6656/6655, 40960000000/40943078891, [-133 50 -7 18 -6⟩

Mapping: [⟨1 118 -107 -212 450], ⟨0 -266 254 496 -1025]]

Optimal tuning (POTE): ~2588443885831192576/1914932769775390625 = 521.856

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

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

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

RMS error: 2.063 cents

Related temperament: superpyth, ultrapyth

Marveltri

Subgroup: 2.5.9/7

Comma list: 225/224

Sval mapping: [⟨1 2 1], ⟨0 1 -2]]

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

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

RMS error: 1.074 cents

Related temperament: minerva, würschmidt