Subgroup temperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular 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-thirteens subgroup temperaments
• No-elevens subgroup temperaments
• No-sevens subgroup temperaments
• No-fives subgroup temperaments
• No-threes subgroup temperaments
• No-twos subgroup temperaments (additionally, 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.

Composite subgroup temperaments

2.9.5.7 subgroup

See also antikythera and isra.

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⟩

Subgroup-val 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

Subgroup-val 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

See also: No-threes subgroup temperaments § Frostburn

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

Subgroup-val 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

See also: Chromatic pairs § Baldy

Baldy is every other step of garibaldi, without the mapping of prime 11. It can be described as the 6 & 35 temperament.

Subgroup: 2.9.5.7.13

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

Subgroup-val mapping: [⟨1 0 15 25 -28], ⟨0 1 -4 -7 10]]

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

gencom: [2 9/8; 225/224 325/324 640/637]

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

Optimal ET sequence: 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd

RMS error: 0.5999 cents

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

Subgroup-val 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

Subgroup-val 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

2.3.25 subgroup

Shrub

This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a shrubmajor third (~425c) in quarter-comma tuning.

Subgroup: 2.3.25

Edo join: 17 & 12

Comma list: 2048/2025

Subgroup-val mapping: [⟨1 1 7], ⟨0 1 -4]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136

2.3.23.25.41 subgroup

See also: Reversed meantone

Edo join: 17 & 12

Comma list: 2048/2025, 576/575, 82/81

Subgroup-val mapping: [⟨1 1 1 7 3], ⟨0 1 6 -4 4]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264

Sburb

This temperament sets the octave-reduced 413th harmonic (413/256, 827.998 ¢) to the diminished seventh.

Subgroup: 2.3.7.23.25.41.59

Edo join: 17 & 12

Comma list: 64/63, 225/224, 162/161, 82/81, 177/175

Subgroup-val mapping: [⟨1 1 4 1 7 3 10], ⟨0 1 -2 6 -4 4 -7]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387

2.9.5.11 subgroup

Glacial

See also: Chromatic pairs § Glacial

Subgroup: 2.9.5.11.13

Comma list: 45/44, 65/64, 81/80

Subgroup-val mapping: [⟨1 0 -4 -6 10], ⟨0 1 2 3 -2]]

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

gencom: [2 9/8; 45/44 65/64 81/80]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 186.151

Optimal ET sequence: 6, 13, 45be, 58bce, 71bce, 84bce

RMS error: 2.887 cents

Music:

• Thundersnow - Sevish (2021)

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

Apparatus

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Subgroup-val 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

Joan

See also: Chromatic pairs § Joan

Joan is related to casablanca as well as to orwell.

Subgroup: 2.9.7.11

Comma list: 99/98, 9317/9216

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

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

gencom: [2 11/8; 99/98 9317/9216]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 542.672 cents

Optimal ET sequence: 11, 20, 31, 42, 115bd, 157bd

RMS error: 1.424 cents

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

Subgroup-val 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

Subgroup-val 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

Scales: penta5, penta8, penta11, penta19

2.9.7.13.17 subgroup

Novisept

Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of gizzard.

Subgroup: 2.9.7.13.17

Comma list: 729/728, 442/441, 833/832

Subgroup-val mapping: [⟨1 1 1 -1 3], ⟨0 6 5 13 3]]

Optimal tuning (CWE): ~2 = 1\1, ~9/7 = 433.836

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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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 slendric; the 2.9.5.21.11 extension below specifically restricts mothra.

Subgroup: 2.9.21

Comma list: 1029/1024

Subgroup-val 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

Lookalike temperament: Dual-3 A-Team

Subgroup: 2.9.5.21

Comma list: 81/80, 1029/1024

Sval mapping: [⟨1 2 0 4], ⟨0 3 6 1]]

Mapping generators: ~2, ~21/16

Optimal (POL2) generator: 464.3865

Optimal ET sequence: 13, 18, 31, 44

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

B-team

B-team (23 & 41) is every other step of rodan.

