Aberschismic 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.

This page collects rank-2 aberschismic temperaments, which temper out the aberschisma (monzo: [10 -6 1 -1⟩, ratio: 5120/5103). These temperaments divide an exact or approximate septimal quartertone, 36/35 into two equal steps, each representing 81/80~64/63, the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same chain of fifths inflected by the same comma to the opposite sides. In addition we may identify 10/7 by the augmented fourth and 50/49 by the Pythagorean comma.

Temperaments belonging to this category and generated by the octave and the fifth are dominant, garibaldi, leapday, kwai, and undecental, which map 5/4 to the M3, d4, 3A1, 4A7, and 5d7, respectively.

Temperaments that slice the octave or fifth into multiple parts are common. Some examples are diaschismic (diploid), hemififths (dicot), hemidromeda (alpha-dicot), misty (triploid), rodan (tricot), alphatrimot (alpha-tricot), undim (tetraploid), monkey (tetracot), and buzzard (alpha-tetracot). Others reach the octave or fifth in more complex ways

Temperaments discussed elsewhere are:

• Dominant (+36/35) → Meantone family
• Garibaldi (+225/224) → Schismatic family
• Leapday (+686/675) → Sengic temperaments
• Diaschismic (+126/125) → Diaschismic family
• Hemififths (+2401/2400) → Breedsmic temperaments
• Misty (+3136/3125) → Misty family
• Rodan (+245/243) → Gamelismic clan
• Alphatrimot (+2430/2401) → Alphatricot family
• Undim (+390625/388962) → Undim family
• Monkey (+875/864) → Tetracot family
• Buzzard (+1728/1715) → Buzzardsmic clan
• Amity (+4375/4374) → Amity family
• Supers (+118098/117649) → Stearnsmic clan
• Countercata (+15625/15552) → Kleismic family
• Abergravity (+177147/175000) → Gravity family
• Warrior (+78732/78125) → Sensipent family
• Alphaquarter (+29360128/29296875) → Escapade family

Considered below are septiquarter, kwai, ketchup, undecental, mystery, hemidromeda, countriton, artoneutral, quintakwai, quinticosiennic, quanic and jorgensen, in the order of increasing badness.

Septiquarter

Septiquarter tempers out 420175/419904 and may be described as the 94 & 99 temperament. Its ploidacot is epsilon-heptacot. 99edo makes for an excellent tuning, and 292edo an even better one. 94edo and 104edo in the 104c val are also among the possibilities.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 420175/419904

Mapping: [⟨1 -4 -28 6], ⟨0 7 38 -4]]

mapping generators: ~2, ~243/140

Optimal tunings:

• WE: ~2 = 1199.7212 ¢, ~243/140 = 957.3250 ¢

error map: ⟨-0.279 +0.435 -0.158 +0.201]

• CWE: ~2 = 1200.0000 ¢, ~243/140 = 957.5424 ¢

error map: ⟨0.000 +0.842 +0.298 +1.004]

Optimal ET sequence: 94, 99, 292, 391, 881bd, 1272bcd

Badness (Sintel): 1.36

Semiseptiquarter

Subgroup: 2.3.5.7.11

Comma list: 5120/5103, 9801/9800, 14641/14580

Mapping: [⟨2 -8 -56 12 -25], ⟨0 7 38 -4 20]]

Optimal tunings:

• WE: ~99/70 = 599.8953 ¢, ~210/121 = 957.3819 ¢
• CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5449 ¢

Optimal ET sequence: 94, 198, 292, 490

Badness (Sintel): 2.12

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1716/1715, 14641/14580

Mapping: [⟨2 -8 -56 12 -25 9], ⟨0 7 38 -4 20 -1]]

Optimal tunings:

• WE: ~99/70 = 599.8565 ¢, ~210/121 = 957.3261 ¢
• CWE: ~99/70 = 600.0000 ¢, ~210/121 = 957.5508 ¢

Optimal ET sequence: 94, 198, 490f

Badness (Sintel): 1.44

Kwai

For the 5-limit version, see Miscellaneous 5-limit temperaments #Kwai.

