Hemimage temperaments: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 29: Line 29:
[[POTE generator]]: ~28/27 = 60.528
[[POTE generator]]: ~28/27 = 60.528


{{Val list|legend=1| 39d, 60, 99, 258, 357, 456 }}
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


[[Badness]]: 0.057499
[[Badness]]: 0.057499
Line 42: Line 42:
POTE generator: ~28/27 = 60.430
POTE generator: ~28/27 = 60.430


Optimal GPV sequence: {{Val list| 60e, 99e, 159, 258, 417d }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}


Badness: 0.050379
Badness: 0.050379
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POTE generator: ~28/27 = 60.428
POTE generator: ~28/27 = 60.428


Optimal GPV sequence: {{Val list| 99ef, 159, 258, 417d }}
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}


Badness: 0.046006
Badness: 0.046006
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POTE generator: ~28/27 = 60.438
POTE generator: ~28/27 = 60.438


Optimal GPV sequence: {{Val list| 99ef, 159, 258, 417dg }}
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}


Badness: 0.031678
Badness: 0.031678
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POTE generator: ~28/27 = 60.378
POTE generator: ~28/27 = 60.378


Optimal GPV sequence: {{Val list| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Badness: 0.043844
Badness: 0.043844
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POTE generator: ~28/27 = 60.377
POTE generator: ~28/27 = 60.377


Optimal GPV sequence: {{Val list| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Badness: 0.030218
Badness: 0.030218
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POTE generator: ~27/26 = 60.456
POTE generator: ~27/26 = 60.456


Optimal GPV sequence: {{Val list| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


Badness: 0.049857
Badness: 0.049857
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POTE generator: ~27/26 = 60.459
POTE generator: ~27/26 = 60.459


Optimal GPV sequence: {{Val list| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


Badness: 0.031043
Badness: 0.031043
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[[POTE generator]]: ~192/175 = 162.806
[[POTE generator]]: ~192/175 = 162.806


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


[[Badness]]: 0.065492
[[Badness]]: 0.065492
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POTE generators: ~11/10 = 162.773
POTE generators: ~11/10 = 162.773


Optimal GPV sequence: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}


Badness: 0.032080
Badness: 0.032080
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[[POTE generator]]: ~81/80 = 20.377
[[POTE generator]]: ~81/80 = 20.377


{{Val list|legend=1| 58, 118, 294, 412d, 530d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}


[[Badness]]: 0.084317
[[Badness]]: 0.084317
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POTE generator: ~81/80 = 20.390
POTE generator: ~81/80 = 20.390


Optimal GPV sequence: {{Val list| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


Badness: 0.030461
Badness: 0.030461
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POTE generator: ~66/65 = 20.427
POTE generator: ~66/65 = 20.427


Optimal GPV sequence: {{Val list| 58, 118, 176f }}
{{Optimal ET sequence|legend=1| 58, 118, 176f }}


Badness: 0.026336
Badness: 0.026336
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POTE generator: ~66/65 = 20.378
POTE generator: ~66/65 = 20.378


Optimal GPV sequence: {{Val list| 58, 118, 294ffg, 412dffgg }}
{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}


Badness: 0.022396
Badness: 0.022396
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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.317
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.317


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217, 258, 475 }}


[[Badness]]: 0.105632
[[Badness]]: 0.105632
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Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303


Optimal GPV sequence: {{Val list| 41, 135c, 176, 217 }}
{{Optimal ET sequence|legend=1| 41, 135c, 176, 217 }}


Badness: 0.050966
Badness: 0.050966
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Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.306
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.306


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=1| 41, 176, 217 }}


Badness: 0.036951
Badness: 0.036951
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Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.307
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.307


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=1| 41, 176, 217 }}


Badness: 0.029495
Badness: 0.029495
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Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.308
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.308


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=1| 41, 176, 217 }}


Badness: 0.021811
Badness: 0.021811
Line 296: Line 296:
[[POTE generator]]: ~3/2 = 703.015
[[POTE generator]]: ~3/2 = 703.015


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b, 920b }}


[[Badness]]: 0.106471
[[Badness]]: 0.106471
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POTE generator: ~3/2 = 703.231
POTE generator: ~3/2 = 703.231


Optimal GPV sequence: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=1| 60e, 80, 140, 360, 500be, 860bde }}


Badness: 0.046770
Badness: 0.046770
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POTE generator: ~3/2 = 703.080
POTE generator: ~3/2 = 703.080


Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
{{Optimal ET sequence|legend=1| 60e, 80, 140, 500be, 640be, 780be }}


Badness: 0.032718
Badness: 0.032718
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[[POTE generator]]: ~98304/78125 = 396.621
[[POTE generator]]: ~98304/78125 = 396.621


{{Val list|legend=1| 118, 593, 711, 829, 947 }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}


[[Badness]]: 0.218314
[[Badness]]: 0.218314
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[[POTE generator]]: ~1125/896 = 396.643
[[POTE generator]]: ~1125/896 = 396.643


{{Val list|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}


[[Badness]]: 0.132821
[[Badness]]: 0.132821
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POTE generator: ~44/35 = 396.644
POTE generator: ~44/35 = 396.644


Optimal GPV sequence: {{Val list| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


