Very low accuracy temperaments

Below are listed some very low accuracy temperaments. Temperaments with exceedingly low accuracy such as these tend to temper out very large "commas" such as 9/8, 32/27, or 15/14, equating wildly different interval sizes with semitone-level or even greater errors. As a result, all of them are on the very edge of what can sensibly be 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". 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

Wedgie: ⟨⟨ 0 0 3 0 5 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]]

Wedgie: ⟨⟨ 1 -3 3 -7 2 15 ]]

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 in the same way decimal is of semaphore.

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