Named by Gene Ward Smith in 2004 for its "bridgeability", kwai is generated by a perfect fifth, and can be described as 41 & 70.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 16875/16807

Mapping: [⟨1 0 -50 -40], ⟨0 1 33 27]]

mapping generators: ~2, ~3

Optimal tunings:

• WE: ~2 = 1199.7337 ¢, ~3/2 = 702.4600 ¢

error map: ⟨-0.266 +0.239 -0.607 +1.055]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6085 ¢

error map: ⟨0.000 +0.653 -0.234 +1.603]

Optimal ET sequence: 41, 111, 152, 345, 497d

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 5120/5103

Mapping: [⟨1 0 -50 -40 32], ⟨0 1 33 27 -18]]

Optimal tunings:

• WE: ~2 = 1199.6672 ¢, ~3/2 = 702.4282 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6189 ¢

Optimal ET sequence: 41, 111, 152, 497de, 649dde

Badness (Sintel): 0.867

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1375/1372

Mapping: [⟨1 0 -50 -40 32 27], ⟨0 1 33 27 -18 -21]]

Optimal tunings:

• WE: ~2 = 1199.4772 ¢, ~3/2 = 702.3379 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6409 ¢

Optimal ET sequence: 41, 111, 152f, 415dff

Badness (Sintel): 1.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 540/539, 715/714, 1089/1088

Mapping: [⟨1 0 -50 -40 32 27 58], ⟨0 1 33 27 -18 -21 -34]]

Optimal tunings:

• WE: ~2 = 1199.3537 ¢, ~3/2 = 702.2850 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6589 ¢

Optimal ET sequence: 41, 70, 111, 152fg, 263dfg

Badness (Sintel): 1.12

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 352/351, 400/399, 456/455, 715/714, 847/845

Mapping: [⟨1 0 -50 -40 32 27 58 -56], ⟨0 1 33 27 -18 -21 -34 38]]

Optimal tunings:

• WE: ~2 = 1199.3401 ¢, ~3/2 = 702.2705 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6548 ¢

Optimal ET sequence: 41, 70h, 111, 152fg, 263dfgh

Badness (Sintel): 1.03

Hemikwai

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 676/675, 1375/1372, 5120/5103

Mapping: [⟨1 0 -50 -40 32 -51], ⟨0 2 66 54 -36 69]]

mapping generators: ~2, ~26/15

Optimal tunings:

• WE: ~2 = 1199.6968 ¢, ~26/15 = 951.0740 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3123 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.82

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 442/441, 540/539, 676/675, 715/714, 5120/5103

Mapping: [⟨1 0 -50 -40 32 -51 -30], ⟨0 2 66 54 -36 69 43]]

Optimal tunings:

• WE: ~2 = 1199.6861 ¢, ~26/15 = 951.0654 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3120 ¢

Optimal ET sequence: 82, 111, 193, 304d

Badness (Sintel): 1.31

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 442/441, 540/539, 676/675, 715/714, 1445/1444

Mapping: [⟨1 0 -50 -40 32 -51 -30 -56], ⟨0 2 66 54 -36 69 43 76]]

Optimal tunings:

• WE: ~2 = 1199.6718 ¢, ~26/15 = 951.0526 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.3103 ¢

Optimal ET sequence: 82, 111, 193, 304dh

Badness (Sintel): 1.16

Ketchup

Ketchup may be described as the 46 & 94 temperament. It has a semi-octave period and a generator for a syntonic~septimal comma, four of which plus a period gives the perfect fifth; its ploidacot is diploid gamma-tetracot. 140edo is an obvious tuning for this temperament.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 1071875/1062882

Mapping: [⟨2 3 4 6], ⟨0 4 15 -9]]

mapping generators: ~1225/864, ~64/63

Optimal tunings:

• WE: ~1225/864 = 599.9685 ¢, ~64/63 = 25.7181 ¢

error map: ⟨-0.063 +0.823 -0.668 -0.478]

• CWE: ~1225/864 = 600.0000 ¢, ~64/63 = 25.7181 ¢

error map: ⟨0.000 +0.917 -0.543 -0.288]

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 2.14

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 2200/2187

Mapping: [⟨2 3 4 6 7], ⟨0 4 15 -9 -2]]

Optimal tunings:

• WE: ~99/70 = 600.0678 ¢, ~64/63 = 25.6963 ¢
• CWE: ~99/70 = 600.0000 ¢, ~64/63 = 25.6956 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1331/1323

