Very low accuracy 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.
Below are listed some very low accuracy temperaments. Temperaments with exceedingly low accuracy such as these tend to temper out very large intervals such as 9/8, 10/9, 32/27, or 15/14, equating wildly different interval sizes with semitone-level or even greater errors. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are exotemperaments.
Antitonic
This temperament is characterized by 9/8 being tempered out and has been termed a "troll temperament" by its namers. Its ploidacot is diploid acot. It is named on account of 4/3 and 3/2 both being represented by the 600-cent half octave, which, in terms of diatonic function, serves as an antitonic. The 7-limit extension tempers out 15/14 and 21/20, equating 5/4 with 7/6 and 6/5 with 8/7. The 11-limit extension tempers out 12/11 and 33/32. The original 5-limit is basically the 3-limit music of 2edo with the addition of harmonic 5 represented by an independent generator.
Subgroup: 2.3.5
Mapping: [⟨2 3 0], ⟨0 0 1]]
- mapping generators: ~3/2, ~5
- WE: ~3/2 = 615.125 ¢, ~5/4 = 321.597 ¢ (~6/5 = 293.528 ¢)
- error map: ⟨+30.250 -56.581 -4.217]
- CWE: ~3/2 = 600.000 ¢, ~5/4 = 336.527 ¢ (~6/5 = 263.473 ¢)
- error map: ⟨0.000 -101.955 -49.787]
Badness (Sintel): 0.508
- Music
- Antitonic(8) (2024) – short composition by Wensik in POTE-tuned 5-limit antitonic using an 8-note ternary scale.
7-limit
Subgroup: 2.3.5.7
Comma list: 9/8, 15/14
Mapping: [⟨2 3 0 1], ⟨0 0 1 1]]
Optimal tunings:
- WE: ~3/2 = 614.759 ¢, ~7/6 = 309.426 ¢ (~6/5 = 305.334 ¢)
- CWE: ~3/2 = 600.000 ¢, ~7/6 = 326.047 ¢ (~6/5 = 273.953 ¢)
Badness (Sintel): 0.490
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9/8, 12/11, 15/14
Mapping: [⟨2 3 0 1 7], ⟨0 0 1 1 0]]
Optimal tunings:
- WE: ~3/2 = 606.293 ¢, ~5/4 = 343.862 ¢ (~8/7 = 262.431 ¢)
- CWE: ~3/2 = 600.000 ¢, ~5/4 = 348.102 ¢ (~8/7 = 251.898 ¢)
Badness (Sintel): 0.721
Antietam
Subgroup: 2.3.5.7.11
Comma list: 9/8, 11/10, 15/14
Mapping: [⟨2 3 0 1 2], ⟨0 0 1 1 1]]
Optimal tunings:
- WE: ~3/2 = 616.135 ¢, ~5/4 = 330.741 ¢ (~6/5 = 285.393 ¢)
- CWE: ~3/2 = 600.000 ¢, ~5/4 = 349.843 ¢ (~8/7 = 250.157 ¢)
Badness (Sintel): 0.628
Antaeus
Subgroup: 2.3.5.7
Comma list: 9/8, 35/32
Mapping: [⟨2 3 0 10], ⟨0 0 1 -1]]
Optimal tunings:
- WE: ~3/2 = 614.854 ¢, ~5/4 = 323.784 ¢ (~6/5 = 291.070 ¢)
- CWE: ~3/2 = 600.000 ¢, ~5/4 = 317.349 ¢ (~6/5 = 282.651 ¢)
Badness (Sintel): 0.950
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9/8, 12/11, 35/32
Mapping: [⟨2 3 0 10 7], ⟨0 0 1 -1 0]]
Optimal tunings:
- WE: ~3/2 = 609.311 ¢, ~5/4 = 323.110 ¢ (~6/5 = 286.200 ¢)
- CWE: ~3/2 = 600.000 ¢, ~5/4 = 318.904 ¢ (~6/5 = 281.096 ¢)
Badness (Sintel): 1.12
Alteraugment
Alteraugment tempers out the Pythagorean minor third (32/27). It is like augmented, but the period represents 4/3 instead of 5/4, and the generator in turn provides 5/4 instead of 3/2. Its ploidacot is triploid acot. Vector Graphics suggests the name kinsborough for this temperament.