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

Optimal ET sequence: 5, 13c, 18, 23, 41, 64, 87, 151

4.3.5 subgroup

Tetrahanson

Main article: Tetrahanson

Subgroup: 4.3.5

Comma list: 15625/15552

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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⟩

Subgroup-val 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⟩

Subgroup-val 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.

Todo: cleanup

 
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

Subgroup-val 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

Subgroup-val mapping: [⟨1 3 0], ⟨0 1 -2]]

Mapping generators: ~4, ~9/64

Optimal tuning (CTE): ~9/4 = 1419.190

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

Subgroup-val mapping: [⟨1 0 2], ⟨0 1 -1]]

sval mapping generators: ~8, ~9

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

Optimal tuning (CTE): ~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

2.5/3.… subgroups

Magicaltet

See also: Chromatic pairs § Magicaltet

Magicaltet is related to keemic, superkleismic, and magic. The tonic and the first three generator steps make a magical seventh chord, hence the name.

Subgroup: 2.5/3.7.11

Comma list: 100/99 ([2 2 0 -1⟩), 385/384 ([-7 1 1 1⟩)

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

mapping generators: ~2, ~5/3

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

gencom: [2 6/5; 100/99 385/384]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~5/3 = 877.343
• subgroup POTE: ~2 = 1200.000, ~5/3 = 877.351

Optimal ET sequence: 4, 7, 11, 15, 26, 67, 93*

* wart for 5/3

RMS error: 1.206 cents

Starlingtet

See also: Chromatic pairs § Starlingtet

Starlingtet, the 4 & 15 temperament in the 2.5/3.7/3 subgroup, is related to starling as well as to myna. The tonic and the first three generator steps make a starling tetrad, hence the name.

Subgroup: 2.5/3.7/3

Comma list: 126/125 ([1 -3 1⟩)

Subgroup-val mapping: [⟨1 0 -1], ⟨0 1 3]]

mapping generators: ~2, ~5/3

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

gencom: [2 6/5; 126/125]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~5/3 = 888.759
• subgroup POTE: ~2 = 1200.000, ~5/3 = 888.846

Optimal ET sequence: 4, 15, 19, 23, 27

RMS error: 0.8398 cents

Greeley

See also: Chromatic pairs § Greeley

Greeley is related to opossum as well as to nusecond.

Subgroup: 2.5/3.7/3.11/3

Comma list: 121/120 ([-3 -1 0 2⟩), 126/125 ([1 -3 1⟩)

Subgroup-val 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 tunings:

• subgroup CTE: ~2 = 1200.000, ~11/10 = 155.696
• subgroup POTE: ~2 = 1200.000, ~11/10 = 155.776

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

* wart for 11/3

RMS error: 1.034 cents

Skateboard

See also: Chromatic pairs § Skateboard

Skateboard is related to thrasher.

Subgroup: 2.5/3.7/3.11.13/9

Comma list: 56/55 ([3 -1 1 -1⟩), 91/90 ([-1 -1 1 0 1⟩), 100/99 ([2 2 0 -1⟩)

Subgroup-val mapping: [⟨1 0 -1 2 2], ⟨0 1 3 2 -2]]

Gencom mapping: [⟨1 -3/7 4/7 11/7 4 -6/7], ⟨0 0 -1 -3 -2 2]]

gencom: [2 6/5; 56/55 91/90 100/99]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~5/3 = 886.158
• subgroup POTE: ~2 = 1200.000, ~5/3 = 886.158

Optimal ET sequence: 11, 15, 19, 23, 42d, 65d

RMS error: 2.396 cents

Gariberttet

Gariberttet is the 2.5/3.7/3 altergene of sirius.

Gariberttet (2.5/3.7/3.13/11 subgroup)

See also: Chromatic pairs § Gariberttet

Gariberttet can be described as the 4 & 29 temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include quasitemp.

