Hemimage temperaments: Difference between revisions

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-cotoneum (addressed in garischismic clan)
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This is a collection of temperaments tempering out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935. These include commatic, chromat, degrees, subfourth, bisupermajor and cotoneum, considered below, as well as the following discussed elsewhere:  
This is a collection of temperaments tempering out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935. These include commatic, chromat, degrees, subfourth, and bisupermajor, considered below, as well as the following discussed elsewhere:  
* ''[[Quasisuper]]'', {64/63, 2430/2401} → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Quasisuper]]'' → [[Archytas clan #Quasisuper|Archytas clan]] (+64/63)
* ''[[Liese]]'', {81/80, 686/675} → [[Meantone family #Liese|Meantone family]]
* ''[[Liese]]'' → [[Meantone family #Liese|Meantone family]] (+81/80)
* ''[[Unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Unicorn]]'' → [[Unicorn family #Septimal unicorn|Unicorn family]] (+126/125)
* [[Magic]], {225/224, 245/243} → [[Magic family #Magic|Magic family]]
* [[Magic]] → [[Magic family #Magic|Magic family]] (+225/224 or 245/243)
* ''[[Guiron]]'', {1029/1024, 10976/10935} → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Guiron]]'' → [[Gamelismic clan #Guiron|Gamelismic clan]] (+1029/1024)
* ''[[Echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[Echidna]]'' → [[Diaschismic family #Echidna|Diaschismic family]] (+1728/1715 or 2048/2025)
* [[Hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Hemififths]] → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]] (+2401/2400 or 5120/5103)
* ''[[Dodecacot]]'', {3125/3087, 10976/10935} → [[Tetracot family #Dodecacot|Tetracot family]]
* ''[[Dodecacot]]'' → [[Tetracot family #Dodecacot|Tetracot family]] (+3125/3087)
* [[Parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]] → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]] (+3136/3125 or 4375/4374)
* ''[[Pluto]]'', {4000/3969, 10976/10935} → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Pluto]]'' → [[Mirkwai clan #Pluto|Mirkwai clan]] (+4000/3969)
* ''[[Hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Hendecatonic]]'' → [[Porwell temperaments #Hendecatonic|Porwell temperaments]] (+6144/6125)
* ''[[Marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Marfifths]]'' → [[Kleismic family #Marfifths|Kleismic family]] (+15625/15552)
* ''[[Yarman I]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]]
* ''[[Cotoneum]]'' → [[Garischismic clan #Cotoneum|Garischismic clan]] (+33554432/33480783)
* ''[[Yarman I]]' → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]] (+244140625/243045684)


== Chromat ==
== Chromat ==
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Badness: 0.022396
Badness: 0.022396
== Cotoneum ==
{{Main| Cotoneum }}
The ''cotoneum'' temperament (41&217, named after the Latin for "[[Wikipedia:quince|quince]]") tempers out the [[Quince clan|quince comma]], 823543/819200 and the [[garischisma]], 33554432/33480783. This temperament is supported by [[41edo|41-]], [[176edo|176-]], [[217edo|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]]: [{{val| 1 0 80 25 }}, {{val| 0 1 -49 -14 }}]
Mapping generators: ~2, ~3
{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.317
{{Optimal ET sequence|legend=1| 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: [{{val| 1 0 80 25 -33 }}, {{val| 0 1 -49 -14 23 }}]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.303
{{Optimal ET sequence|legend=1| 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: [{{val| 1 0 80 25 -33 -93 }}, {{val| 0 1 -49 -14 23 61 }}]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.306
{{Optimal ET sequence|legend=1| 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: [{{val| 1 0 80 25 -33 -93 -137 }}, {{val| 0 1 -49 -14 23 61 89 }}]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.307
{{Optimal ET sequence|legend=1| 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: [{{val| 1 0 80 25 -33 -93 -137 74 }}, {{val| 0 1 -49 -14 23 61 89 -44 }}]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.308
{{Optimal ET sequence|legend=1| 41, 176, 217 }}
Badness: 0.021811


== Degrees ==
== Degrees ==

Revision as of 09:11, 5 July 2023

This is a collection of temperaments tempering out the hemimage comma, [5 -7 -1 3⟩ = 10976/10935. These include commatic, chromat, degrees, subfourth, and bisupermajor, 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

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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