Subgroup: 2.3.5
Mapping: [⟨3 5 0], ⟨0 0 1]]
- mapping generators: ~4/3, ~5
- WE: ~4/3 = 389.212 ¢, ~5/4 = 447.530 ¢ (~10/9 = 58.318 ¢)
- error map: ⟨-32.364 +44.105 -3.512]
- CWE: ~4/3 = 400.000 ¢, ~5/4 = 434.191 ¢ (~15/16 = 34.191 ¢)
- error map: ⟨0.000 +98.045 +47.878]
Optimal ET sequence: 3, 12bcc, 15bbcc
Badness (Sintel): 1.83
Yo (2c & 3)
This temperament family is characterized by the yo 2nd 10/9 being tempering out.
Subgroup: 2.3.5
Mapping: [⟨1 0 -1], ⟨0 1 2]]
- WE: ~2 = 1187.236 ¢, ~3/2 = 767.716 ¢
- error map: ⟨-12.764 +52.997 -63.645]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 767.718 ¢
- error map: ⟨0.000 +65.763 -50.877]
Badness (Sintel): 0.392
Brutus
Subgroup: 2.3.5.7
Comma list: 10/9, 28/25
Mapping: [⟨1 0 -1 -4], ⟨0 1 2 4]]
Optimal tunings:
- WE: ~2 = 1158.982 ¢, ~3/2 = 819.228 ¢
- CWE: ~2 = 1200.000 ¢, ~3/2 = 831.346 ¢
Badness (Sintel): 1.35
Phlegyas
Subgroup: 2.3.5.7
Comma list: 10/9, 35/32
Mapping: [⟨1 0 -1 6], ⟨0 1 2 -2]]
Optimal tunings:
- WE: ~2 = 1206.510 ¢, ~3/2 = 747.166 ¢
- CWE: ~2 = 1200.000 ¢, ~3/2 = 743.797 ¢
Optimal ET sequence: 3, 5c, 8c
Badness (Sintel): 1.30
Charon
Subgroup: 2.3.5.7
Comma list: 10/9, 49/45
Mapping: [⟨2 0 -2 -1], ⟨0 1 2 2]]
Optimal tunings:
- WE: ~7/5 = 593.832 ¢, ~3/2 = 774.559 ¢ (~15/14 = 180.726 ¢)
- CWE: ~7/5 = 600.000 ¢, ~3/2 = 774.466 ¢ (~15/14 = 174.466 ¢)
Badness (Sintel): 1.43
Nessus
Subgroup: 2.3.5.7
Comma list: 10/9, 49/48
Mapping: [⟨1 0 -1 2], ⟨0 2 4 1]]
- mapping generators: ~2, ~7/4
Optimal tunings:
- WE: ~2 = 1189.201 ¢, ~3/2 = 978.002 ¢
- CWE: ~2 = 1200.000 ¢, ~3/2 = 983.918 ¢
Badness (Sintel): 1.49
Ternary
Ternary is identical to 3et in the 5-limit, but has an independent generator for prime 7.
