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. All temperaments listed temper out an interval of at least size 15/14. As a result, every temperament here is an exotemperament.

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:

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

Optimal ET sequence: 2, 4

Badness: 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:

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

Optimal ET sequence: 2, 4

Badness: 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:

• POTE: ~3/2 = 600.000, ~5/4 = 340.293 (~6/5 = 259.707)

Optimal ET sequence: 2, 4

Badness: 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:

• POTE: ~3/2 = 600.000, ~5/4 = 322.080 (~6/5 = 277.920)

Optimal ET sequence: 2, 4e

Badness: 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:

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

Optimal ET sequence: 2d, 4

Badness: 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:

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

Optimal ET sequence: 2d, 4

Badness: 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:

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

Optimal ET sequence: 3, 12bcc, 15bbcc, 18bbcc

Badness: 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:

• POTE: ~2 = 1200.000, ~3/2 = 775.970

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

Badness: 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:

• POTE: ~2 = 1200.000, ~3/2 = 848.221

Optimal ET sequence: 3, 7bc, 10bc

Badness: 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:

• POTE: ~2 = 1200.000, ~3/2 = 743.135

Optimal ET sequence: 3, 5c, 8c, 13bcc

Badness: 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:

• POTE: ~2 = 1200.000, ~3/2 = 782.604 (~15/14 = 182.604)

Optimal ET sequence: 6, 14bc, 20bb

Badness: 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]]

Optimal tunings:

• POTE: ~2 = 1200.000, ~7/6 = 213.117

Optimal ET sequence: 5c, 6, 11bc, 17bb

Badness: 0.059070

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:

• POTE: ~2 = 1200.000, ~5/4 = 385.440

Optimal ET sequence: 3d, ⟨16 26 37 48], ⟨19 31 44 57], ⟨22 36 51 66]

Badness: 0.051009

Malacoda

Subgroup: 2.3.5.7

Comma list: 15/14, 35/32

Mapping: [⟨1 0 3 2], ⟨0 2 -1 1]]

Optimal tunings:

• POTE: ~2 = 1200.000, ~7/6 = 276.283

Optimal ET sequence: 1, 4, 9c, 13bcc

Badness: 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:

• POTE: ~2 = 1200.000, ~6/5 = 278.360

Optimal ET sequence: 4, 9d, 13bd

Badness: 0.043758

Uncle

See also: Trienstonic clan § Uncle

Subgroup: 2.3.5

Comma list: 4096/3645

Mapping: [⟨1 0 12], ⟨0 1 -6]]

mapping generators: ~2, ~3

Optimal tunings:

• CWE: ~2 = 1200.000, ~3/2 = 731.732

Optimal ET sequence: 5, 13, 18, 23bc

Badness (Sintel): 6.332

Medusa

See also: Pelogic family

Subgroup: 2.3.5.7

Comma list: 15/14, 64/63

Mapping: [⟨1 0 7 6], ⟨0 1 -3 -2]]

Optimal tunings:

• POTE: ~2 = 1200.000, ~3/2 = 685.632

Optimal ET sequence: 2, 5c, 7

Badness: 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:

• POTE: ~2 = 1200.000, ~3/2 = 685.809

Optimal ET sequence: 2, 5c, 7

Badness: 0.026828

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:

• POTE: ~2 = 1200.000, ~3/2 = 682.895

Optimal ET sequence: 2cd, 5d, 7

Badness: 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:

• POTE: ~2 = 1200.000, ~3/2 = 685.234

Optimal ET sequence: 2cde, 5de, 7

Badness: 0.032521

Involution

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

Optimal ET sequence: 2c, 5bc, 7bbcc

Badness: 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

Optimal ET sequence: none

Badness: 0.118764

Devolution

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

Optimal ET sequence: b3, b8cdd

Badness (Sintel): 0.751

Codex

See also: Bug family

This temperament is named as an abbreviation of "co-decimal", as it resembles decimal in many ways, but exchanges the roles of 5s and 7s. 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

Optimal ET sequence: 10cd, 14d

Badness: 0.077065