Subgroup: 2.5/3.7/3.13/11

Comma list: 275/273 ([0 2 -1 -1⟩), 847/845 ([0 -1 1 -2⟩)

Subgroup-val mapping: [⟨1 0 0 0], ⟨0 3 5 1]]

Gencom mapping: [⟨1 0 0 0 0 0], ⟨0 -8/3 1/3 7/3 -1/2 1/2]]

gencom: [2 13/11; 275/273 847/845]

Optimal tunings:

• subgroup CTE and POTE: ~2 = 1200.000, ~13/11 = 293.679

Optimal ET sequence: 29, 33, 37, 41, 45, 49, 78, 94, 143*

* wart for 13/11

RMS error: 0.6914 cents

Indium

See also: Chromatic pairs § Indium

Indium can be described as the 8 & 33 temperament in the 2.5/3.7/3.11/3 subgroup.

Subgroup: 2.5/3.7/3.11/3

Comma list: 3025/3024 ([-4 2 -1 2⟩), 3125/3087 ([0 5 -3⟩)

Subgroup-val mapping: [⟨1 0 0 2], ⟨0 6 10 -1]]

Gencom mapping: [⟨1 -1/2 -1/2 -1/2 3/2], ⟨0 -15/4 9/4 25/4 -19/4]]

gencom: [2 12/11; 3025/3024 3125/3087]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~12/11 = 146.978
• subgroup POTE: ~2 = 1200.000, ~12/11 = 147.010

Optimal ET sequence: 8, 33, 41, 49, 204*^†

* wart for 7/3

^† wart for 11/3

RMS error: 0.7788 cents

Ammon

See also: Chromatic pairs § Ammon

Ammon can be described as the 8 & 29 temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends tridec, and is related to ammonite. It is generated by a semidiminished fourth, hence the old name semidim, which has been rejected since 2025 to avoid confusion with another temperament of the same name.

Subgroup: 2.5/3.7/3.11/3.13/3

Comma list: 121/120 ([-3 -1 0 2⟩), 169/168 ([-3 0 -1 0 2⟩), 275/273 ([0 2 -1 1 -1⟩)

Subgroup-val mapping: [⟨1 3 5 3 4], ⟨0 -6 -10 -3 -5]]

Gencom mapping: [⟨1 -3 0 2 0 1], ⟨0 24/5 -6/5 -26/5 9/5 -1/5]]

gencom: [2 13/10; 121/120 169/168 275/273]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~13/10 = 453.121
• subgroup POTE: ~2 = 1200.000, ~13/10 = 453.242

Optimal ET sequence: 8, 29, 37, 45

RMS error: 1.052 cents

Sentry

See also: Chromatic pairs § Sentry

Sentry, the 3 & 5 temperament in the 2.5/3.9/7 subgroup, is related to sensi.

Subgroup: 2.5/3.9/7

Comma list: 245/243 ([0 1 -2⟩)

Subgroup-val mapping: [⟨1 0 0], ⟨0 2 1]]

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

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

Optimal tunings:

• subgroup CTE and POTE: ~2 = 1200.000, ~9/7 = 440.902

Optimal ET sequence: 8, 11, 19, 30, 41, 49, 52, 145*, 166^†, 197*^†, 215^†, 264*^†

* wart for 5/3

^† wart for 9/7

RMS error: 0.7105 cents

Marveltwintri

See also: Chromatic pairs § Marveltwintri

Marveltwintri can be described as the 3 & 4 temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a marveltwin triad, hence the name. Cata is a very natural extension of this temperament to the 2.3.5.13-subgroup.

Subgroup: 2.5/3.13/9

Comma list: 325/324 ([-2 2 1⟩)

Subgroup-val mapping: [⟨1 0 2], ⟨0 1 -2]]

Gencom mapping: [⟨1 -1/6 5/6 0 0 -1/3], ⟨0 -1/2 -3/2 0 0 1]]

gencom: [2 6/5; 325/324]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~5/3 = 882.886
• subgroup POTE: ~2 = 1200.000, ~5/3 = 882.861

Optimal ET sequence: 3, 4, 11, 15, 19, 34, 53, 87, 140

RMS error: 0.2444 cents

2.….7/3.… subgroups

Guanyintet

See also: Chromatic pairs § Guanyintet

Guanyintet, the 4 & 9 temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of guanyin and a restriction of orwell. It is defined by tempering out 1728/1715 (S6/S7) and 540/539 (S12/S14), which imply 176/175 (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a guanyin tetrad, hence the name.