Mapping: [⟨2 3 4 6 7 8], ⟨0 4 15 -9 -2 -14]]

Optimal tunings:

• WE: ~99/70 = 600.0612 ¢, ~66/65 = 25.7000 ¢
• CWE: ~99/70 = 600.0000 ¢, ~66/65 = 25.6978 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 1.03

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 385/384, 442/441

Mapping: [⟨2 3 4 6 7 8 8], ⟨0 4 15 -9 -2 -14 4]]

Optimal tunings:

• WE: ~17/12 = 600.0896 ¢, ~66/65 = 25.7048 ¢
• CWE: ~17/12 = 600.0000 ¢, ~66/65 = 25.7017 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 0.845

2.3.5.7.11.13.17.23 subgroup

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 253/252, 289/288, 325/324, 352/351, 385/384, 391/390

Mapping: [⟨2 3 4 6 7 8 8 9], ⟨0 4 15 -9 -2 -14 4 1]]

Optimal tunings:

• WE: ~17/12 = 600.1139 ¢, ~66/65 = 25.7053 ¢
• CWE: ~17/12 = 600.0000 ¢, ~66/65 = 25.7013 ¢

Optimal ET sequence: 46, 94, 140

Badness (Sintel): 0.772

Undecental

Undecental adds the triwellisma to the comma list and may be described as the 29 & 70 temperament. 5/4 is mapped to the quintuple-diminished seventh or equivalently the perfect fourth minus three dieses. 58\99 is an almost perfect generator, just as the name suggests. Another interesting tuning choice is the argent fifth, 2^{(2 - sqrt (2))}.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 235298/234375

Mapping: [⟨1 0 61 71], ⟨0 1 -37 -43]]

mapping generators: ~2, ~3

Optimal tunings:

• WE: ~2 = 1199.6543 ¢, ~3/2 = 702.8370 ¢

error map: ⟨-0.346 +0.536 +0.423 -0.494]

• CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.0465 ¢

error map: ⟨0.000 +1.092 +0.966 +0.175]

Optimal ET sequence: 29, 70, 99, 722bc, 821bc, 920bc, 1019bc

Badness (Sintel): 2.39

Mystery

Main article: Mystery

For the 5-limit version, see 29th-octave temperaments #Mystery.

Mystery tempers out 50421/50000 and may be described as the 29 & 58 temperament. It has a 1\29 period and primes 5, 7, 11 and 13 are all reached by one generator step; its ploidacot is 29-ploid acot. 145edo or 232edo are good candidates for tunings.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 50421/50000

Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]

mapping generators: ~50/49, ~5

Optimal tunings:

• WE: ~50/49 = 41.3652 ¢, ~5/4 = 388.5128 ¢

error map: ⟨-0.410 +0.842 +1.378 -2.022]

• CWE: ~50/49 = 41.3793 ¢, ~5/4 = 388.3030 ¢

error map: ⟨0.000 +1.493 +1.989 -1.213]

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 2.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3388/3375

Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]

Optimal tunings:

• WE: ~45/44 = 41.3637 ¢, ~5/4 = 388.3136 ¢
• CWE: ~45/44 = 41.3793 ¢, ~5/4 = 388.0598 ¢

Optimal ET sequence: 29, 58, 87, 145

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 676/675

Mapping: [⟨29 46 0 14 33 40], ⟨0 0 1 1 1 1]]

Optimal tunings:

• WE: ~45/44 = 41.3623 ¢, ~5/4 = 388.1942 ¢
• CWE: ~40/39 = 41.3793 ¢, ~5/4 = 387.9017 ¢

Optimal ET sequence: 29, 58, 87, 145, 232

Badness (Sintel): 0.768

Hemidromeda

Hemidromeda may be described as the 29 & 111 temperament. Named by Xenllium in 2023, hemidromeda comes from hemi- (Ancient Greek for "one half") and andromeda, because the generator is 1/2 of andromeda's perfect twelfth (~3/1, about 1902.4 cents); the ploidacot for this temperament is alpha-dicot.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 52734375/52706752

Mapping: [⟨1 0 38 48], ⟨0 2 -45 -57]]

mapping generator: ~2, ~12500/7203

Optimal tunings:

• WE: ~2 = 1199.7236 ¢, ~12500/7203 = 951.1864 ¢

error map: ⟨-0.276 +0.418 -0.205 +0.282]

• CWE: ~2 = 1200.0000 ¢, ~12500/7203 = 951.4098 ¢

error map: ⟨0.000 +0.865 +0.243 +0.813]

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391, 1424bbcdd

Badness (Sintel): 2.93

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 5120/5103

Mapping: [⟨1 0 38 48 32], ⟨0 2 -45 -57 -36]]

Optimal tunings:

• WE: ~2 = 1199.8767 ¢, ~400/231 = 951.3065 ¢
• CWE: ~2 = 1200.0000 ¢, ~400/231 = 951.4063 ¢

Optimal ET sequence: 29, 82cd, 111, 140, 251, 391e

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 676/675, 847/845, 1331/1323

Mapping: [⟨1 0 38 48 32 37], ⟨0 2 -45 -57 -36 -42]]

Optimal tunings:

• WE: ~2 = 1199.8753 ¢, ~26/15 = 951.3054 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4064 ¢

Optimal ET sequence: 29, 82cdf, 111, 140, 251, 391e

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 561/560, 676/675, 715/714

Mapping: [⟨1 0 38 48 32 37 58], ⟨0 2 -45 -57 -36 -42 -68]]

Optimal tunings:

• WE: ~2 = 1199.8770 ¢, ~26/15 = 951.3039 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4035 ¢

Optimal ET sequence: 29g, 82cdfg, 111, 140, 251, 391e

Badness (Sintel): 0.971

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 352/351, 363/361, 442/441, 476/475, 561/560

Mapping: [⟨1 0 38 48 32 37 58 32], ⟨0 2 -45 -57 -36 -42 -68 -35]]

Optimal tunings:

• WE: ~2 = 1199.7534 ¢, ~26/15 = 951.2024 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4020 ¢

Optimal ET sequence: 29g, 82cdfgh, 111, 140

Badness (Sintel): 1.01

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 352/351, 363/361, 391/390, 442/441, 460/459

Mapping: [⟨1 0 38 48 32 37 58 32 18], ⟨0 2 -45 -57 -36 -42 -68 -35 -17]]

Optimal tunings:

• WE: ~2 = 1199.9128 ¢, ~26/15 = 951.3371 ¢
• CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.4076 ¢

Optimal ET sequence: 29g, 82cdfgh, 111, 140

Badness (Sintel): 1.10

Countriton

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic.

Countriton may be described as the 51c & 53 temperament. It splits the 24th harmonic into nine tritone generators; its ploidacot is thus delta-enneacot. Among the possible tunings are 157edo and 210edo, as well as 104edo in the 104c val.

Countriton was named by Xenllium in 2022 as a counterpart of untriton.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 7558272/7503125

Mapping: [⟨1 -3 -15 13], ⟨0 9 34 -20]]

mapping generators: ~2, ~1225/864

Optimal tunings:

• WE: ~2 = 1199.4179 ¢, ~1225/864 = 611.1213 ¢

error map: ⟨-0.582 -0.117 +0.541 +1.181]

• CWE: ~2 = 1200.0000 ¢, ~1225/864 = 611.4120 ¢

error map: ⟨0.000 +0.753 +1.695 +2.934]

Optimal ET sequence: 51c, 53, 157, 210, 473cdd

Badness (Sintel): 3.32

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 5120/5103, 41503/41472

Mapping: [⟨1 -3 -15 13 -21], ⟨0 9 34 -20 48]]

Optimal tunings:

• WE: ~2 = 1199.5178 ¢, ~77/54 = 611.2097 ¢
• CWE: ~2 = 1200.0000 ¢, ~77/54 = 611.4495 ¢

Optimal ET sequence: 51ce, 53, 104c, 157

Badness (Sintel): 2.80

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 2197/2187

Mapping: [⟨1 -3 -15 13 -21 -7], ⟨0 9 34 -20 48 21]]

Optimal tunings:

• WE: ~2 = 1199.5944 ¢, ~77/54 = 611.2491 ¢
• CWE: ~2 = 1200.0000 ¢, ~77/54 = 611.4506 ¢

Optimal ET sequence: 51ce, 53, 104c, 157

Badness (Sintel): 1.75

Artoneutral

Artoneutral can be described as the 87 & 94 temperament. It is generated by an artoneutral third of ~11/9 (or a tendoneutral sixth of ~18/11), nine of which make the 12th harmonic; its ploidacot is thus beta-enneacot. 181edo may be recommended as a tuning.

