Subgroup temperaments

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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.

2.3.7

Semaphore

Comma: 49/48

POL2 generator: ~15/13= 250.385

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

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

Map: [<1 2 3|, <0 -2 -1|]

EDOs: 5, 14, 19, 24, 67c, 91c

RMS error: 2.523 cents

Archy

Archy can be thought of as "no-5's dominant" or "no-5's superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Comma: 64/63

POL2 generator: ~3/2 = 709.321

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

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

Map: [<1 2 2|, <0 -1 2|]

EDOs: 5, 12, 17, 22, 27, 137bc

RMS error: 1.856 cents

Slendric

Comma: 1029/1024

POL2 generator: ~8/7 = 233.688

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

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

Map: [<1 1 3|, <0 3 -1|]

EDOs: 5, 21, 26, 31, 36, 77, 113, 190

RMS error: 0.3202 cents

Lee

Comma: 177147/175616

POL2 generator: ~13/9 = 633.525

Map: [<1 0 -3|, <0 3 11|]

EDOs: 17, 36, 89, 125, 161, 358, 519b

RMS error: 0.3519 cents

Skwares

Comma: 19683/19208

POL2 generator: ~23/18 = 425.365 cents

Map: [<1 3 6|, <0 -4 -9|]

EDOs: 14, 17, 31, 48, 79, 189b, 268bc, 347bc

RMS error: 1.149 cents

2.3.11

Neutral

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

Comma: 243/242

POL2 generator: ~11/9 = 350.525

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

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

Map: [<1 1 2|, <0 2 5|]

EDOs: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

Seven note albitonic scale

Ten note chromatic scale

Seventeen note mega chromatic scale

2.3.5.11

Porkypine

Commas: 55/54, 100/99

POL2 generator: ~11/10 = 164.078

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

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

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

EDOs: 7, 15, 22, 37, 73cd, 95cd, 117bcd

RMS error: 2.287 cents

Mohaha

Commas: 81/80, 121/120

POL2 generator: ~11/9 = 348.094

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

Gencom mapping: [<1 1 0 0 2|, <0 2 8 0 5|]

Map: [<1 1 0 2|, <0 2 8 5|]

EDOs: 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd

RMS error: 1.392 cents

Music

Mohaha10ping2 by Billy Stiltner

2.3.7.11

Ennea

Commas: 41503/41472, 43923/43904

POL2 generator: ~99/98 = 17.6258

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

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

Sval map: [<9 0 11 24|, <0 2 2 1|]

EDOs: 54, 63, 68, 72, 135, 342, 477, 525, 613, 681, 749, 817, 2159bcd, 3336bcd, 4153bcd

RMS error: 0.0383 cents

Supra

Commas: 64/63, 99/98

POTE generator: ~3/2 = 707.192

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

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

Sval map: [<1 0 6 13|, <0 1 -2 -6|]

EDOs: 5, 12, 17, 39c, 56c

RMS error: 1.977 cents

Skwares

Commas: 99/98, 243/242

POTE generator: ~23/18 = 425.244

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

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

Sval map: [<1 3 6 7|, <0 -4 -9 -10|]

EDOs: 14, 17, 31, 48, 79, 127, 206bcd

RMS error: 1.099 cents

Hemif

Commas: 243/242, 896/891

POTE generator: ~11/9 = 351.535

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

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

Sval map: [<1 1 -1 2|, <0 2 13 5|]

EDOs: 7, 17, 41, 58, 99d

RMS error: 0.6108 cents

Bleu

Commas: 99/98, 864/847

POTE generator: ~13/12 = 140.005

Map: [<1 1 2 3|, <0 5 7 4|]

EDOs: 9, 17, 43, 60c

RMS error: 1.829 cents

2.3.7.11.13

Suhajira

Commas: 64/63, 78/77, 144/143

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

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

POT2 generator: ~11/9 = 353.775

Map: [<1 1 4 2 4|, <0 2 -4 5 -1|]

EDOs: 7, 10, 17, 44d, 61cd, 78cd

RMS error: 1.953 cents

Hemif

Commas: 144/143, 243/242, 364/363

Related temperament: Hemififths, namo

POT2 generator: ~11/9 = 351.691

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

Gencom map: [<1 1 0 -1 2 4|, <0 2 0 13 5 -1|]

Map: [<1 1 -1 2 4|, <0 2 13 5 -1|]

EDOs: 7, 10, 17, 24, 41, 58

RMS error: 0.7167 cents

Skwares

Commas: 78/77, 99/98, 243/242

POT2 generator: ~23/18 = 424.457

Map: [<1 3 6 7 9|, <0 -4 -9 -10 -15|]

EDOs: 17, 48e, 65de, 82c, 147ce

RMS error: 1.769 cents

Leapfrog

Commas: 169/168, 352/351, 364/363

POL2 generator: ~3/2 = 704.745 cents

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 map: [<1 0 -21 -14 -9|, <0 1 15 11 8|]

EDOs: 17, 46, 63

RMS error: 0.7541 cents

Bleu

Group: 2.3.7.11.13

Commas: 78/77, 99/98, 144/143

Sval name: 9&17

POL2 generator: ~13/12 = 139.990

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

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

Map: [<1 1 2 3 3|, <0 5 7 4 6|]

EDOs: 17, 43, 60c

RMS error: 1.752 cents

Parapyth (Rank 3)

Commas: 352/351, 364/363

POL2 tuning: <1200.000 1903.856 (2787.774) 3369.907 4154.482 4442.913|

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

EDOs: 17, 41, 46, 58, 87, 104

RMS error: 0.3789 cents

2.3.5.11

Larry

Commas: 243/242, 4000/3993

Related temperaments: gravity, harry

POTE generator: ~40/27 = 683.166

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

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

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

EDOs: 7, 58, 65, 137, 202

RMS error: 0.3025 cents

2.9.7.11

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

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

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

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

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

Didacus

Subgroup: 2.5.7

Commas: 3136/3125

Related temperaments: roulette

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