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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.


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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.


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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.


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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.


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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.


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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 }}
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