Very low accuracy temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
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 (interval size measure)|semitone]]-level or even greater [[error]]s. 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 [[exotemperament]]s.
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s. 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 [[exotemperament]]s.


== Antitonic ==
== Antitonic ==
This temperament is characterized by [[9/8]] being tempering out, and has been termed a "troll temperament" by its namers. 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. Surprisingly, it has extensions all the way up to the 11-limit, as confirmed by the data from [http://x31eq.com/cgi-bin/rt.cgi?limit=11&ets=2_4p&tuning=po x31eq]. The original 5-limit is basically the 3-limit music of [[2edo]] with the addition of harmonic 5 represented by an independent generator.  
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
[[Subgroup]]: 2.3.5
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== Alteraugment ==
== Alteraugment ==
Alteraugment is like [[augmented]], but the generator provides 5/4 instead of 3/2. [[User:VectorGraphics|Vector Graphics]] suggests the name ''kinsborough'' for this temperament.
Alteraugment tempers out the [[32/27|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. [[User:VectorGraphics|Vector Graphics]] suggests the name ''kinsborough'' for this temperament.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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== Involution ==
== Involution ==
Involution was named by [[User:CompactStar|CompactStar]] in 2023.  
Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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{{See also| Bug family }}
{{See also| Bug family }}


This temperament was named by [[User:Jerdle|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 [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]].  
Codex was named by [[User:Jerdle|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 [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid beta-dicot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7

Revision as of 14:38, 10 August 2025

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

Comma list: 9/8

Mapping: [⟨2 3 0], ⟨0 0 1]]

mapping generators: ~3/2, ~5

Optimal tunings:

  • 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]

Optimal ET sequence: 2, 4

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 ¢)

Optimal ET sequence: 2, 4

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 ¢)

Optimal ET sequence: 2, 4

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 ¢)

Optimal ET sequence: 2, 4e

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 ¢)

Optimal ET sequence: 2d, 4

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 ¢)

Optimal ET sequence: 2d, 4

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

Comma list: 32/27

Mapping: [⟨3 5 0], ⟨0 0 1]]

mapping generators: ~4/3, ~5

Optimal tunings:

  • 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

Comma list: 10/9

Mapping: [⟨1 0 -1], ⟨0 1 2]]

Optimal tunings:

  • 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]

Optimal ET sequence: 2c, 3

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 ¢

Optimal ET sequence: 3, 7bc

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 ¢)

Optimal ET sequence: 4bcd, 6

Badness (Sintel): 1.43

Nessus

See also: Semaphoresmic clan

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 ¢

Optimal ET sequence: 5c, 6

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

Optimal tunings:

  • 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 ¢)

Optimal ET sequence: 3de, 6

Badness (Sintel): 0.846

Geryon

See also: Dicot family

Subgroup: 2.3.5.7

Comma list: 8/7, 25/21

Mapping: [⟨1 1 2 3], ⟨0 2 1 0]]

Optimal tunings:

  • 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

See also: Semaphoresmic clan

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

Optimal tunings:

  • 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

See also: Bug family

Subgroup: 2.3.5.7

Comma list: 15/14, 27/25

Mapping: [⟨1 0 0 -1], ⟨0 2 3 5]]

Optimal tunings:

  • 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

See also: Archytas clan and Mavila family

Subgroup: 2.3.5.7

Comma list: 15/14, 64/63

Mapping: [⟨1 0 7 6], ⟨0 1 -3 -2]]

Optimal tunings:

  • 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 ¢

Optimal ET sequence: 2, 5c, 7

Badness (Sintel): 0.887

Wallaby

See also: Trienstonic clan and Mavila family

Subgroup: 2.3.5.7

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

Optimal tunings:

  • 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

See also: Meantone family

Subgroup: 2.3.5.7

Comma list: 15/14, 81/80

Mapping: [⟨1 0 -4 -5], ⟨0 1 4 5]]

Optimal tunings:

  • 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

Comma list: 45/32

Mapping: [⟨1 0 5], ⟨0 1 -2]]

mapping generators: ~2, ~3

Optimal tunings:

  • 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

Comma list: 35/27

Mapping: [⟨1 0 3], ⟨0 1 -1]]

mapping generators: ~3, ~5

Optimal tunings:

  • 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

See also: Bug family

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

Comma list: 27/25, 54/49

Mapping: [⟨2 0 0 1], ⟨0 2 3 3]]

Optimal tunings:

  • 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

See also: Porcupine family

Subgroup: 2.3.5.7

Comma list: 21/20, 175/162

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -2]]

Optimal tunings:

  • 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

Main article: Bixby

Subgroup: 2.3.5

Comma list: 4/3

Mapping: [⟨1 2 0], ⟨0 0 1]]

Optimal tunings:

  • 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

Comma list: 5/4

Mapping: [⟨1 0 2], ⟨0 1 0]]

Optimal tunings:

  • 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]

Optimal ET sequence: 2c

Badness (Sintel): 0.474

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