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.

Integer subgroup temperaments

2.3.35 subgroup

Shaka

See also: Kalismic temperaments

Two commas that split 2/1 in half, corresponding to convergents to sqrt(2), are the shaftesburisma S29/S41 and the kalisma S99, prompting to temper out {S29, S41, S99}, approximating /29 and /41 primodal chords well.

Subgroup: 2.3.35.11.29.41

Comma list: 841/840, 1189/1188, 1681/1680

Sval mapping: [⟨2 2 6 5 7 8], ⟨0 1 1 -1 1 1], ⟨0 0 2 2 1 1]]

Optimal tuning (CTE): ~41/29 = 1\2, ~3/2 = 702.031, ~41/24 = 926.693

Supporting ETs: 22, 26, 36, 48, 70, 96, 106, 118, 140, 154, 176, 188, 224, 272, 294, 342

Scale: Shaka10

2.9.5.7 subgroup

Commatose

Commatose is a dual-fifth 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 & 1329 temperament. In the 13-limit extension 24 generators are equal to ~13/9.

Subgroup: 2.9.5.7

Comma list: [28 -2 -19 8⟩, [9 -25 23 6⟩

Sval mapping: [⟨1 9 6 13], ⟨0 -298 -188 -521]]

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4765

Optimal ET sequence: 460, 869, 1329

Badness: 0.611

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⟩

Sval mapping: [⟨1 9 6 13 16], ⟨0 -298 -188 -521 -641]]

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

Optimal ET sequence: 460, 869e, 1329, 1789, 3118

Badness: 0.165

2.9.5.7.11.13

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): ~2 = 1\1, ~3575/3528 = 23.4767

Optimal ET sequence: 460, 869e, 1329, 1789, 3118

Badness: 0.0564

Daemotertiaschis

See also: Schismatic family § Tertiaschis

Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a daemotonic 7L 4s scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

Sval mapping: [⟨1 1 11 -16 13 -18 20], ⟨0 3 -12 26 -11 30 -22]]

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

Supporting ETs: 47, 65f, 112, 159, 206, 253

Baldy

See also: Schismatic family § Garibaldi

Baldy results from taking every other generator of the garibaldi temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

Subgroup: 2.9.5.7

Comma list: 225/224, 3125/3087

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

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.170

Optimal ET sequence: 6, 29, 35, 41, 47

Related temperament: Garibaldi

2.9.5.7.13

Subgroup: 2.9.5.7.13

Comma list: 225/224, 325/324, 640/637

Sval mapping: [⟨1 3 3 4 2], ⟨0 1 -4 -7 10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.090

Optimal ET sequence: 6, 29f, 35, 41, 47

Related temperament: Cassandra

Baldanders

Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

Sval mapping: [⟨1 3 3 4 5], ⟨0 1 -4 -7 -9]]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

Optimal ET sequence: 6, 23de, 29, 35, 41

Related temperament: Andromeda

2.9.5.7.11.13

Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

Sval mapping: [⟨1 3 3 4 5 2], ⟨0 1 -4 -7 -9 10]]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

Optimal ET sequence: 6, 23def, 29f, 35, 41, 47

2.9.7 subgroup

Mabon

Derived from a calendar leap cycle built for the autumn equinox, hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [⟨1 1 -3], ⟨0 3 8]]

Optimal tuning (CTE): ~729/448 = 870.792

Optimal ET sequence: 7d, 11, 18d, 29, 40, 62, ...

2.9.7.11 subgroup

Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [⟨1 1 -3 2], ⟨0 3 8 2]]

Optimal tuning (CTE): ~16/11 = 870.966

Optimal ET sequence: 7d, 11, 40, 51, 62

2.9.7.11 subgroup

Machine

Machine is every other step of supra, most interesting for its scale patterns.