Artoneutral was named by Flora Canou in 2023 for its generator's quality.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 3828125/3779136

Mapping: [⟨1 -1 -4 12], ⟨0 9 22 -32]]

mapping generators: ~2, ~128/105

Optimal tunings:

• WE: ~2 = 1200.1400 ¢, ~128/105 = 344.7929 ¢

error map: ⟨+0.140 +1.041 -1.430 -0.518]

• CWE: ~2 = 1200.0000 ¢, ~128/105 = 344.7531 ¢

error map: ⟨0.000 +0.823 -1.746 -0.925]

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 3.98

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 4000/3993

Mapping: [⟨1 -1 -4 12 -2], ⟨0 9 22 -32 19]]

Optimal tunings:

• WE: ~2 = 1200.1668 ¢, ~11/9 = 344.8027 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7557 ¢

Optimal ET sequence: 87, 181

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 1575/1573

Mapping: [⟨1 -1 -4 12 -2 6], ⟨0 9 22 -32 19 -8]]

Optimal tunings:

• WE: ~2 = 1200.0662 ¢, ~11/9 = 344.7804 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7617 ¢

Optimal ET sequence: 87, 181

Badness (Sintel): 1.08

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 385/384, 595/594

Mapping: [⟨1 -1 -4 12 -2 6 -12], ⟨0 9 22 -32 19 -8 56]]

Optimal tunings:

• WE: ~2 = 1200.0346 ¢, ~11/9 = 344.7589 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7492 ¢

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 1.16

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 352/351, 375/374, 385/384, 400/399, 595/594

Mapping: [⟨1 -1 -4 12 -2 6 -12 -15], ⟨0 9 22 -32 19 -8 56 67]]

Optimal tunings:

• WE: ~2 = 1200.0282 ¢, ~11/9 = 344.7532 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7453 ¢

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 1.19

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 300/299, 325/324, 352/351, 375/374, 385/384, 400/399, 484/483

Mapping: [⟨1 -1 -4 12 -2 6 -12 -15 -13], ⟨0 9 22 -32 19 -8 56 67 61]]

Optimal tunings:

• WE: ~2 = 1200.0163 ¢, ~11/9 = 344.7461 ¢
• CWE: ~2 = 1200.0000 ¢, ~11/9 = 344.7416 ¢

Optimal ET sequence: 87, 94, 181

Badness (Sintel): 1.17

Quintakwai

For the 5-limit version, see Schismic–commatic equivalence continuum #Quindromeda.

Quintakwai may be described as the 12 & 181 temperament with a ploidacot signature of omega-pentacot. It tempers out the compass comma in the 7-limit; 1375/1372 and 4375/4356 in the 11-limit. It also allows a natural extension to the 2.3.5.7.11.17.19 subgroup, where 225/224 (S15), 256/255 (S16), 289/288 (S17), 324/323 (S18), and 361/360 (S19) are tempered together, and 400/399 (S20) is tempered out.

Quintakwai was named by Xenllium in 2021 after the fact that the generator is 1/5 of a perfect fourth of about 497.4 cents, which can serve as a tuning for kwai.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 9765625/9680832

Mapping: [⟨1 2 0 -2], ⟨0 -5 28 58]]

mapping generators: ~2, ~625/588

Optimal tunings:

• WE: ~2 = 1199.8761 ¢, ~625/588 = 99.4732 ¢

error map: ⟨-0.124 +0.431 -1.065 +0.867]

• CWE: ~2 = 1200.0000 ¢, ~625/588 = 99.4817 ¢

error map: ⟨0.000 +0.637 -0.827 +1.110]

Optimal ET sequence: 12, …, 169, 181, 193

Badness (Sintel): 3.94

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4356, 5120/5103

Mapping: [⟨1 2 0 -2 -4], ⟨0 -5 28 58 90]]

Optimal tunings:

• WE: ~2 = 1199.9057 ¢, ~35/33 = 99.4638 ¢
• CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.4704 ¢

Optimal ET sequence: 12, …, 169e, 181, 193, 374

Badness (Sintel): 2.42

Quintakwoid

This 13-limit extension follows from tridecimal compass, using the identity 4375/4356 = (364/363)⋅(625/624).

