Subgroup temperaments

A subgroup temperament is a regular temperament defined on a just intonation subgroup that is not a full p-limit group.

Below are some 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.

2.3.5.11 subgroup

Porkypine

Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

Gencom: [2 10/9; 55/54, 100/99]

Gencom mapping: [⟨1 2 3 0 4], ⟨0 -3 -5 0 -4]]

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

POL2 generator: ~11/10 = 164.078

{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd ]}

RMS error: 2.287 cents

Mohaha

Subgroup: 2.3.5.11

Comma list: 81/80, 121/120

Gencom: [2 11/9; 81/80 121/120]

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]

Sval mapping: [⟨1 1 0 2], ⟨0 2 8 5]]

POL2 generator: ~11/9 = 348.094

RMS error: 1.392 cents

Music

Mohaha10ping2 by Billy Stiltner

Larry

Subgroup: 2.3.5.11

Comma list: 243/242, 4000/3993

Related temperaments: gravity, harry

Gencom: [2 40/27; 243/242 4000/3993]

Gencom mapping: [<1 5 12 0 12|, <0 -6 -17 0 -15|]

Sval mapping: [<1 5 12 12|, <0 -6 -17 -15|]

POL2 generator: ~40/27 = 683.166

RMS error: 0.3025 cents

2.3.7 subgroup

Semaphore

Subgroup: 2.3.7

Comma: 49/48

Gencom: [2 8/7; 49/48]

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

Sval mapping: [⟨1 2 3], ⟨0 -2 -1]]

POL2 generator: ~7/6 = 250.385

RMS error: 2.523 cents

Archy

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives dominant" or "no-fives superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Subgroup: 2.3.7

Comma: 64/63

Gencom: [2 3/2; 64/63]

Gencom mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]

Sval mapping: [⟨1 2 2], ⟨0 -1 2]]

POL2 generator: ~3/2 = 709.321

RMS error: 1.856 cents

Slendric

Subgroup: 2.3.7

Comma: 1029/1024

Gencom: [2 8/7; 1029/1024]

Gencom mapping: [⟨1 1 0 3], ⟨0 3 0 -1]]

Sval mapping: [⟨1 1 3], ⟨0 3 -1]]

POL2 generator: ~8/7 = 233.688

RMS error: 0.3202 cents

Lee

Subgroup: 2.3.7

Comma: 177147/175616

Gencom: [2 81/56; 177147/175616]

Gencom mapping: [⟨1 0 0 -3], ⟨0 3 0 11]]

Sval mapping: [⟨1 0 -3], ⟨0 3 11]]

POL2 generator: ~81/56 = 633.525

RMS error: 0.3519 cents

Skwares

Subgroup: 2.3.7

Comma: 19683/19208

Gencom: [2 9/7; 19683/19208]

Gencom mapping: [⟨1 3 6], ⟨0 -4 -9]]

Sval mapping: [⟨1 3 6], ⟨0 -4 -9]]

POL2 generator: ~9/7 = 425.365 cents

RMS error: 1.149 cents

2.3.7.11 subgroup

Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Gencom: [2 99/98; 41503/41472, 43923/43904]

Gencom mapping: [⟨1 14/9 0 25/9 31/9], ⟨0 2 0 2 1]]

Sval mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

POL2 generator: ~99/98 = 17.6258

RMS error: 0.0383 cents

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

POL2 generator: ~3/2 = 707.192

RMS error: 1.977 cents

Skwares

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Gencom: [2 9/7; 99/98 243/242]

Gencom mapping: [⟨1 3 0 6 7], ⟨0 -4 0 -9 -10]]

Sval mapping: [⟨1 3 6 7], ⟨0 -4 -9 -10]]

POL2 generator: ~23/18 = 425.244

RMS error: 1.099 cents

Hemif

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Gencom: [2 11/9; 243/242 896/891]

Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]

Sval mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]

POL2 generator: ~11/9 = 351.535

RMS error: 0.6108 cents

Bleu

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Gencom: [2 12/11; 99/98 864/847]

Gencom mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]

Sval mapping: [⟨1 1 2 3], ⟨0 5 7 4]]

POL2 generator: ~12/11 = 140.005

RMS error: 1.829 cents

2.3.7.11.13 subgroup

Suhajira

Comma list: 64/63, 78/77, 144/143

Gencom: [2 11/9; 64/63 78/77 144/143]

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

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

POL2 generator: ~11/9 = 353.775

RMS error: 1.953 cents

Hemif

Comma list: 144/143, 243/242, 364/363

Related temperament: Hemififths, namo

Gencom: [2 11/9; 144/143 243/242 364/363]

Gencom mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]

Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]

POL2 generator: ~11/9 = 351.691

RMS error: 0.7167 cents

Skwares

Comma list: 78/77, 99/98, 243/242

Gencom: [2 9/7; 78/77, 99/98, 243/242]

Gencom mapping: [{{val|1 3 0 6 7 9}, ⟨0 -4 0 -9 -10 -15]]

Sval mapping: [{{val|1 3 6 7 9}, ⟨0 -4 -9 -10 -15]]

POL2 generator: ~9/7 = 424.457

RMS error: 1.769 cents

Leapfrog

See also: Gentle region

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1], ⟨0 1 0 15 11 8]]

Sval mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]

POL2 generator: ~3/2 = 704.745 cents

RMS error: 0.7541 cents

Music

• Suite for Harpsichord in A Locrian, tuning: Eb-G# in 46edo by IlL (in progress):
  • I. Prelude
  • II. Allemande
  • III. Courante
  • IV. Sarabande (score, 17edo version)
  • V. Menuet and Trio
  • VI. Gavotte I and II
  • VII. Gigue

Bleu

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Gencom: [2 13/12; 78/77 99/98 144/143]

Gencom mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]

Sval mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]

POL2 generator: ~13/12 = 139.990

RMS error: 1.752 cents

Parapyth (Rank 3)

Comma list: 352/351, 364/363

The gencom below gives Margo Schulter's favored basis

Gencom: [2 3/2 28/27; 352/351 364/363]

Gencom mapping: [<1 1 0 1 4 6|, <0 1 0 3 -1 -4|, <0 0 0 1 1 1|]

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

POL2 tuning: ~3 = 1903.856, ~7 = 3369.907

RMS error: 0.3789 cents

2.3.11 subgroup

Neutral

Neutral can be thought of as the 2.3.11 version of either mohajira or maqamic, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.

