Subgroup temperaments

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

For temperaments that omit various prime harmonics, see:

• No-elevens subgroup temperaments
• No-sevens subgroup temperaments
• No-fives subgroup temperaments
• No-threes subgroup temperaments
• 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

2.3.35 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

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⟩

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

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

Sval 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

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

2.9.5.11 subgroup

Glacial

See also: Chromatic pairs § Glacial

Subgroup: 2.9.5.11.13

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

Sval 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

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

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

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

Sval 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

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

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

2.5/3… subgroups

Magicaltet

See also: Chromatic pairs § Magicaltet

Magicaltet is related to supermagic, superkleismic, and magic.

Subgroup: 2.5/3.7.11

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

Sval 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 = 1\1, ~5/3 = 877.3426
• subgroup POTE: ~2 = 1\1, ~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.

Subgroup: 2.5/3.7/3

Comma list: 126/125

Sval mapping: [⟨1 0 -1], ⟨0 1 3]]

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

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 311.154

Optimal ET sequence: 15, 19, 23, 27, 239b, 266b, 293b, 320b, 347b, 374b, 401b

RMS error: 0.8398 cents

Greeley

See also: Chromatic pairs § 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

Skateboard

See also: Chromatic pairs § Skateboard

Skateboard is related to thrasher.

Subgroup: 2.5/3.7/3.11.13/9

Comma list: 56/55, 91/90, 100/99

Sval 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 tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 313.842

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.

Subgroup: 2.5/3.7/3.13/11

Comma list: 275/273, 847/845

Sval 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 tuning (subgroup POTE): ~2 = 1\1, ~13/11 = 293.679

Optimal ET sequence: 29, 33, 37, 41, 45, 49, 78, 94, 143d, 237bd, 380bd

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, 3125/3087

Sval 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 tuning (subgroup POTE): ~2 = 1\1, ~12/11 = 147.010

Optimal ET sequence: 33, 41, 49, 57, 106, 204, 253

RMS error: 0.7788 cents

Semidim

See also: Chromatic pairs § Semidim

Semidim can be described as the 8 & 29 temperament in the 2.5/3.7/3.11/3.13/3 subgroup.

Subgroup: 2.5/3.7/3.11/3.13/3

Comma list: 121/120, 169/168, 275/273

Sval 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 tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 453.242

Optimal ET sequence: 5, 8, 13, 21, 29, 37, 44, 45, 188bde

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

Sval mapping: [⟨1 0 0], ⟨0 2 1]]

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

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~9/7 = 440.902

Optimal ET sequence: 8, 11, 19, 30, 41, 49, 52, 145b, 166c, 197bc, 215c, 264bc

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.

Subgroup: 2.5/3.13/9

Comma list: 325/324

Sval 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 tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 317.139

Optimal ET sequence: 11, 15, 19, 34, 53, 87, 140, 185, 219, 253, 287b, 321b

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 related to guanyin as well as to orwell.

Subgroup: 2.5.7/3.11/3

Comma list: 176/175, 540/539

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

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 tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 270.093

Optimal ET sequence: 9, 31, 40, 49, 89, 191bc, 227bc, 231bc, 271bc, 311bc, 316bcd

RMS error: 0.6028 cents

Laz

See also: Chromatic pairs § Laz

Laz is related to georgian as well as to winston.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 144/143, 176/175, 196/195

Sval 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 tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 269.300

Optimal ET sequence: 9, 31, 40, 49, 58, 156bde, 205bde

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, 78/77, 91/90

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

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 tuning (subgroup POTE): ~2 = 1\1, ~13/12 = 132.428

Optimal ET sequence: 9, 63, 82bd, 91bde

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, 245/242

Sval mapping: [⟨1 1 1 0], ⟨0 -2 3 4]]

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 tuning (subgroup POTE): ~2 = 1\1, ~21/20 = 87.892

Optimal ET sequence: 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd

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.

Subgroup: 2.7/3.11

Comma list: 99/98

Sval mapping: [⟨1 0 1], ⟨0 1 2]]

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

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

Optimal tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 273.174

Optimal ET sequence: 9, 22, 40, 49c, 58c, 67c, 76c, 79, 101b, 123bc

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.

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: 2.5.9/7.11/7

Comma list: 99/98, 176/175

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

Optimal tuning (subgroup POTE): ~5/4 = 386.558

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

RMS error: 1.074 cents

2.….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

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

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

Edson

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

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⟩

Sval 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

Sval 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.

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

Terrain

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

Sval 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

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

Sval 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

2.….11/5… subgroups

Petrtri

See also: Chromatic pairs § 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

Sval 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

Sval 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, 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

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

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

2.….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

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

Sval 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

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

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

2.….49/5… subgroups

Direct breedsmic

Related temperament: hemithirds, newt

Subgroup: 2.3.49/5

Comma list: 2401/2400

Sval mapping: [⟨1 1 3], ⟨0 2 1]]

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

Optimal ET sequence: ?

RMS error: ?

3/2.5/2… subgroups

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

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

Sval 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

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

5/2-equave 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

Related temperament collections

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