Hemimage temperaments: Difference between revisions

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== Commatic ==
== 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.
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
Subgroup: 2.3.5.7
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== Chromat ==
== 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 family|amity extension]] with third-octave period.
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 family|amity extension]] with third-octave period.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
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== Degrees ==
== 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.
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
Subgroup: 2.3.5.7
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{{Multival|legend=1| 16 -10 34 -53 9 107 }}
{{Multival|legend=1| 16 -10 34 -53 9 107 }}


[[POTE generator]]: ~192/175 = 162.8061
[[POTE generator]]: ~192/175 = 162.806


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
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Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]


POTE generators: ~11/10 = 162.7733
POTE generators: ~11/10 = 162.773


Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
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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|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.
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|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
Subgroup: 2.3.5.7
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[[POTE generator]]: ~3/2 = 702.317
[[POTE generator]]: ~3/2 = 702.317
[[Minimax tuning]]:
* 7-odd-limit: ~3/2 = {{Monzo| 3/5 1/50 -1/50 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 8/5 1/50 -1/50 0 }}, {{Monzo| 8/5 -49/50 49/50 0 }}, {{Monzo| 13/5 -7/25 7/25 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 6/5
* 9-odd-limit: ~3/2 = {{Monzo| 29/51 2/51 -1/51 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 80/51 2/51 -1/51 0 }}, {{Monzo| 160/51 -98/51 49/51 0 }}, {{Monzo| 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)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.9550, 702.3575]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.9550, 702.3575]


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
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POTE generator: ~3/2 = 702.303
POTE generator: ~3/2 = 702.303
Minimax tuning:
* 11-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}
: Eigenmonzos (unchanged intervals): 2, 11/10
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [702.1277, 702.4390] (55\94 to 24\41)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3575]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [702.1277, 702.3575]


Vals: {{Val list| 41, 135c, 176, 217 }}
Vals: {{Val list| 41, 135c, 176, 217 }}
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POTE generator: ~3/2 = 702.306
POTE generator: ~3/2 = 702.306
Minimax tuning:
* 13-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}
: Eigenmonzos (unchanged intervals): 2, 11/10
* 15-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}
: Eigenmonzos (unchanged intervals): 2, 15/11
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~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: {{Val list| 41, 176, 217 }}
Vals: {{Val list| 41, 176, 217 }}
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POTE generator: ~3/2 = 702.307
POTE generator: ~3/2 = 702.307
Minimax tuning:
* 17-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}
: Eigenmonzos (unchanged intervals): 2, 15/11
Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
* 17-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3693]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3693]


Vals: {{Val list| 41, 176, 217 }}
Vals: {{Val list| 41, 176, 217 }}
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POTE generator: ~3/2 = 702.308
POTE generator: ~3/2 = 702.308
Minimax tuning:
* 19- and 21-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}
: Eigenmonzos (unchanged intervals): 2, 15/11
Tuning ranges:
* 19- and 21-odd-limit diamond monotone: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
* 19- and 21-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3771]
* 19- and 21-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3771]


Vals: {{Val list| 41, 176, 217 }}
Vals: {{Val list| 41, 176, 217 }}
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== Squarschmidt ==
== Squarschmidt ==
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&amp;239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.
A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&amp;239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&amp;239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.


Subgroup: 2.3.5
Subgroup: 2.3.5
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[[Mapping]]: [{{val| 1 -8 1 }}, {{val| 0 29 4 }}]
[[Mapping]]: [{{val| 1 -8 1 }}, {{val| 0 29 4 }}]


[[POTE generator]]: ~98304/78125 = 396.6208
[[POTE generator]]: ~98304/78125 = 396.621


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

Revision as of 13:34, 19 October 2021

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:

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

Template:Val list

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

Template:Val list

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

Template:Val list

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

Template:Val list

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 = 162.806

Template:Val list

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

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

Template:Val list

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

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

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

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

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 = 396.621

Template:Val list

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

Template:Val list

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

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