Subgroup: 2.5.7/3.11/3

Comma list: 176/175 ([4 -2 -1 1⟩), 540/539 ([2 1 -2 -1⟩)

Subgroup-val mapping: [⟨1 0 1 3], ⟨0 -3 1 -5]]

mapping generators: ~2, ~7/6

Gencom mapping: [⟨1 -4/3 3 -1/3 5/3], ⟨0 4/3 -3 7/3 -11/3]]

gencom: [2 7/6; 176/175 540/539]

Optimal tunings:

• (subgroup CTE): ~2 = 1200.000, ~7/6 = 270.455
• (subgroup POTE): ~2 = 1200.000, ~7/6 = 270.093

Optimal ET sequence: 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c*

* wart for 7/3

RMS error: 0.6028 cents

Tridecimal guanyintet

Guanyintet can extend to the 13th harmonic by the equivalences (12/11)³ = 13/10 and (15/14)³ = 16/13, therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate winston (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. 40edo remains an excellent tuning.

Subgroup: 2.5.7/3.11/3.13

Comma list: 176/175 ([4 -2 -1 1 0⟩), 540/539 ([2 1 -2 -1 0⟩), 1573/1568 ([-5 0 -2 2 1⟩)

Subgroup-val mapping: [⟨1 0 1 3 1], ⟨0 -3 1 -5 12]]

mapping generators: ~2, ~12/7

Optimal tunings:

• (subgroup CTE): ~2 = 1200.000, ~7/6 = 270.152
• (subgroup POTE): ~2 = 1200.000, ~7/6 = 270.218

Optimal ET sequence: 9, 22, 31, 40, 71, 111, 151, 262c* using subgroup TE

* wart for 7/3

Badness (Sintel): 0.329

Laz

See also: Chromatic pairs § Laz

Laz is related to avalokita as well as to winston.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 144/143 ([4 0 0 -1 -1⟩), 176/175 ([4 -2 -1 1⟩), 196/195 ([2 -1 2 0 -1⟩

Subgroup-val mapping: [⟨1 0 2 -2 6], ⟨0 3 -1 5 -5]]

Gencom mapping: [⟨1 -5/4 3 -1/4 7/4 -1/4], ⟨0 -1/4 -3 3/4 -21/4 19/4]]

gencom: [2 7/6; 144/143 176/175 196/195]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~12/7 = 930.598
• subgroup POTE: ~2 = 1200.000, ~12/7 = 930.700

Optimal ET sequence: 9, 31, 40, 49, 156c*†, 205c*†

* wart for 7/3

† wart for 11/3

RMS error: 0.8790 cents

Kryptonite

See also: Chromatic pairs § Kryptonite

Kryptonite is related to krypton.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 56/55 ([3 -1 1 -1⟩), 78/77 ([1 0 -1 -1 1⟩), 91/90 ([-1 -2 1 0 1⟩)

Subgroup-val mapping: [⟨1 2 1 2 2], ⟨0 3 2 -1 1]]

mapping generators: ~2, ~13/12

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

gencom: [2 13/12; 56/55 78/77 91/90]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~13/12 = 130.945
• subgroup POTE: ~2 = 1200.000, ~13/12 = 132.428

Optimal ET sequence: 1, …, 8, 9

RMS error: 2.545 cents

Kiribati

See also: Chromatic pairs § Kiribati

Kiribati is related to nakika as well as to octacot.

Subgroup: 2.9/5.7/3.11/9

Comma list: 100/99 ([2 -2 0 -1⟩), 245/242 ([-1 -1 2 -2⟩)

Subgroup-val mapping: [⟨1 1 1 0], ⟨0 -2 3 4]]

mapping generators: ~2, ~21/20

Gencom mapping: [⟨1 1/10 -4/5 11/10 1/5], ⟨0 -3/2 -1 3/2 1]]

gencom: [2 21/20; 100/99 245/242]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~21/20 = 87.776
• subgroup POTE: ~2 = 1200.000, ~21/20 = 87.892

Optimal ET sequence: 13, 14, 27, 41

RMS error: 1.245 cents

Mothwelltri

See also: Chromatic pairs § Mothwelltri

Mothwelltri, the 1 & 4 temperament in the 2.7/3.11 subgroup, is related to orwell. The tonic and the first two generator steps make a mothwellsmic triad, hence the name.