Subgroup: 2.9.7.11

Comma list: 64/63, 99/98

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

sval mapping generators: ~2, ~9

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

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

Optimal tunings:

CTE: ~2 = 1\1, ~9/8 = 216.9128
POTE: ~2 = 1\1, ~9/8 = 214.3843

Optimal ET sequence: 5, 6, 11, 17, 28

Badness: 0.00233

Penta a.k.a. mechanism

Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

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

sval mapping generators: ~2, ~14/9

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

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

Optimal tuning (POTE): ~2 = 1\1, ~14/9 = 761.3782

Optimal ET sequence: 8, 11, 30, 41, 52

RMS error: 0.4262 cents

Badness: 0.00439

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

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

mapping generators: ~2, ~77/72

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

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

Optimal tuning (CTE): ~77/72 = 115.5685

Optimal ET sequence: 10e, 21, 31, 52, 83, 135, 353, 488, 623

Badness: 0.00263

2.9.11 subgroup

Demon

Demon is a temperament which equates 3 11/9 with 16/9, or equivalently 3 18/11 with 9/8, tempering out 1331/1296. This results in 11/9 being tuned flat to a supraminor third, and 27/22 being tuned sharp to a submajor third. It was discovered by CompactStar while searching for temperaments assosciated with the 7L 4s ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed 18edo supports demon temperament.

Subgroup: 2.9.11

Comma list: 1331/1296

Sval mapping: [⟨1 1 2], ⟨0 3 2]]

Optimal tuning (CTE): ~18/11 = 870.060

Optimal ET sequence: 4, 7, 11, 18, 29, 76e

Genius

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

Subgroup: 2.9.11

Comma list: 131769/131072

Sval mapping: [⟨1 1 4], ⟨0 4 -1]]

Optimal tuning (CTE): ~16/11 = 650.863

Optimal ET sequence: 9, 11, 24, 59, 83, 142, 225, 367[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

2.9.15.7 subgroup

Stacks (a.k.a. 2magic)

Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of magic.

Subgroup: 2.9.15.7

Comma list: 225/224, 245/243

Sval mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]

sval mapping generators: ~2, ~14/9

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

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 760.704

Optimal ET 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 mapping: [⟨1 5/2 5/2 5 2], ⟨0 -5/2 -1/2 -6 4]]

gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: 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 mapping: [⟨1 5/2 5/2 5 2 7], ⟨0 -5/2 -1/2 -6 4 -9]]

gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: 11, 30, 41, 153cdef, 194cdef, 235cdef

RMS error: 1.540 cents

2.9.21 subgroup

A-team

A-team is every other step of mothra.

Subgroup: 2.9.21

Comma list: 1029/1024

Sval mapping: [⟨1 2 4], ⟨0 3 1]]

sval mapping generators: ~2, ~21/16

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

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 467.375

Optimal ET 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 mapping: [⟨1 1 0 3 5], ⟨0 3/2 6 -1/2 -4]]

gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

Optimal ET sequence: 5, 13, 31

2.15.55 subgroup

Spog

This temperament produces superpelog-like semiquartal scales while being more accurate (see rational approximations to their intervals).

Subgroup: 2.15.55

Comma list: 100663296/100656875

Sval mapping: [⟨1 0 5], ⟨0 5 1]]

Optimal tuning (subgroup CTE): ~55/32 = 937.655

Optimal ET sequence: 5, 9, 23, 32, 151, 183, 215, 247, 956, 1203, 1450, 3147, 4597

2.15.55.325

Subgroup: 2.15.55.325

Comma list: 4225/4224, 6656/6655

Sval mapping: [⟨1 0 5 6], ⟨0 5 1 3]]

Optimal tuning (subgroup CTE): ~55/32 = 937.647

Supporting ETs: 5, 9, 13[-15], 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311

2.15.189.55.325

Related temperament: lux

Subgroup: 2.15.189.55.325

Comma list: 2080/2079, 3025/3024, 4096/4095

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

Optimal tuning (subgroup CTE): ~55/32 = 937.677

Supporting ETs: 5, 9, 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311

2.15.189.55.325.725

Subgroup: 2.15.189.55.325.725

Comma list: 1625/1624, 2080/2079, 3025/3024, 4096/4095

Sval mapping: [⟨1 0 6 5 6 -3], ⟨0 5 2 1 3 16]]

Optimal tuning (subgroup CTE): ~55/32 = 937.649

Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 142[+725], 151, 183, 206[+725], 311

2.15.189.55.325.725.279

Here are rational approximations to the intervals of the semiquartal scale.

Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.

Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.

Subgroup: 2.15.189.55.325.725.279

Comma list: 1625/1624, 2016/2015, 2080/2079, 3025/3024, 4096/4095

Sval mapping: [⟨1 0 6 5 6 -3 5], ⟨0 5 2 1 3 16 4]]

Optimal tuning (subgroup CTE): ~55/32 = 937.638

Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311

4.3.5 subgroup

Tetrahanson

Main article: Tetrahanson

Subgroup: 4.3.5

Comma list: 15625/15552

Sval mapping: [⟨1 3 3], ⟨0 -6 -5]]

Mapping generators: ~4, ~5/3

Optimal tuning (CTE): ~4 = 2\1, ~5/3 = 882.941

Supporting ETs: 19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79

Tetrameantone

Main article: Tetrameantone

Subgroup: 4.3.5

Comma list: 81/80

Sval mapping: [⟨1 1 2], ⟨0 -1 -4]]

Mapping generators: ~4, ~4/3

Optimal tuning (POTE): 4 = 2400.0, ~4/3 = 503.761

Supporting ETs: 5, 9, 14, 19, 24, 43, 62, 81, 100

Tetramagic

Subgroup: 4.3.5

Comma list: 3125/3072

Sval mapping: [⟨1 0 1], ⟨0 5 1]]

Mapping generators: ~4, ~5/4

Optimal tuning (POTE): 4 = 2400.0, ~5/4 = 380.059

Supporting ETs: 6, 13, 19, 25, 38, 44, 63, 82

Blacktetra

Subgroup: 4.3.5

Comma list: 256/243

Sval mapping: [⟨5 4 6], ⟨0 0 -1]]

Mapping generators: ~4, ~16/15

Optimal tuning (POTE): 1\5ed4 = 480.0, ~16/15 = 80.4062

Supporting ETs: 5, 10, 15, 20, 25, 30, 55, 85, 115

4.6.5 subgroup

Meanquad

Main article: Meanquad

Subgroup: 4.6.5

Comma list: 81/80 = [-4 4 -1⟩

Sval mapping: [⟨1 0 -4], ⟨0 1 4]]

mapping generators: ~4, ~6

Optimal tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 697.214

Supporting ETs: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

* wart for 4

4.6.5.7 subgroup (tetrominant)

Subgroup: 4.6.5.7

Comma list: 36/35 = [0 2 -1 -1⟩, 64/63 = [4 -2 0 -1⟩

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

Optimal tuning (subgroup CTE): ~4 = 2\1, ~3/2 = 699.622

Supporting ETs: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

* wart for 4

Fourwar

The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related hemiwar temperament.

 
Reduced Mapping
4	6	5	
[ ⟨	1	0	1	]
⟨	0	16	2	] ⟩
 
TE Generator Tunings (cents)
⟨2399.3973, 193.8643]
 
TE Step Tunings (cents)
⟨25.21211, 47.81337]
 
TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]
 
TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]
 
Complexity	1.369085
Adjusted Error	0.692892 cents
TE Error	0.268047 cents/octave
 
Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235

4.6.5.7

Reduced Mapping
4	6	5	7	
[ ⟨	1	0	1	1	]
⟨	0	16	2	5	] ⟩
 
TE Generator Tunings (cents)
⟨2399.4195, 193.8654]
 
TE Step Tunings (cents)
⟨25.23883, 47.79592]
 
TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]
 
TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]
 
Complexity	1.192044
Adjusted Error	0.653313 cents
TE Error	0.232715 cents/octave
 
Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235

4.6.5.7.11

Reduced Mapping
4	6	5	7	11	
[ ⟨	1	0	1	1	1	]
⟨	0	16	2	5	9	] ⟩
 
TE Generator Tunings (cents)
⟨2400.1097, 193.9498]
 
