Hemimage temperaments

This is a collection of temperaments tempering out the hemimage comma, [5 -7 -1 3⟩ = 10976/10935. These include commatic, chromat, degrees, subfourth, bisupermajor and cotoneum, considered below, as well as the following discussed elsewhere:

• quasisuper, {64/63, 2430/2401} → Archytas clan #Quasisuper
• liese, {81/80, 686/675} → Meantone family #Liese
• unicorn, {126/125, 10976/10935} → Unicorn family #Septimal unicorn
• magic, {225/224, 245/243} → Magic family #Magic
• guiron, {1029/1024, 10976/10935} → Gamelismic clan #Guiron
• echidna, {1728/1715, 2048/2025} → Diaschismic family #Echidna
• hemififths, {2401/2400, 5120/5103} → Breedsmic temperaments #Hemififths
• dodecacot, {3125/3087, 10976/10935} → Tetracot family #Dodecacot
• parakleismic, {3136/3125, 4375/4374} → Ragismic microtemperaments #Parakleismic
• pluto, {4000/3969, 10976/10935} → Mirkwai clan #Pluto
• hendecatonic, {6144/6125, 10976/10935} → Porwell temperaments #Hendecatonic
• marfifths, {10976/10935, 15625/15552} → Kleismic family #Marfifths
• yarman, {10976/10935, 244140625/243045684} → Turkish maqam music temperaments #Yarman

Commatic

The commatic temperament has a period of half octave and a generator of 20.4 cents. It is so named because the generator is a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 50421/50000

Mapping: [⟨2 3 4 5], ⟨0 5 19 18]]

Wedgie: ⟨⟨ 10 38 36 37 29 -23 ]]

POTE generator: ~81/80 = 20.377

Badness: 0.084317

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: [⟨2 3 4 5 6], ⟨0 5 19 18 27]]

POTE generator: ~81/80 = 20.390

Vals: Template:Val list

Badness: 0.030461

Chromat

The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an amity extension with third-octave period.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 235298/234375

Mapping: [⟨3 4 5 6], ⟨0 5 13 16]]

Wedgie: ⟨⟨ 15 39 48 27 34 2 ]]

POTE generator: ~28/27 = 60.528

Badness: 0.057499

Degrees

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 390625/388962

Mapping: [⟨20 0 -17 -39], ⟨0 1 2 3]]

Wedgie: ⟨⟨ 20 40 60 17 39 27 ]]

POTE generator: ~3/2 = 703.015

Badness: 0.106471

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: [⟨20 0 -17 -39 -26], ⟨0 1 2 3 3]]

POTE generator: ~3/2 = 703.231

Vals: Template:Val list

Badness: 0.046770

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

Mapping: [⟨20 0 -17 -39 -26 74], ⟨0 1 2 3 3 0]]

POTE generator: ~3/2 = 703.080

Vals: Template:Val list

Badness: 0.032718

Subfourth

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65536/64827

Mapping: [⟨1 0 17 4], ⟨0 4 -37 -3]]

Wedgie: ⟨⟨ 4 -37 -3 -68 -16 97 ]]

POTE generator: ~21/16 = 475.991

Badness: 0.140722

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 12005/11979

Mapping: [⟨1 0 17 4 11], ⟨0 4 -37 -3 -19]]

POTE generator: ~21/16 = 475.995

Vals: Template:Val list

Badness: 0.045323

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: [⟨1 0 17 4 11 16], ⟨0 4 -37 -3 -19 -31]]

POTE generator: ~21/16 = 475.996

Vals: Template:Val list

Badness: 0.023800

Bisupermajor

See also: Very high accuracy temperaments § Kwazy

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65625/65536

Mapping: [⟨2 1 6 1], ⟨0 8 -5 17]]

Wedgie: ⟨⟨ 16 -10 34 -53 9 107 ]]

POTE generator: ~192/175 = 162.8061

Badness: 0.065492

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 9801/9800

Mapping: [⟨2 1 6 1 8], ⟨0 8 -5 17 -4]]

POTE generators: ~11/10 = 162.7733

Vals: Template:Val list

Badness: 0.032080

Cotoneum

Main article: Cotoneum

The cotoneum temperament (41&217, named after the Latin for "quince") tempers out the quince comma, 823543/819200 and the garischisma, 33554432/33480783. This temperament is supported by 41, 176, 217, and 258 EDOs, and can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 823543/819200

Mapping: [⟨1 2 -18 -3], ⟨0 -1 49 14]]

Wedgie: ⟨⟨ 1 -49 -14 -80 -25 105 ]]

POTE generator: ~3/2 = 702.317

Minimax tuning:

• 7-odd-limit: ~3/2 = [3/5 1/50 -1/50⟩

[[1 0 0 0⟩, [8/5 1/50 -1/50 0⟩, [8/5 -49/50 49/50 0⟩, [13/5 -7/25 7/25 0⟩]

Eigenmonzos (unchanged intervals): 2, 6/5

• 9-odd-limit: ~3/2 = [29/51 2/51 -1/51⟩

[[1 0 0 0⟩, [80/51 2/51 -1/51 0⟩, [160/51 -98/51 49/51 0⟩, [155/51 -28/51 14/51 0⟩]

Eigenmonzos (unchanged intervals): 2, 10/9

Tuning ranges:

• 7-odd-limit diamond monotone: ~3/2 = [701.5385, 702.8571] (38\65 to 41\70)
• 9-odd-limit diamond monotone: ~3/2 = [701.8868, 702.8571] (31\53 to 41\70)
• Diamond tradeoff range: ~3/2 = [701.9550, 702.3575]
• Diamond monotone and tradeoff: ~3/2 = [701.9550, 702.3575]

Badness: 0.105632

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 10976/10935, 16384/16335

Mapping: [⟨1 2 -18 -3 13], ⟨0 -1 49 14 -23]]

POTE generator: ~3/2 = 702.303

Minimax tuning:

• 11-odd-limit: ~3/2 = [41/72 0 -1/72 0 1/72⟩

Eigenmonzos (unchanged intervals): 2, 11/10

Tuning ranges:

• Diamond monotone range: ~3/2 = [702.1277, 702.4390] (55\94 to 24\41)
• Diamond tradeoff range: ~3/2 = [701.9550, 702.3575]
• Diamond monotone and tradeoff: ~3/2 = [702.1277, 702.3575]

Vals: Template:Val list

Badness: 0.050966

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3584/3575, 10976/10935

Mapping: [⟨1 2 -18 -3 13 29], ⟨0 -1 49 14 -23 -61]]

POTE generator: ~3/2 = 702.306

Minimax tuning:

• 13-odd-limit: ~3/2 = [41/72 0 -1/72 0 1/72⟩

Eigenmonzos (unchanged intervals): 2, 11/10

• 15-odd-limit: ~3/2 = [42/71 -1/71 -1/71 0 1/71⟩

Eigenmonzos (unchanged intervals): 2, 15/11

Tuning ranges:

• Diamond monotone range: ~3/2 = [702.2222, 702.4390] (79\135 to 24\41)
• 13-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3575]
• 15-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3693]
• 13-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2222, 702.3575]
• 15-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2222, 702.3693]

Vals: Template:Val list

Badness: 0.036951

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262

Mapping: [⟨1 2 -18 -3 13 29 41], ⟨0 -1 49 14 -23 -61 -89]]

POTE generator: ~3/2 = 702.307

Minimax tuning:

• 17-odd-limit: ~3/2 = [42/71 -1/71 -1/71 0 1/71⟩

Eigenmonzos (unchanged intervals): 2, 15/11

Tuning ranges:

• Diamond monotone range: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
• Diamond tradeoff range: ~3/2 = [701.9550, 702.3693]
• Diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3693]

Vals: Template:Val list

Badness: 0.029495

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728

Mapping: [⟨1 2 -18 -3 13 29 41 -14], ⟨0 -1 49 14 -23 -61 -89 44]]

POTE generator: ~3/2 = 702.308

Minimax tuning:

• 19- and 21-odd-limit: ~3/2 = [42/71 -1/71 -1/71 0 1/71⟩

Eigenmonzos (unchanged intervals): 2, 15/11

Tuning ranges:

• Diamond monotone range: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
• Diamond tradeoff range: ~3/2 = [701.9550, 702.3771]
• Diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3771]

Vals: Template:Val list

Badness: 0.021811

Squarschmidt

A generator for the squarschimidt temperament is the fourth root of 5/2, (5/2)^1/4, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.

Subgroup: 2.3.5

Comma: [61 4 -29⟩

Mapping: [⟨1 -8 1], ⟨0 29 4]]

POTE generator: ~98304/78125 = 396.6208

Badness: 0.218314

7-limit

Subgroup: 2.3.5.7

Comma list: 10976/10935, 29360128/29296875

Mapping: [⟨1 -8 1 -20], ⟨0 29 4 69]]

Wedgie: ⟨⟨ 29 4 69 -61 28 149 ]]

POTE generator: ~1125/896 = 396.643

Badness: 0.132821

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 10976/10935

Mapping: [⟨1 -8 1 -20 -21], ⟨0 29 4 69 74]]

POTE generator: ~44/35 = 396.644

Vals: Template:Val list

Badness: 0.038186