Subgroup: 2.7/3.11

Comma list: 99/98 ([-1 -2 1⟩)

Subgroup-val mapping: [⟨1 0 1], ⟨0 1 2]]

mapping generators: ~2, ~7/3

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

gencom: [2 7/6; 99/98]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~7/6 = 273.695
• subgroup POTE: ~2 = 1200.000, ~7/6 = 273.174

Optimal ET sequence: 4, 9, 13, 22, 79

RMS error: 1.064 cents

2.….9/7.… subgroups

Marveltri

See also: Chromatic pairs § Marveltri

Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament. The tonic and the first two generator steps make a marvel triad, hence the name.

Subgroup: 2.5.9/7

Comma list: 225/224 ([-5 2 1⟩)

Subgroup-val mapping: [⟨1 0 5], ⟨0 1 -2]]

mapping generators: ~2, ~5

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

gencom: [2 5; 225/224]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~5/4 = 384.208
• subgroup POTE: ~2 = 1200.000, ~5/4 = 383.638

Optimal ET sequence: 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c*

* wart for 9/7

RMS error: 0.4801 cents

Sulis

Sulis is related to minerva and würschmidt.

Subgroup: 2.5.9/7.11/9

Comma list: 99/98 ([-1 0 2 1⟩), 176/175 ([4 -2 1 1⟩)

Subgroup-val mapping: [⟨1 0 5 -9], ⟨0 1 -2 4]]]

Optimal tunings:

• subgroup CTE: ~2 = 1200.000, ~5/4 = 386.617
• subgroup POTE: ~2 = 1200.000, ~5/4 = 386.558

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

RMS error: 1.074 cents

2.….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

Subgroup-val 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

Argentic

Argentic is the 2.3.7/5 subgroup temperament tempering out 5120/5103.

Subgroup: 2.3.7/5

Comma list: 5120/5103 = [10 -6 -1⟩

Subgroup-val mapping: [⟨1 0 10], ⟨0 1 -6]]

mapping generators: ~2, ~3

Optimal tunings:

• subgroup CTE: ~2 = 1\1, ~3/2 = 702.792
• subgroup POTE: ~2 = 1\1, ~3/2 = 702.830

Optimal ET sequence: 12, 29, 41, 70, 321, 391, 461, 531, 601 based on subgroup TE

Badness (Sintel): 0.119

Edson (2.3.7/5.11/5.13/5 subgroup)

See also: Chromatic pairs § Edson

Edson is related to pele and andromeda.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 196/195 = [2 -1 2 0 -1⟩, 352/351 = [5 -3 0 1 -1⟩, 364/363 = [2 -1 1 -2 1⟩

Subgroup-val mapping: [⟨1 0 10 17 22], ⟨0 1 -6 -10 -13]]

mapping generators: ~2, ~3

Gencom mapping: [⟨1 1 -5 -1 2 4], ⟨0 1 29/4 5/4 -11/4 -23/4]]

gencom: [2 3/2; 196/195, 352/351, 364/363]

Optimal tunings:

• subgroup CTE: ~2 = 1\1, ~3/2 = 703.4398
• subgroup POTE: ~2 = 1\1, ~3/2 = 703.414

Optimal ET sequence: 12, 17, 29

RMS error: 0.5102 cents

Haumea

See also: Chromatic pairs § Haumea

Related temperaments include bridgetown, namaka, hemigari, barbados, and parizekmic.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 352/351, 676/675, 847/845

Subgroup-val mapping: [⟨1 0 10 -6 -1], ⟨0 2 -12 9 3]]

Gencom mapping: [⟨1 2 -3/4 -11/4 9/4 5/4], ⟨0 -2 0 12 -9 -3]]

gencom: [2 15/13; 352/351 676/675 847/845]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.491

Optimal ET sequence: 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd

RMS error: 0.2668 cents

Historical

Not to be confused with Historical temperaments.