Comma: 243/242

Gencom: [2 11/9; 243/242]

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

Sval mapping: [<1 1 2|, <0 2 5|]

POL2 generator: ~11/9 = 350.525

RMS error: 0.3021 cents

Seven note albitonic scale

Ten note chromatic scale

Seventeen note mega chromatic scale

2.9.7.11 subgroup

Machine

Commas: 64/63, 99/98

POTE generator: ~17/15 = 214.384

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

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

Sval map: [<1 0 6 13|, <0 1 -1 -3|]

EDOs: 5, 6, 11, 17, 28

RMS error: 1.977 cents

Apparatus

Commas: 41503/41472, 322102/321489

POTE generator: ~31/29 = 115.570

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

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

Sval map: [<1 5 3 5|, <0 -19 -2 -16|]

EDOs: 10, 21, 31, 52, 83, 135, 353, 488, 623

RMS error: 0.0673 cents

Mechanism

Commas: 896/891, 26411/26244

POTE generator: ~9/7 = 438.465

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

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

Sval map: [<1 5 5 2|, <0 -5 -6 4|]

EDOs: 8, 11, 30, 41, 52

RMS error: 0.4262 cents

2.9.15.7.11.13 subgroup

Stacks (aka 2magic)

Commas: 100/99, 105/104, 144/143, 196/195

POL2 generator: ~9/7 = 438.977

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 map: [<1 0 2 -1 6 -2|, <0 5 3 6 -4 9|]

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

RMS error: 1.540 cents

2.9.15.7.11 stacks

Commas: 100/99, 225/224, 245/243

POL2 generator: ~9/7 = 438.607

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 map: [<1 0 2 -1 6|, <0 5 3 6 -4|]

EDOs: 8, 11, 30, 41, 52, 93, 145, 342bce,

RMS error: 1.226 cents

2.9.15.7 stacks

Commas: 225/224, 245/243

POL2 generator: ~9/7 = 439.296

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

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

Sval map: [<1 0 2 -1|, <0 5 3 6|]

EDOs: 8, 11, 30, 41, 71, 93, 112c, 134c, 175c

RMS error: 1.074 cents

2.9.21 subgroup

A-team

Commas: 1029/1024

POL2 generator: ~17/13 = 467.375

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

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

Sval map: [<1 2 4|, <0 3 1|]

EDOs: 5, 13, 18, 41, 59, 77, 95

RMS error: 0.3202 cents

2.3.7/5.11/5.13/5 subgroup

Historical

Subgroup: 2.3.7/5.11/5.13/5

Commas: 364/363, 441/440, 1001/1000

POL2 generator: ~21/20 = 83.016

Gencom: [2 21/20; 364/363 441/440 1001/1000]

Gencom map: [<1 2 -3/4 -3/4 1/4 5/4|, <0 -6 0 7 2 -9|]

Sval map: [<1 2 0 1 2|, <0 -6 7 2 -9|]

EDOs: 14, 29, 72, 101, 130, 159

RMS error: 0.2562 cents

2.3.7.11/5.13 subgroup

Hypnosis

Subgroup: 2.3.7.11/5.13

Commas: 169/168, 540/539, 729/728

Related temperament: hypnos, tricot

Gencom: [2 13/9; 169/168, 540/539, 729/728]

Gencom map: [<1 0 -4 -3 4 0|, <0 3 13/2 11 -13/2 7|]

Sval map: [<1 0 -3 8 0|, <0 3 11 -13 7|]

EDOs: 17, 36, 118e, 125e, 161e, 197e

RMS error: 0.5379 cents

2.5.7 subgroup

Didacus

Subgroup: 2.5.7

Commas: 3136/3125

Related temperaments: roulette, hemithirds

Gencom: [2 28/25; 3136/3125]

Gencom map: [<1 0 2 2|, <0 0 2 5|]

Sval map: [<1 2 2|, <0 2 5|]

EDOs: 6, 19, 25, 31, 37, 99, 130, 161, 353

RMS error: 0.2138 cents

Llywelyn

Subgroup: 2.5.7

Commas: 4194304/4117715

Gencom: [2 8/7; 4194304/4117715]

Gencom map: [<1 0 1 3|, <0 0 7 -1|]

Sval map: [<1 1 3|, <0 7 -1|]

EDOs: 16, 37

RMS error: 0.5391 cents

Rainy

Three generators make an 8/7; five generators make a 5/4. This is the no-threes version of tertiaseptal.

Subgroup: 2.5.7

Commas: 2100875/2097152

Gencom: [2 256/245; 2100875/2097152]

Gencom map: [⟨1 0 2 3], ⟨0 0 5 -3]]

Sval map: [⟨1 2 3], ⟨0 5 -3]]

POTE generator: 77.2047 cents

EDOs: 31, 47, 62, 78, 93, 109, 140, 171, 202, 233

RMS error: 0.058596 cents