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 625/624, 1375/1372, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6], ⟨0 -5 28 58 90 117]]

Optimal tunings:

• WE: ~2 = 1199.8443 ¢, ~35/33 = 99.4713 ¢
• CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.4825 ¢

Optimal ET sequence: 12f, 181f, 193

Badness (Sintel): 2.37

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 442/441, 595/594, 5120/5103

Mapping: [⟨1 2 0 -2 -4 -6 5], ⟨0 -5 28 58 90 117 -11]]

Optimal tunings:

• WE: ~2 = 1199.8384 ¢, ~18/17 = 99.4710 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.4832 ¢

Optimal ET sequence: 12f, 181f, 193

Badness (Sintel): 2.00

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 364/363, 375/374, 400/399, 442/441, 595/594, 1216/1215

Mapping: [⟨1 2 0 -2 -4 -6 5 4], ⟨0 -5 28 58 90 117 -11 3]]

Optimal tunings:

• WE: ~2 = 1199.8261 ¢, ~18/17 = 99.4704 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.4837 ¢

Optimal ET sequence: 12f, 181f, 193

Badness (Sintel): 1.71

Quinkwai

This 13-limit extension tempers out 352/351 and 847/845 like most aberschismic temperaments do, mapping 13/11 to the diatonic minor third.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 1375/1372, 4375/4356

Mapping: [⟨1 2 0 -2 -4 -5], ⟨0 -5 28 58 90 105]]

Optimal tunings:

• WE: ~2 = 1199.9637 ¢, ~35/33 = 99.4527 ¢
• CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.4553 ¢

Optimal ET sequence: 12f, …, 169e, 181

Badness (Sintel): 2.56

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 375/374, 595/594, 833/832, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5], ⟨0 -5 28 58 90 105 -11]]

Optimal tunings:

• WE: ~2 = 1199.9007 ¢, ~18/17 = 99.4502 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.4576 ¢

Optimal ET sequence: 12f, 169e, 181

Badness (Sintel): 2.22

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 375/374, 400/399, 495/494, 595/594, 1375/1372

Mapping: [⟨1 2 0 -2 -4 -5 5 4], ⟨0 -5 28 58 90 105 -11 3]]

Optimal tunings:

• WE: ~2 = 1199.8793 ¢, ~18/17 = 99.4492 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.4584 ¢

Optimal ET sequence: 12f, 169e, 181

Badness (Sintel): 1.90

Quinticosiennic

For the 5-limit version, see Schismic–commatic equivalence continuum #Quintaleap.

Quinticosiennic may be described as the 12 & 145 temperament with a ploidacot signature of omega-pentacot.

The 11-limit extension tempers out 441/440 and 896/891, making it a pele temperament. It should be noted that 145 & 157 and 157 & 169cd are viable alternative extensions that favor tunings closer to the 7-limit optimum.

Quinticosiennic was named by Xenllium in 2021, referring to 5 (quint-) × 29 (είκοσι εννέα) = 145, because 1/5 of 29edo fourth, i.e. 12\145, is a possible generator.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 395136/390625

Mapping: [⟨1 2 1 -1], ⟨0 -5 16 46]]

mapping generators: ~2, ~135/128

Optimal tunings:

• WE: ~2 = 1199.3522 ¢, ~135/128 = 99.2915 ¢

error map: ⟨-0.648 +0.292 +1.702 -0.769]

• CWE: ~2 = 1200.0000 ¢, ~135/128 = 99.3345 ¢

error map: ⟨0.000 +1.373 +3.038 +0.560]

Optimal ET sequence: 12, 121d, 133, 145, 157, 302c, 459bcc

Badness (Sintel): 4.00

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 78408/78125

Mapping: [⟨1 2 1 -1 -2], ⟨0 -5 16 46 66]]

Optimal tunings:

• WE: ~2 = 1199.3799 ¢, ~35/33 = 99.2665 ¢
• CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.3084 ¢