Subgroup: 2.3.5.7
Comma list: 10/9, 16/15
Mapping: [⟨3 5 7 0], ⟨0 0 0 1]]
- mapping generators: ~5/4, ~7
- WE: ~5/4 = 391.796 ¢, ~7/4 = 1012.806 ¢ (~15/14 = 162.582 ¢)
- error map: ⟨-24.612 +57.026 -43.741 -5.243]
- CWE: ~5/4 = 400.000 ¢, ~7/4 = 1016.378 ¢ (~8/7 = 183.622 ¢)
- error map: ⟨0.000 +98.045 +13.686 +47.552]
Optimal ET sequence: 3d, 6, 9bd
Badness (Sintel): 0.726
11-limit
Subgroup: 2.3.5.7.11
Comma list: 10/9, 16/15, 22/21
Mapping: [⟨3 5 7 0 2], ⟨0 0 0 1 1]]
Optimal tunings:
- WE: ~5/4 = 391.788 ¢, ~7/4 = 1011.942 ¢ (~12/11 = 163.422 ¢)
- CWE: ~5/4 = 400.000 ¢, ~7/4 = 1013.973 ¢ (~12/11 = 186.027 ¢)
Badness (Sintel): 0.846
Geryon
Subgroup: 2.3.5.7
Comma list: 8/7, 25/21
Mapping: [⟨1 1 2 3], ⟨0 2 1 0]]
- WE: ~2 = 1164.885 ¢, ~5/4 = 374.162 ¢
- error map: ⟨-35.115 +11.253 -82.382 +125.830]
- CWE: ~2 = 1200.000 ¢, ~5/4 = 375.277 ¢
- error map: ⟨0.000 +48.600 -11.036 +231.174]
Optimal ET sequence: 1b, 2b, 3d
Badness (Sintel): 1.29
Malacoda
Subgroup: 2.3.5.7
Comma list: 15/14, 35/32
Mapping: [⟨1 0 3 2], ⟨0 2 -1 1]]
- mapping generators: ~2, ~7/4
- WE: ~2 = 1223.542 ¢, ~7/4 = 941.838 ¢
- error map: ⟨+23.542 -18.278 -57.528 +20.096]
- CWE: ~2 = 1200.000 ¢, ~7/4 = 927.096 ¢
- error map: ⟨0.000 -47.763 -113.410 -41.730]
Optimal ET sequence: 1, 3b, 4, 9c, 13bcc
Badness (Sintel): 0.942
Ugolino
Subgroup: 2.3.5.7
Comma list: 15/14, 27/25
Mapping: [⟨1 0 0 -1], ⟨0 2 3 5]]
- WE: ~2 = 1206.628 ¢, ~7/4 = 926.730 ¢
- error map: ⟨+6.628 -48.494 -6.122 +58.198]
- CWE: ~2 = 1200.000 ¢, ~7/4 = 923.776 ¢
- error map: ⟨0.000 -54.403 -14.986 +50.054]
Optimal ET sequence: 1cdd, 3bcdd, 4, 9d
Badness (Sintel): 1.11
Medusa
Subgroup: 2.3.5.7
Comma list: 15/14, 64/63
Mapping: [⟨1 0 7 6], ⟨0 1 -3 -2]]
- WE: ~2 = 1200.960 ¢, ~3/2 = 686.181 ¢
- error map: ⟨+0.960 -14.814 -41.014 +62.655]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 685.511 ¢
- error map: ⟨0.000 -16.443 -42.849 +60.150]
Optimal ET sequence: 2, 5c, 7
Badness (Sintel): 1.08
11-limit
Subgroup: 2.3.5.7.11
Comma list: 15/14, 22/21, 33/32
Mapping: [⟨1 0 7 6 5], ⟨0 1 -3 -2 -1]]
Optimal tunings:
- WE: ~2 = 1202.757 ¢, ~3/2 = 687.384 ¢
- CWE: ~2 = 1200.000 ¢, ~3/2 = 685.462 ¢
Badness (Sintel): 0.887
Wallaby
Subgroup: 2.3.5.7
Comma list: 28/27, 35/32
Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]
- WE: ~2 = 1216.024 ¢, ~3/2 = 700.583 ¢
- error map: ⟨+16.024 +14.652 -23.967 -51.053]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 691.757 ¢
- error map: ⟨0.000 -10.198 -61.585 -93.555]
Optimal ET sequence: 2d, 5c, 7d, 19ccdd
Badness (Sintel): 1.48
Plutus
Subgroup: 2.3.5.7
Comma list: 15/14, 81/80
Mapping: [⟨1 0 -4 -5], ⟨0 1 4 5]]
- WE: ~2 = 1203.936 ¢, ~3/2 = 685.135 ¢
- error map: ⟨+3.936 -12.884 -45.774 +56.849]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 683.935 ¢
- error map: ⟨0.000 -18.020 -50.573 +50.850]
Optimal ET sequence: 2cd, 5d, 7
Badness (Sintel): 1.14
11-limit
Subgroup: 2.3.5.7.11
Comma list: 15/14, 22/21, 81/80
Mapping: [⟨1 0 -4 -5 -6], ⟨0 1 4 5 6]]
Optimal tunings:
- WE: ~2 = 1203.293 ¢, ~3/2 = 687.114 ¢
- CWE: ~2 = 1200.000 ¢, ~3/2 = 686.078 ¢
Optimal ET sequence: 2cde, 5de, 7
Badness (Sintel): 1.08
Involution
Involution tempers out the ptolemaic augmented fourth (45/32). Its ploidacot is monocot. It was named by CompactStar in 2023.