Not to be confused with History (temperament)., which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of miracle that splits 4/3 in six rather than 3/2. It tempers out the comma S10/S11 = 4000/3993 to set 11/10 equal to one-third of 4/3, and S13/S15 = 676/675 to equate 15/13 to one-half of 4/3, and tempers out S21 = 441/440 to split 11/10 into two instances of 22/21~21/20. Sextilifourths adds the schismic mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 364/363, 441/440, 1001/1000

Subgroup-val 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

Terrain

"Terrain" redirects here. For the scale, see Terrain (scale).

See also: Chromatic pairs § Terrain

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to domain. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

Subgroup: 2.7/5.9/5

Comma list: 250047/250000

Subgroup-val mapping: [⟨3 1 3], ⟨0 1 -1]]

Gencom mapping: [⟨3 10/9 -7/9 2/9], ⟨0 -2/3 -1/3 2/3]]

gencom: [63/50 10/9; 250047/250000]

Optimal tuning (subgroup POTE): ~63/50 = 1\3, ~10/9 = 182.461

Optimal ET sequence: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558

RMS error: 0.00844 cents

Tridec

See also: Chromatic pairs § Tridec

See also: Non-over-1 temperament § Tridec

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends #Petrtri.

Subgroup: 2.7/5.11/5.13/5

Comma list: 847/845, 1001/1000

Subgroup-val mapping: [⟨1 2 0 1], ⟨0 -4 3 1]]

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

gencom: [2 13/10; 847/845 1001/1000]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 454.556

Optimal ET sequence: 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc

RMS error: 0.1613 cents

Naiadec

Subgroup: 2.7/5.11/5.13/5.17/5

Comma list: 170/169, 221/220, 847/845

Subgroup-val mapping: [⟨1 2 0 1 1], ⟨0 -4 3 1 2]]

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

gencom: [2 13/10; 170/169 221/220 847/845]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 454.882

Optimal ET sequence: 5, 8, 21, 29, 95^t, 124^t

^t wart for 17/5

RMS error: 0.7521 cents

2.….11/5.… subgroups

Petrtri

See also: Chromatic pairs § Petrtri

See also: 5L 3s/Temperaments § Petrtri

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

Subgroup: 2.11/5.13/5

Comma list: 2200/2197

Subgroup-val mapping: [⟨1 0 1], ⟨0 3 1]]

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

gencom: [2 13/10; 2200/2197]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 455.012

Optimal ET sequence: 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c

RMS error: 0.0749 cents

Bridgetown

See also: Chromatic pairs § Bridgetown

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to haumea and barbados.

Subgroup: 2.3.11/5.13/5

Comma list: 352/351, 676/675

Subgroup-val mapping: [⟨1 0 -6 -1], ⟨0 2 9 3]]

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

gencom: [2 15/13; 352/351 676/675]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.399

Optimal ET sequence: 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314

RMS error: 0.2513 cents

Hypnosis

Related temperaments: hypnos, alphatricot

Subgroup: 2.3.7.11/5.13

Comma list: 169/168, 540/539, 729/728

Subgroup-val 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

Trisect

Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of trisected.

Extending this temperament to the full 11-, 13-, or 17-limit through portent or landscape results in the weak extension known as tritikleismic.

Subgroup: 2.3.7.11/5

Comma list: 1029/1024, 4000/3993

Subgroup-val mapping: [⟨3 0 10 5], ⟨0 3 -1 -1]]

Optimal tuning (subgroup POTE): ~44/35 = 1\3, ~13/9 = 633.742

Optimal ET sequence: 15, 21, 36, 123, 159, 195, 231

RMS error: ???

2.3.7.11/5.13 subgroup

Subgroup: 2.3.7.11/5.13

Comma list: 1029/1024, 1575/1573, 2080/2079

Subgroup-val mapping: [⟨3 0 10 5 0], ⟨0 3 -1 -1 7]]

Optimal tuning (subgroup POTE): ~44/35 = 1\3, ~13/9 = 633.918

Optimal ET sequence: 15, 21f, 36, 87, 123, 159

RMS error: ???