Optimal ET sequence: 12, 121de, 133, 145

Badness (Sintel): 2.67

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 364/363, 78408/78125

Mapping: [⟨1 2 1 -1 -2 -3], ⟨0 -5 16 46 66 81]]

Optimal tunings:

• WE: ~2 = 1199.4063 ¢, ~35/33 = 99.2575 ¢
• CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.2985 ¢

Optimal ET sequence: 12f, 121def, 133, 145

Badness (Sintel): 2.17

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 196/195, 256/255, 352/351, 364/363, 3757/3750

Mapping: [⟨1 2 1 -1 -2 -3 5], ⟨0 -5 16 46 66 81 -11]]

Optimal tunings:

• WE: ~2 = 1199.3895 ¢, ~18/17 = 99.2571 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.3017 ¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 1.89

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 196/195, 256/255, 352/351, 361/360, 364/363, 476/475

Mapping: [⟨1 2 1 -1 -2 -3 5 4], ⟨0 -5 16 46 66 81 -11 3]]

Optimal tunings:

• WE: ~2 = 1199.4875 ¢, ~18/17 = 99.2609 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.2987 ¢

Optimal ET sequence: 12f, 133, 145

Badness (Sintel): 1.73

Quanic

Quanic may be described as the 94 & 111 temperament. It splits the perfect fifth into five generators which in the 13-limit extension may be taken as ~13/12; its ploidacot is thus pentacot. 205edo may be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 5832000/5764801

Mapping: [⟨1 1 -4 0], ⟨0 5 54 24]]

mapping generators: ~2, ~160/147

Optimal tunings:

• WE: ~2 = 1199.6159 ¢, ~160/147 = 140.4483 ¢

error map: ⟨-0.384 -0.098 -0.570 +1.933]

• CWE: ~2 = 1200.0000 ¢, ~160/147 = 140.4862 ¢

error map: ⟨0.000 +0.476 -0.061 +2.842]

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 4.54

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1331/1323, 5120/5103

Mapping: [⟨1 1 -4 0 1], ⟨0 5 54 24 21]]

Optimal tunings:

• WE: ~2 = 1199.7834 ¢, ~88/81 = 140.4635 ¢
• CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.4850 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.94

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 729/728, 1331/1323

Mapping: [⟨1 1 -4 0 1 3], ⟨0 5 54 24 21 6]]

Optimal tunings:

• WE: ~2 = 1199.6639 ¢, ~13/12 = 140.4562 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4904 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.34

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 442/441, 540/539, 715/714, 847/845

Mapping: [⟨1 1 -4 0 1 3 -2], ⟨0 5 54 24 21 6 52]]

Optimal tunings:

• WE: ~2 = 1199.6699 ¢, ~13/12 = 140.4586 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4920 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.08

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 352/351, 400/399, 442/441, 456/455, 495/494, 715/714

Mapping: [⟨1 1 -4 0 1 3 -2 -5], ⟨0 5 54 24 21 6 52 79]]

Optimal tunings:

• WE: ~2 = 1199.6745 ¢, ~13/12 = 140.4574 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.4908 ¢

Optimal ET sequence: 94, 111, 205

Badness (Sintel): 1.05

Jorgensen

For the 5-limit version, see Miscellaneous 5-limit temperaments #Jorgensen.

Jorgensen tempers out the linus comma in addition to the aberschisma, and may be described as the 70 & 140 temperament, with a 70th-octave period. Its ploidacot is 70-ploid acot.

It is the natural 7-limit extension of the 5-limit temperament tempering out the 70-comma, named by Mike Battaglia in 2012 for historical interests.

Subgroup: 2.3.5.7

Comma list: 5120/5103, 578509309952/576650390625

Mapping: [⟨70 111 0 34], ⟨0 0 1 1]]

mapping generators: ~50421/50000, ~5

Optimal tunings:

• WE: ~50421/50000 = 17.1387 ¢, ~5/4 = 386.8071 ¢

error map: ⟨-0.288 +0.445 -0.084 +0.121]

• CWE: ~50421/50000 = 17.1429 ¢, ~5/4 = 386.6593 ¢

error map: ⟨0.000 +0.902 +0.346 +0.690]

Optimal ET sequence: 70, 140, 350, 490

Badness (Sintel): 5.40