TE Step Tunings (cents)
⟨24.18752, 48.52491]
 
TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]
 
TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]
 
Complexity	1.068792
Adjusted Error	2.926965 cents
TE Error	0.846083 cents/octave
 
Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124

4.6.5.7.11.13

Reduced Mapping
4	6	5	7	11	13	
[ ⟨	1	0	1	1	1	0	]
⟨	0	16	2	5	9	23	] ⟩
 
TE Generator Tunings (cents)
⟨2401.2305, 193.5378]
 
TE Step Tunings (cents)
⟨42.79107, 35.98524]
 
TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]
 
TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]
 
Complexity	1.219191
Adjusted Error	6.699599 cents
TE Error	1.810487 cents/octave
 
Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff

4.6.5.7.11.13.17

Reduced Mapping
4	6	5	7	11	13	17	
[ ⟨	1	0	1	1	1	0	1	]
⟨	0	16	2	5	9	23	13	] ⟩
 
TE Generator Tunings (cents)
⟨2400.4701, 193.4599]
 
TE Step Tunings (cents)
⟨43.39350, 35.55764]
 
TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]
 
TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]
 
Complexity	1.129881
Adjusted Error	8.082725 cents
TE Error	1.977443 cents/octave
 
Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg

4.6.5.7.11.13.17.19

Reduced Mapping
4	6	5	7	11	13	17	19	
[ ⟨	1	0	1	1	1	0	1	1	]
⟨	0	16	2	5	9	23	13	14	] ⟩
 
TE Generator Tunings (cents)
⟨2399.9219, 193.3952]
 
TE Step Tunings (cents)
⟨44.14256, 35.03670]
 
TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]
 
TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]
 
Complexity	1.058472
Adjusted Error	8.712222 cents
TE Error	2.050935 cents/octave
 
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh

4.6.5.7.11.13.17.19.23

Reduced Mapping
4	6	5	7	11	13	17	19	23	
[ ⟨	1	0	1	1	1	0	1	1	0	]
⟨	0	16	2	5	9	23	13	14	28	] ⟩
 
TE Generator Tunings (cents)
⟨2399.3286, 193.5316]
 
TE Step Tunings (cents)
⟨37.31613, 39.63311]
 
TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]
 
TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]
 
Complexity	1.115920
Adjusted Error	9.502017 cents
TE Error	2.100561 cents/octave
 
Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii

4.9.25 subgroup

Meansquared

Subgroup: 4.9.25

Comma list: 6561/6400

Sval mapping: [⟨1 3 4], ⟨0 1 4]]

Mapping generators: ~4, ~9/64

Optimal tuning (CTE): ~4 = 2\1, ~9/4 = 1394.429

Supporting ETs: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

4.9.49 subgroup

Archsquared

Subgroup: 4.9.49

Comma list: 4096/3969

Sval mapping: [⟨1 3 0], ⟨0 1 -2]]

Mapping generators: ~4, ~9/64

Optimal tuning (CTE): ~9/8 = 219.190

Supporting ETs: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

8.9.7 subgroup

Sixscared

Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

Subgroup: 8.9.7

Comma list: 64/63

Sval mapping: [⟨1 0 2], ⟨0 1 -1]]

sval mapping generators: ~8, ~9

gencom: [8 9/8; 64/63]

Optimal tuning (CTE): 1\3ed8 = 1600.0, ~9/8 = 219.1898

Optimal ET sequence: ⟨16 17 15], ⟨33 35 31], ⟨148 …], ⟨181 …], ⟨214 …], ⟨247 …]

Badness: 0.0215 × 10^-3

Fractional subgroup temperaments

3/2.5/2… subgroup

Main article: Half-prime subgroup

Hemihemi

Subgroup: 3/2.5/2.7/2

Comma list: 10976/10935

Sval mapping: [⟨1 2 3], ⟨0 3 1]]