2.3.7.11/5.13.17 subgroup

Subgroup: 2.3.7.11/5.13.17

Comma list: 273/272, 833/832, 1575/1573, 2080/2079

Subgroup-val mapping: [⟨3 0 10 5 0 -2], ⟨0 3 -1 -1 7 9]]

Optimal tuning (subgroup POTE): ~34/27 = 1\3, ~13/9 = 633.820

Optimal ET sequence: 15, 21fg, 36, 123, 159

RMS error: ???

Trisector

Subgroup: 2.3.7.11/5.13.17.19

Comma list: 210/209, 273/272, 286/285, 595/594, 2080/2079

Subgroup-val mapping: [⟨3 0 10 5 0 -2 8], ⟨0 3 -1 -1 7 9 3]]

Optimal tuning (subgroup POTE): ~34/27 = 1\3, ~13/9 = 633.894

Optimal ET sequence: 15, 21fg, 36, 123h, 159h

RMS error: ???

2.3.7.11/5.13.17.19.23 subgroup

Subgroup: 2.3.7.11/5.13.17.19.23

Comma list: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

Subgroup-val mapping: [⟨3 0 10 5 0 -2 8 12], ⟨0 3 -1 -1 7 9 3 1]]

Optimal tuning (subgroup POTE): ~34/27 = 1\3, ~13/9 = 634.038

Optimal ET sequence: 15g, 21fg, 36, 87, 123hi

RMS error: ???

2.3.7.11/5.13.17.19.23.29 subgroup

Subgroup: 2.3.7.11/5.13.17.19.23.29

Comma list: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

Subgroup-val mapping: [⟨3 0 10 5 0 -2 8 12 13], ⟨0 3 -1 -1 7 9 3 1 1]]

Optimal tuning (subgroup POTE): ~29/23 = 1\3, ~13/9 = 634.102

Optimal ET sequence: 15g, 21fg, 36, 87, 123hi

RMS error: ???

2.….11/7.… subgroups

Pepperoni

Main article: Parapyth

See also: Chromatic pairs § Pepperoni

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of parapyth. 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

Subgroup-val 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, 12f, 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

2.….13/5.… subgroups

Barbados

The minimax tuning for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are 24edo, 29edo, 53edo and 111edo, with mos scales of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

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

Music

• Desert Island Rain in 313edo tuned Barbados[9], by Sevish

Tobago

See also: Chromatic pairs § Tobago

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends neutral and barbados.

Subgroup: 2.3.11.13/5

Comma list: 243/242, 676/675

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

Gencom mapping: [⟨2 4 -2 0 9 2], ⟨0 -2 3/2 0 -5 -3/2]]

gencom: [55/39 15/13; 243/242 676/675]

Optimal tuning (subgroup POTE): ~55/39 = 1\2, ~15/13 = 249.312

Optimal ET sequence: 10, 14, 24, 58, 82, 130

RMS error: 0.3533 cents

Pakkanian hemipyth

Subgroup: 2.3.11.13/5.17

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

Subgroup-val 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

Oceanfront

Related temperaments: superpyth, ultrapyth

Subgroup: 2.3.7.13/5

Comma list: 64/63, 91/90

Subgroup-val 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

2.….49/5.… subgroups

Direct breedsmic

Related temperament: hemithirds, newt

Subgroup: 2.3.49/5

Comma list: 2401/2400

Subgroup-val mapping: [⟨1 1 3], ⟨0 2 1]]

Optimal tuning (subgroup POTE): ~49/40 = 350.966

Optimal ET sequence: 7, 10, 17

RMS error: ?

