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Hemimage temperaments: Difference between revisions

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* ''[[Marfifths]]'' → [[Kleismic family #Marfifths|Kleismic family]] (+15625/15552)
* ''[[Marfifths]]'' → [[Kleismic family #Marfifths|Kleismic family]] (+15625/15552)
* ''[[Cotoneum]]'' → [[Garischismic clan #Cotoneum|Garischismic clan]] (+33554432/33480783)
* ''[[Cotoneum]]'' → [[Garischismic clan #Cotoneum|Garischismic clan]] (+33554432/33480783)
* ''[[Yarman I]]' → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]] (+244140625/243045684)
* ''[[Yarman I]]'' → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]] (+244140625/243045684)


== 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


[[Comma list]]: 10976/10935, 235298/234375
[[Comma list]]: 10976/10935, 235298/234375


[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}


{{Multival|legend=1| 15 39 48 27 34 2 }}
{{Multival|legend=1| 15 39 48 27 34 2 }}


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


[[POTE generator]]: ~28/27 = 60.528
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528


{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
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Comma list: 441/440, 4375/4356, 10976/10935
Comma list: 441/440, 4375/4356, 10976/10935


Mapping: [{{val| 3 4 5 6 6 }}, {{val| 0 5 13 16 29 }}]
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


POTE generator: ~28/27 = 60.430
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430


{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
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Comma list: 364/363, 441/440, 625/624, 10976/10935
Comma list: 364/363, 441/440, 625/624, 10976/10935


Mapping: [{{val| 3 4 5 6 6 4 }}, {{val| 0 5 13 16 29 47 }}]
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


POTE generator: ~28/27 = 60.428
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428


{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
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Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757


Mapping: [{{val| 3 4 5 6 6 4 10 }}, {{val| 0 5 13 16 29 47 15 }}]
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


POTE generator: ~28/27 = 60.438
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438


{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
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Comma list: 325/324, 441/440, 1001/1000, 10976/10935
Comma list: 325/324, 441/440, 1001/1000, 10976/10935


Mapping: [{{val| 3 4 5 6 6 12 }}, {{val| 0 5 13 16 29 -6 }}]
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


POTE generator: ~28/27 = 60.378
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378


{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
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Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913


Mapping: [{{val| 3 4 5 6 6 12 10 }}, {{val| 0 5 13 16 29 -6 15 }}]
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


POTE generator: ~28/27 = 60.377
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377


{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
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Comma list: 196/195, 352/351, 729/728, 1875/1859
Comma list: 196/195, 352/351, 729/728, 1875/1859


Mapping: [{{val| 3 4 5 6 6 9 }}, {{val| 0 5 13 16 29 14 }}]
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


POTE generator: ~27/26 = 60.456
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456


{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
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Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


Mapping: [{{val| 3 4 5 6 6 9 10 }}, {{val| 0 5 13 16 29 14 15 }}]
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


POTE generator: ~27/26 = 60.459
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459


{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
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== Bisupermajor ==
== Bisupermajor ==
{{see also| Very high accuracy temperaments #Kwazy }}
{{See also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 65625/65536
[[Comma list]]: 10976/10935, 65625/65536


[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
 
: mapping generators: ~1225/864, ~192/175


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
{{Multival|legend=1| 16 -10 34 -53 9 107 }}


[[POTE generator]]: ~192/175 = 162.806
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806


{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
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Comma list: 385/384, 3388/3375, 9801/9800
Comma list: 385/384, 3388/3375, 9801/9800


Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}


POTE generators: ~11/10 = 162.773
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773


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


[[Comma list]]: 10976/10935, 50421/50000
[[Comma list]]: 10976/10935, 50421/50000


[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
 
: mapping generators: ~567/400, ~81/80


{{Multival|legend=1| 10 38 36 37 29 -23 }}
{{Multival|legend=1| 10 38 36 37 29 -23 }}


[[POTE generator]]: ~81/80 = 20.377
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377


{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
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Comma list: 441/440, 3388/3375, 8019/8000
Comma list: 441/440, 3388/3375, 8019/8000


Mapping: [{{val| 2 3 4 5 6 }}, {{val| 0 5 19 18 27 }}]
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}


POTE generator: ~81/80 = 20.390
Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390


{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
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Comma list: 196/195, 352/351, 729/728, 1001/1000
Comma list: 196/195, 352/351, 729/728, 1001/1000


Mapping: [{{val| 2 3 4 5 6 7 }}, {{val| 0 5 19 18 27 12 }}]
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


POTE generator: ~66/65 = 20.427
Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427


{{Optimal ET sequence|legend=1| 58, 118, 176f }}
{{Optimal ET sequence|legend=1| 58, 118, 176f }}
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Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560


Mapping: [{{val| 2 3 4 5 6 7 8 }}, {{val| 0 5 19 18 27 12 5 }}]
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


POTE generator: ~66/65 = 20.378
Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378


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


[[Comma list]]: 10976/10935, 390625/388962
[[Comma list]]: 10976/10935, 390625/388962


[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
 
: mapping generators: ~28/27, ~3


{{Multival|legend=1| 20 40 60 17 39 27 }}
{{Multival|legend=1| 20 40 60 17 39 27 }}


[[POTE generator]]: ~3/2 = 703.015
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015


{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b, 920b }}
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Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 1331/1323, 1375/1372, 2200/2187


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


POTE generator: ~3/2 = 703.231
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231


{{Optimal ET sequence|legend=1| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=1| 60e, 80, 140, 360, 500be, 860bde }}
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Comma list: 325/324, 352/351, 1001/1000, 1331/1323
Comma list: 325/324, 352/351, 1001/1000, 1331/1323


Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


POTE generator: ~3/2 = 703.080
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080


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


[[Comma]]: {{monzo| 61 4 -29 }}
[[Comma list]]: {{monzo| 61 4 -29 }}


[[Mapping]]: [{{val| 1 -8 1 }}, {{val| 0 29 4 }}]
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}


[[POTE generator]]: ~98304/78125 = 396.621
: mapping generators: ~2, ~98304/78125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621


{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
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=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 29360128/29296875
[[Comma list]]: 10976/10935, 29360128/29296875


[[Mapping]]: [{{val| 1 -8 1 -20 }}, {{val| 0 29 4 69 }}]
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}


{{Multival|legend=1| 29 4 69 -61 28 149 }}
{{Multival|legend=1| 29 4 69 -61 28 149 }}


[[POTE generator]]: ~1125/896 = 396.643
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1125/896 = 396.643


{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}
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Comma list: 3025/3024, 5632/5625, 10976/10935
Comma list: 3025/3024, 5632/5625, 10976/10935


Mapping: [{{val| 1 -8 1 -20 -21 }}, {{val| 0 29 4 69 74 }}]
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


POTE generator: ~44/35 = 396.644
Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644


{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
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== Subfourth ==
== Subfourth ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 65536/64827
[[Comma list]]: 10976/10935, 65536/64827


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}
 
: mapping generators: ~2, ~21/16


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}


[[POTE generator]]: ~21/16 = 475.991
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991


{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}
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Comma list: 540/539, 896/891, 12005/11979
Comma list: 540/539, 896/891, 12005/11979


Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}


POTE generator: ~21/16 = 475.995
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995


{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}
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Comma list: 352/351, 364/363, 540/539, 676/675
Comma list: 352/351, 364/363, 540/539, 676/675


Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}


POTE generator: ~21/16 = 475.996
Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996


{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemimage]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"