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/27 = 60.909

Supporting ETs: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

Halftone

Main article: Halftone

Subgroup: 3/2.5/2.7/2

Comma list: 9604/9375

Sval mapping: [⟨1 3 4], ⟨0 -4 -5]]

sval mapping generators: ~3/2, ~15/14

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49

* wart for 3/2

3/2.5/2.7/2.11/2

Subgroup: 3/2.5/2.7/2.11/2

Comma list: 1232/1215, 27783/27500

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

sval mapping generators: ~3/2, ~15/14

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 129.186

Supporting ETs: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]

* wart for 3/2

3/2.5/2.7/2.11/2.13/2

Subgroup: 3/2.5/2.7/2.11/2.13/2

Comma list: 275/273, 1232/1215, 1323/1300

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

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~15/14 = 129.381

Supporting ETs: *11, *5, *16, *6, *27[-11/2]

* wart for 3/2

Semiwolf

Subgroup: 3/2.5/2.7/4

Comma list: 245/243

Sval mapping: [⟨1 1 2], ⟨0 2 -1]]

sval mapping generators: ~3/2, ~9/7

Optimal tuning (subgroup POTE): ~7/6 = 262.1728

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

Semilupine

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 100/99, 245/243

Sval mapping: [⟨1 1 2 0], ⟨0 2 -1 4]]

Optimal tuning (subgroup POTE): ~7/6 = 264.3771

Optimal ET sequence: 8edf, 13edf

Hemilycan

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 245/243, 441/440

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

Optimal tuning (subgroup POTE): ~7/6 = 261.5939

Optimal ET sequence: 8edf, 11edf

5/3.7/3… subgroup

Greeley

Related temperaments: Opossum, Nusecond

Subgroup: 2.5/3.7/3.11/3

Comma list: 121/120, 126/125

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

Gencom mapping: [⟨1 -5/4 -1/4 3/4 3/4], ⟨0 9/4 1/4 -15/4 5/4]]

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

Optimal tuning (subgroup POTE): ~11/10 = 155.776

Optimal ET sequence: 8, 15, 23, 54, 77, 100, 131†, 208*†

* wart for 5/3

† wart for 11/3

RMS error: 1.034 cents

7/5.11/5… subgroup

Historical

Not to be confused with Historical temperaments.

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 ET sequence: 14, 29, 72, 101, 130, 159

RMS error: 0.2562 cents

11/7.13/7… subgroup

Pepperoni

Main article: Parapyth

Pepperoni is the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. The Pepper fifth, which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

Subgroup: 2.3.11/7.13/7

Comma list: 352/351, 364/363

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

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

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

Optimal tuning (subgroup POTE): ~3/2 = 703.856

Optimal ET sequence: 5, 7, 12, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*†

* wart for 11/7

† wart for 13/7

RMS error: 0.3789 cents

Other 3/2 subgroups

Auk

Subgroup: 3/2.7.13

Comma list: 87808/85293

Sval mapping: [⟨1 0 -8], ⟨0 1 3]]

sval mapping generators: ~3/2, ~7

Optimal tuning (subgroup CTE): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]

* wart for 3/2

Doubleton

Subgroup: 3/2.7.13

Comma list: 1352/1323

Sval mapping: [⟨2 0 3], ⟨0 1 1]]

sval mapping generators: ~26/21, ~7

Optimal tuning (subgroup CTE): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]

* wart for 3/2

Other 5/2 subgroups

Hyperion

Subgroup: 5/2.7.11

Comma list: [11 1 -5⟩

Sval mapping: [⟨1 4 3], ⟨0 -5 -1]]

gencom: [5/2 125/88; 341796875/329832448]

Optimal tuning (subgroup POTE): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]

* wart for 5/2

Other 7/5 subgroups

Hydrothermal

A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

Subgroup: 2.3.7/5

Comma list: 50/49

Sval mapping: [⟨2 3 1], ⟨0 1 0]]

Optimal tuning (inharmonic TE): ~1\2 = 590.998, ~10/7-1\2 = 128.962

Supporting ETs: 4, 6, 8, 10, 18, 28, 46, 64, 110

Other 11/5 subgroups

Hypnosis

Related temperaments: 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]]

