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 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 x31eq. 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:

• CTE: ~3/2 = 600.000, ~5/4 = 386.314 (~6/5 = 213.686)

error map: ⟨0.000 -101.955 0.000]

• POTE: ~3/2 = 600.000, ~5/4 = 313.690 (~6/5 = 286.310)

error map: ⟨0.000 -101.955 -72.624]

Optimal ET sequence: 2, 4

Badness (Smith): 0.021647

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:

• CTE: ~3/2 = 600.000, ~5/4 = 379.210 (~6/5 = 220.890)
• POTE: ~3/2 = 600.000, ~7/6 = 301.997 (~6/5 = 298.003)

Optimal ET sequence: 2, 4

Badness (Smith): 0.019373

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:

• CTE: ~3/2 = 600.000, ~5/4 = 379.210 (~11/10 = 220.890)
• POTE: ~3/2 = 600.000, ~5/4 = 340.293 (~8/7 = 259.707)

Optimal ET sequence: 2, 4

Badness (Smith): 0.021810

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:

• CTE: ~3/2 = 600.000, ~5/4 = 415.533 (~12/11 = 184.467)
• POTE: ~3/2 = 600.000, ~7/6 = 322.080 (~6/5 = 277.920)

Optimal ET sequence: 2, 4e

Badness (Smith): 0.018993

Antaeus

Subgroup: 2.3.5.7

Comma list: 9/8, 35/32

Mapping: [⟨2 3 0 10], ⟨0 0 1 -1]]

Optimal tunings:

• CTE: ~3/2 = 600.000, ~5/4 = 323.296 (~7/6 = 276.704)
• POTE: ~3/2 = 600.000, ~5/4 = 315.962 (~7/6 = 284.038)

Optimal ET sequence: 2d, 4

Badness (Smith): 0.037537

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:

• CTE: ~3/2 = 600.000, ~5/4 = 323.296 (~7/6 = 276.704)
• POTE: ~3/2 = 600.000, ~5/4 = 318.173 (~7/6 = 281.827)

Optimal ET sequence: 2d, 4

Badness (Smith): 0.033786

Alteraugment

Alteraugment is like augmented, but the generator provides 5/4 instead of 3/2. 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:

• CTE: ~4/3 = 400.000, ~5/4 = 386.314 (~16/15 = 13.686)

error map: ⟨0.000 +98.045 0.000]

• POTE: ~4/3 = 400.000, ~5/4 = 459.935 (~10/9 = 59.935)

error map: ⟨0.000 +98.045 +73.621]

Optimal ET sequence: 3, 12bcc, 15bbcc

Badness (Smith): 0.078042

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:

• CTE: ~2 = 1200.000, ~3/2 = 761.310

error map: ⟨0.000 +74.015 -34.374]

• POTE: ~2 = 1200.000, ~3/2 = 775.970

error map: ⟨0.000 +65.763 -50.878]

Optimal ET sequence: 3, 11bc, 14bc, 17bb

Badness (Smith): 0.016712

Brutus

Subgroup: 2.3.5.7

Comma list: 10/9, 28/25

Mapping: [⟨1 0 -1 -4], ⟨0 1 2 4]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 813.116
• POTE: ~2 = 1200.000, ~3/2 = 848.221

Optimal ET sequence: 3, 7bc

Badness (Smith): 0.053389

Phlegyas

Subgroup: 2.3.5.7

Comma list: 10/9, 35/32

Mapping: [⟨1 0 -1 6], ⟨0 1 2 -2]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 747.225
• POTE: ~2 = 1200.000, ~3/2 = 743.135

Optimal ET sequence: 3, 5c, 8c

Badness (Smith): 0.051293

Charon

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

Mapping: [⟨2 0 -2 -1], ⟨0 1 2 2]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 768.427 (~15/14 = 168.427)
• POTE: ~7/5 = 600.000, ~3/2 = 782.604 (~15/14 = 182.604)

Optimal ET sequence: 4bcd, 6

Badness (Smith): 0.056404

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:

• CTE: ~2 = 1200.000, ~7/4 = 980.335
• POTE: ~2 = 1200.000, ~7/4 = 986.883

Optimal ET sequence: 5c, 6

Badness (Smith): 0.059070

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:

• CTE: ~5/4 = 400.000, ~7/4 = 968.826 (~28/25 = 168.826)

error map: ⟨0.000 +98.045 +13.686 0.000]

• POTE: ~5/4 = 400.000, ~7/4 = 1034.013 (~15/14 = 165.987)

error map: ⟨0.000 +98.045 +13.686 +65.187]

Optimal ET sequence: 3d, 6, 9bd

Badness (Smith): 0.028673

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:

• CTE: ~5/4 = 400.000, ~7/4 = 961.874 (~11/10 = 161.874)
• POTE: ~5/4 = 400.000, ~7/4 = 1033.153 (~12/11 = 166.847)

Optimal ET sequence: 3de, 6

Badness (Smith): 0.025592

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:

• CTE: ~2 = 1200.000, ~5/4 = 354.664

error map: ⟨0.000 +7.373 -31.649 +231.174]

• POTE: ~2 = 1200.000, ~5/4 = 385.440

error map: ⟨0.000 +68.926 -0.873 +231.174]

Optimal ET sequence: 1b, 2b, 3d

Badness (Smith): 0.051009

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:

• CTE: ~2 = 1200.000, ~7/4 = 938.796

error map: ⟨0.000 -24.362 -125.110 -30.029]

• POTE: ~2 = 1200.000, ~7/4 = 923.717

error map: ⟨0.000 -54.521 -110.031 -45.109]

Optimal ET sequence: 1, 3b, 4, 9c, 13bcc

Badness (Smith): 0.037207

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:

• CTE: ~2 = 1200.000, ~5/3 = 926.868

error map: ⟨0.000 -48.218 -5.708 +65.517]

• POTE: ~2 = 1200.000, ~5/3 = 921.640

error map: ⟨0.000 -58.675 -21.394 +39.374]

Optimal ET sequence: 1cdd, 3bcdd, 4, 9d

Badness (Smith): 0.043758

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:

• CTE: ~2 = 1200.000, ~3/2 = 684.722

error map: ⟨0.000 -17.233 -40.480 +61.730]

• POTE: ~2 = 1200.000, ~3/2 = 685.632

error map: ⟨0.000 -16.323 -43.209 +59.911]

Optimal ET sequence: 2, 5c, 7

Badness (Smith): 0.042712

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:

• CTE: ~2 = 1200.000, ~3/2 = 683.589
• POTE: ~2 = 1200.000, ~3/2 = 685.809

Optimal ET sequence: 2, 5c, 7

Badness (Smith): 0.026828

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:

• CTE: ~2 = 1200.000, ~3/2 = 691.757

error map: ⟨0.000 -8.515 -66.635 -88.505]

• POTE: ~2 = 1200.000, ~3/2 = 691.351

error map: ⟨0.000 -10.604 -60.368 -94.772]

Optimal ET sequence: 2d, 5c, 7d, 19ccdd

Badness (Smith): 0.058468

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:

• CTE: ~2 = 1200.000, ~3/2 = 685.837

error map: ⟨0.000 -16.118 -42.965 +60.359]

• POTE: ~2 = 1200.000, ~3/2 = 682.895

error map: ⟨0.000 -19.060 -54.734 +45.649]

Optimal ET sequence: 2cd, 5d, 7

Badness (Smith): 0.045275

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:

• CTE: ~2 = 1200.000, ~3/2 = 687.743
• POTE: ~2 = 1200.000, ~3/2 = 685.234

Optimal ET sequence: 2cde, 5de, 7

Badness (Smith): 0.032521

Involution

Involution 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:

• CTE: ~2 = 1200.000, ~3/2 = 509.892

error map: ⟨0.000 -192.063 -206.097]

• CWE: ~2 = 1200.000, ~3/2 = 520.625

error map: ⟨0.000 -181.330 -227.565]

Optimal ET sequence: 2c, 5bc, 7bbcc

Badness (Smith): 0.133021

7-limit

Subgroup: 2.3.5.7

Comma list: 8/7, 45/28

Mapping: [⟨1 0 5 3], ⟨0 1 -2 0]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~3/2 = 509.892
• CWE: ~2 = 1200.000, ~3/2 = 515.099

Optimal ET sequence: none

Badness (Smith): 0.118764

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:

• CTE: ~3 = 1901.955, ~5/3 = 701.863

error map: ⟨0.000 -182.496 -266.779]

• CWE: ~3 = 1901.995, ~5/3 = 691.468

error map: ⟨0.000 -192.891 -256.384]

Optimal ET sequence: b1, b2d, b3, b8cdd

Badness:

• Smith: 0.00406
• Dirichlet: 0.751

Codex

See also: Bug family

This temperament 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.

Subgroup: 2.3.5.7

Comma list: 27/25, 54/49

Mapping: [⟨2 0 0 1], ⟨0 2 3 3]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~5/3 = 935.289

error map: ⟨0.000 -31.377 +19.554 +37.041]

• 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 (Smith): 0.077065

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:

• CTE: ~2 = 1200.000, ~10/9 = 161.341

error map: ⟨0.000 +14.023 +6.982 -91.507]

• POTE: ~2 = 1200.000, ~10/9 = 169.112

error map: ⟨0.000 -9.291 -31.873 -107.050]

Optimal ET sequence: 1c, …, 6bcd, 7d

Badness (Smith): 0.059866

Bixby

Main article: Bixby

See also: Circular temperament

Subgroup: 2.3.5

Comma list: 4/3

Mapping: [[1 2 2][0 0 1]]

Optimal tunings:

• CWE: ~2 = 1200.000, ~5/4 = 629.521

Archon

Subgroup: 2.3.5

Comma list: 5/4

Mapping: [[1 1 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)