Badness: 0.038186
Badness: 0.038186
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[[POTE generator]]: ~21/16 = 475.991
[[POTE generator]]: ~21/16 = 475.991


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}


[[Badness]]: 0.140722
[[Badness]]: 0.140722
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POTE generator: ~21/16 = 475.995
POTE generator: ~21/16 = 475.995


Optimal GPV sequence: {{Val list| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}


Badness: 0.045323
Badness: 0.045323
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POTE generator: ~21/16 = 475.996
POTE generator: ~21/16 = 475.996


Optimal GPV sequence: {{Val list| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}


Badness: 0.023800
Badness: 0.023800

Revision as of 16:52, 7 May 2023

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:

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 ]]

Mapping generators: ~63/50, ~28/27

POTE generator: ~28/27 = 60.528

Optimal ET sequence: 39d, 60, 99, 258, 357, 456

Badness: 0.057499

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 10976/10935

Mapping: [⟨3 4 5 6 6], ⟨0 5 13 16 29]]

POTE generator: ~28/27 = 60.430

Optimal ET sequence: 60e, 99e, 159, 258, 417d

Badness: 0.050379

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 10976/10935

Mapping: [⟨3 4 5 6 6 4], ⟨0 5 13 16 29 47]]

POTE generator: ~28/27 = 60.428

Optimal ET sequence: 99ef, 159, 258, 417d

Badness: 0.046006

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757

Mapping: [⟨3 4 5 6 6 4 10], ⟨0 5 13 16 29 47 15]]

POTE generator: ~28/27 = 60.438

Optimal ET sequence: 99ef, 159, 258, 417dg

Badness: 0.031678

Catachrome

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 1001/1000, 10976/10935

Mapping: [⟨3 4 5 6 6 12], ⟨0 5 13 16 29 -6]]

POTE generator: ~28/27 = 60.378

Optimal ET sequence: 60e, 99e, 159

Badness: 0.043844

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913

Mapping: [⟨3 4 5 6 6 12 10], ⟨0 5 13 16 29 -6 15]]

POTE generator: ~28/27 = 60.377

Optimal ET sequence: 60e, 99e, 159

Badness: 0.030218

Chromic

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1875/1859

Mapping: [⟨3 4 5 6 6 9], ⟨0 5 13 16 29 14]]

POTE generator: ~27/26 = 60.456

Optimal ET sequence: 60e, 99ef, 159f, 258ff

Badness: 0.049857

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 352/351, 375/374, 595/594

Mapping: [⟨3 4 5 6 6 9 10], ⟨0 5 13 16 29 14 15]]

POTE generator: ~27/26 = 60.459

Optimal ET sequence: 60e, 99ef, 159f, 258ff

Badness: 0.031043

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 = 162.806

Optimal ET sequence: 22, 74d, 96d, 118, 140, 258, 398, 656d

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 = 162.773

Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de

Badness: 0.032080

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

Optimal ET sequence: 58, 118, 294, 412d, 530d

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

Optimal ET sequence: 58, 118, 294, 412d

Badness: 0.030461

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1001/1000

Mapping: [⟨2 3 4 5 6 7], ⟨0 5 19 18 27 12]]

POTE generator: ~66/65 = 20.427

Optimal ET sequence: 58, 118, 176f

Badness: 0.026336

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 289/288, 352/351, 561/560

Mapping: [⟨2 3 4 5 6 7 8], ⟨0 5 19 18 27 12 5]]

POTE generator: ~66/65 = 20.378

Optimal ET sequence: 58, 118, 294ffg, 412dffgg

Badness: 0.022396

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 258edo, 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 0 80 25], ⟨0 1 -49 -14]]

Mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -49 -14 -80 -25 105 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.317

Optimal ET sequence: 41, 135c, 176, 217, 258, 475

Badness: 0.105632

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 80 25 -33], ⟨0 1 -49 -14 23]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303

Optimal ET sequence: 41, 135c, 176, 217

Badness: 0.050966

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 80 25 -33 -93], ⟨0 1 -49 -14 23 61]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.306

Optimal ET sequence: 41, 176, 217

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 0 80 25 -33 -93 -137], ⟨0 1 -49 -14 23 61 89]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.307

Optimal ET sequence: 41, 176, 217

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 0 80 25 -33 -93 -137 74], ⟨0 1 -49 -14 23 61 89 -44]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.308

Optimal ET sequence: 41, 176, 217

Badness: 0.021811

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

Optimal ET sequence: 60, 80, 140, 640b, 780b, 920b

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

Optimal ET sequence: 60e, 80, 140, 360, 500be, 860bde

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

Optimal ET sequence: 60e, 80, 140, 500be, 640be, 780be

Badness: 0.032718

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 = 396.621

Optimal ET sequence: 118, 593, 711, 829, 947

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

Optimal ET sequence: 118, 239, 357, 596, 1549bd

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

Optimal ET sequence: 118, 239, 357, 596

Badness: 0.038186

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

Optimal ET sequence: 58, 121, 179, 300bd, 479bcd

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

Optimal ET sequence: 58, 121, 179e, 300bde

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

Optimal ET sequence: 58, 121, 179ef, 300bdef

Badness: 0.023800

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