Optimal tuning (subgroup POTE): ~13/9 = 633.518

Optimal ET sequence: 17, 36, 118f, 125f, 161f, 197f

RMS error: 0.5379 cents

Other 13/5 subgroups

Barbados

Subgroup: 2.3.13/5

Comma list: 676/675 = [2 -3 2⟩

Sval mapping: [⟨1 0 -1], ⟨0 2 3]]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.621

Optimal ET sequence: 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362

Badness: 0.002335

* wart for 3/2

Oceanfront

Related temperaments: superpyth, ultrapyth

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 ET sequence: 5, 22, 27, 32, 37

RMS error: 2.063 cents

Scales: Oceanfront scales

Pakkanian hemipyth

Subgroup: 2.3.11.13/5.17

Comma list: 221/220, 243/242, 289/288

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

Optimal tunings:

subgroup CTE: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
subgroup CWE: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

Optimal ET sequence: 10, 14, 24, 106, 130, 154, 178*, 202*

* wart for 13/5

Other 9/7 subgroups

Marveltri

Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament.

Subgroup: 2.5.9/7

Comma list: 225/224

Sval mapping: [⟨1 2 1], ⟨0 1 -2]]

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

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

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

Optimal ET sequence: 12, 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269c*, 660c*

* wart for 9/7

RMS error: 0.4801 cents

Sulis

Related temperament: minerva, würschmidt

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 ET sequence: 3, …, 22, 25, 28, 31, 59

RMS error: 1.074 cents

Other 15/11 subgroups

Poggers

Related temperaments: pogo, supers

Subgroup: 2.9.7.15/11.13

Comma list: 540/539, 1716/1715, 2080/2079

Sval mapping: [⟨1 1 1 -1 -1], ⟨0 6 5 4 13]]

Optimal tuning (subgroup CTE): ~9/7 = 433.888

Supporting ETs: 8[+9, +7, +13], 11, 14[-13], 19[+9, +7, ++13], 25[-13], 36, 47, 58, 61[-13], 69[+13], 80[+13], 83, 91[+9, +7, +13], 105

Related temperament collections

Subgroup temperaments

Integer subgroup temperaments

2.3.35 subgroup

Shaka

2.9.5.7 subgroup

Commatose

2.9.5.7.11

2.9.5.7.11.13

Daemotertiaschis

Baldy

2.9.5.7.13

Baldanders

2.9.5.7.11.13

2.9.7 subgroup

Mabon

2.9.7.11 subgroup

2.9.7.11 subgroup

Machine

Penta a.k.a. mechanism

Apparatus

2.9.11 subgroup

Demon

Genius

2.9.15.7 subgroup

Stacks (a.k.a. 2magic)

2.9.15.7.11

2.9.15.7.11.13

2.9.21 subgroup

A-team

2.9.5.21.11

2.15.55 subgroup

Spog

2.15.55.325

2.15.189.55.325

2.15.189.55.325.725

2.15.189.55.325.725.279

4.3.5 subgroup

Tetrahanson

Tetrameantone

Tetramagic

Blacktetra

4.6.5 subgroup

Meanquad

4.6.5.7 subgroup (tetrominant)

Fourwar

4.6.5.7

4.6.5.7.11

4.6.5.7.11.13

4.6.5.7.11.13.17

4.6.5.7.11.13.17.19

4.6.5.7.11.13.17.19.23

4.9.25 subgroup

Meansquared

4.9.49 subgroup

Archsquared

8.9.7 subgroup

Sixscared

Fractional subgroup temperaments

3/2.5/2… subgroup

Hemihemi

Halftone

3/2.5/2.7/2.11/2

3/2.5/2.7/2.11/2.13/2

Semiwolf

Semilupine

Hemilycan

5/3.7/3… subgroup

Greeley

7/5.11/5… subgroup

Historical

11/7.13/7… subgroup

Pepperoni

Other 3/2 subgroups

Auk

Doubleton

Other 5/2 subgroups

Hyperion

Other 7/5 subgroups

Hydrothermal

Other 11/5 subgroups