2.….17/5.… subgroups

Fiventeen

Fiventeen tempers out 136/135 ([3 -3 1⟩) in 2.3.17/5. It equates 17/15 with 9/8, so it implies a supersoft pentic pentad of ~30:34:40:45:51. 17edo makes a good tuning especially for its size, which gives a supersoft pentic scale corresponding approximately to a just 20/17 tuning, although 80edo might be preferred for an approximately just 51/40 to optimize plausibility slightly more, and 97edo (= 80 + 17) and 114edo (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the optimal ET sequence is much more characteristic of optimized tunings, finding 34edo, then 80edo, then 114edo (= 34 + 80) and even 194bc-edo (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting 63edo and 143edo (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

Subgroup: 2.3.17/5

Comma list: 136/135 ([3 -3 1⟩)

Subgroup-val mapping: [⟨1 0 -3], ⟨0 1 3]]

mapping generators: ~2, ~3

Optimal tunings:

• Subgroup WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢
• Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢

Optimal ET sequence: 5, 12, 17, 46, 63, 143

2.….19/7.… subgroups

Surprise

This temperament was named by Vector in 2025, as he was surprised that the temperament of 57/56 did not have a name. This is the rank-2 version of the temperament; Vector surmises that the name hendrix would be more thoughtfully given to the rank-3 version.

Subgroup: 2.3.19/7

Comma list: 57/56 ([-3 1 1⟩)

Subgroup-val mapping: [⟨1 0 3], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tunings:

• Subgroup WE: ~2 = 1202.4345 ¢, ~3/2 = 697.4314 ¢
• Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3981 ¢

Optimal ET sequence: 5, 7, 12, 19, 31*, 50*

* wart for 19/7

Badness (Sintel): 0.082

Supramin

This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is 25edo, which provides an accurate yet tone-efficient tuning of this temperament. It was named by Overthink in 2026 after the fact that the generator is a 17/14 supraminor third, two of which reach 28/19.

Subgroup: 2.17/7.19/7

Comma list: 5491/5488 ([-4 2 1⟩)

Subgroup-val mapping: [⟨1 0 4], ⟨0 1 -2]]

mapping generators: ~2, ~17/7

Optimal tunings:

• Subgroup WE: ~2 = 1200.022 ¢, ~17/14 = 335.793 ¢
• Subgroup CWE: ~2 = 1200.000 ¢, ~17/14 = 335.785 ¢

Optimal ET sequence: 7, 18, 25

Badness (Sintel): 0.005

Supramine

This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

Subgroup: 2.17/7.19/7.23/7

Comma list: 323/322, 392/391

Subgroup-val mapping: [⟨1 0 4 3], ⟨0 1 -2 -1]]

Optimal tunings:

• Subgroup WE: ~2 = 1199.871 ¢, ~17/14 = 336.243 ¢
• Subgroup CWE: ~2 = 1200.000 ¢, ~17/14 = 336.296 ¢

Optimal ET sequence: 7, 18, 25

Badness (Sintel): 0.029

2.25/7.17/7.19/7.23/7 subgroup

Subgroup: 2.25/7.17/7.19/7.23/7

Comma list: 323/322, 392/391, 476/475

Subgroup-val mapping: [⟨1 -2 0 4 3], ⟨0 3 1 -2 -1]]

Optimal tunings:

• Subgroup WE: ~2 = 1199.757 ¢, ~17/14 = 335.428 ¢
• Subgroup CWE: ~2 = 1200.000 ¢, ~17/14 = 335.479 ¢

Optimal ET sequence: 7, 18, 25

Badness (Sintel): 0.053

3/2.5/2.… subgroups

Main article: Half-prime subgroup

Hemihemi

Subgroup: 3/2.5/2.7/2

Comma list: 10976/10935

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

Subgroup-val 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

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

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

Optimal ET sequence: 8edf, 11edf

3/2.5/4.… subgroups

Poseidon

This temperament will be subjected to renaming due to a conflict.

Subgroup: 3/2.5/4.11/8

Comma list: 121/120

Subgroup-val mapping: [⟨1 1 1], ⟨0 2 -1]]]

gencom: [3/2 12/11; 121/120]

Optimal tuning (subgroup POTE): ~3/2, ~12/11 = 158.29

Optimal ET sequence: 9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49

Other 3/2-equave subgroups

Auk

Subgroup: 3/2.7.13

Comma list: 87808/85293

Subgroup-val 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

Subgroup-val 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

5/2-equave subgroups

Hyperion

Subgroup: 5/2.7.11

Comma list: [11 1 -5⟩

Subgroup-val 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

Related temperament collections

• Dual-fifth temperaments
• Equalizer subgroup temperaments
• Substitute harmonic temperaments