Subgroup: 2.3.5
Mapping: [⟨1 0 5], ⟨0 1 -2]]
- mapping generators: ~2, ~3
- WE: ~2 = 1265.406 ¢, ~3/2 = 552.346 ¢
- error map: ⟨+65.405 -84.203 -94.789]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 520.626 ¢
- error map: ⟨0.000 -181.329 -227.565]
Optimal ET sequence: 2c, 5bc, 7bbcc
Badness (Sintel): 3.12
7-limit
Subgroup: 2.3.5.7
Comma list: 8/7, 45/28
Mapping: [⟨1 0 5 3], ⟨0 1 -2 0]]
Optimal tunings:
- WE: ~2 = 1205.230 ¢, ~3/2 = 517.557 ¢
- CWE: ~2 = 1200.000 ¢, ~3/2 = 515.099 ¢
Optimal ET sequence: none
Badness (Sintel): 3.01
Devolution
Devolution was named by Akselai in 2024.
Subgroup: 3.5.7
Mapping: [⟨1 0 3], ⟨0 1 -1]]
- mapping generators: ~3, ~5
- WE: ~3 = 1988.549 ¢, ~3/2 = 719.679 ¢
- error map: ⟨+86.594 -78.085 -111.407]
- CWE: ~3 = 1901.995 ¢, ~5/3 = 691.468 ¢
- error map: ⟨0.000 -192.891 -256.384]
Optimal ET sequence: b1, b2d, b3, b8cdd
Badness (Sintel): 0.751
Codex
Codex was named by Jerdle as an abbreviation of "co-decimal", as it resembles decimal in many ways, but exchanges the roles of 5's and 7's. While decimal equates 8/7 and 7/6, as well as 6/5 and 5/4, this equates 10/9 and 6/5, as well as 7/6 and 9/7. It is an extension of bug and mujannabic in the same way decimal is of semaphore and dicot. Its ploidacot is diploid beta-dicot.
Subgroup: 2.3.5.7
Mapping: [⟨2 0 0 1], ⟨0 2 3 3]]
- WE: ~7/5 = 598.589 ¢, ~5/3 = 934.978 ¢
- error map: ⟨-2.821 -31.998 +18.621 +34.699]
- CWE: ~7/5 = 600.000 ¢, ~5/3 = 936.030 ¢
- error map: ⟨0.000 -29.895 +21.776 +39.264]
Optimal ET sequence: 4, 10cd, 14d
Badness (Sintel): 1.95
Oxygen
Subgroup: 2.3.5.7
Comma list: 21/20, 175/162
Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -2]]
- WE: ~2 = 1213.695 ¢, ~10/9 = 171.042 ¢
- error map: ⟨+13.695 +12.309 -0.438 -69.825]
- CWE: ~2 = 1200.000 ¢, ~10/9 = 166.042 ¢
- error map: ⟨0.000 -0.083 -16.526 -100.911]
Optimal ET sequence: 1c, …, 6bcd, 7d
Badness (Sintel): 1.52
Bixby
Subgroup: 2.3.5
Mapping: [⟨1 2 0], ⟨0 0 1]]
- WE: ~2 = 1020.058 ¢, ~5/4 = 674.394 ¢
- error map: ⟨-179.942 +138.161 -71.803]
- CWE: ~2 = 1200.000 ¢, ~5/4 = 629.521 ¢
- error map: ⟨0.000 +498.045 +243.208]
Optimal ET sequence: 1c, 2b, 3bbcc
Badness (Sintel): 0.424
Archon
Subgroup: 2.3.5
Mapping: [⟨1 0 2], ⟨0 1 0]]
- WE: ~2 = 1268.274 ¢, ~3/2 = 612.921 ¢
- error map: ⟨+68.274 -20.760 -249.765]
- CWE: ~2 = 1200.000 ¢, ~3/2 = 614.055 ¢
- error map: ⟨0.000 -87.900 -386.314]
Badness (Sintel): 0.474