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Hemimage temperaments: Difference between revisions

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* ''[[Marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Yarman I]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]]
* ''[[Yarman I]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]]
== 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]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]
{{Multival|legend=1| 10 38 36 37 29 -23 }}
[[POTE generator]]: ~81/80 = 20.377
{{Val list|legend=1| 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: [{{val| 2 3 4 5 6 }}, {{val| 0 5 19 18 27 }}]
POTE generator: ~81/80 = 20.390
Optimal GPV sequence: {{Val list| 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: [{{val| 2 3 4 5 6 7 }}, {{val| 0 5 19 18 27 12 }}]
POTE generator: ~66/65 = 20.427
Optimal GPV sequence: {{Val list| 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: [{{val| 2 3 4 5 6 7 8 }}, {{val| 0 5 19 18 27 12 5 }}]
POTE generator: ~66/65 = 20.378
Optimal GPV sequence: {{Val list| 58, 118, 294ffg, 412dffgg }}
Badness: 0.022396


== Chromat ==
== Chromat ==
Line 80: Line 24:


{{Multival|legend=1| 15 39 48 27 34 2 }}
{{Multival|legend=1| 15 39 48 27 34 2 }}
Mapping generators: ~63/50, ~28/27


[[POTE generator]]: ~28/27 = 60.528
[[POTE generator]]: ~28/27 = 60.528
Line 124: Line 70:
Optimal GPV sequence: {{Val list| 99ef, 159, 258, 417dg }}
Optimal GPV sequence: {{Val list| 99ef, 159, 258, 417dg }}


Badness: 0.046006
Badness: 0.031678


==== Catachromat ====
==== Catachrome ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 152: Line 98:
Badness: 0.030218
Badness: 0.030218


== Degrees ==
==== Chromic ====
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: 2.3.5.7.11.13


Subgroup: 2.3.5.7
Comma list: 196/195, 352/351, 729/728, 1875/1859


[[Comma list]]: 10976/10935, 390625/388962
Mapping: [{{val| 3 4 5 6 6 9 }}, {{val| 0 5 13 16 29 14 }}]


[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
POTE generator: ~27/26 = 60.456


{{Multival|legend=1| 20 40 60 17 39 27 }}
Optimal GPV sequence: {{Val list| 60e, 99ef, 159f, 258ff }}


[[POTE generator]]: ~3/2 = 703.015
Badness: 0.049857


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Badness]]: 0.106471
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


=== 11-limit ===
Mapping: [{{val| 3 4 5 6 6 9 10 }}, {{val| 0 5 13 16 29 14 15 }}]
Subgroup: 2.3.5.7.11


Comma list: 1331/1323, 1375/1372, 2200/2187
POTE generator: ~27/26 = 60.459


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Optimal GPV sequence: {{Val list| 60e, 99ef, 159f, 258ff }}


POTE generator: ~3/2 = 703.231
Badness: 0.031043


Optimal GPV sequence: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
== Bisupermajor ==
{{see also| Very high accuracy temperaments #Kwazy }}


Badness: 0.046770
Subgroup: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 10976/10935, 65625/65536
Subgroup: 2.3.5.7.11.13


Comma list: 325/324, 352/351, 1001/1000, 1331/1323
[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]


Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
{{Multival|legend=1| 16 -10 34 -53 9 107 }}


POTE generator: ~3/2 = 703.080
[[POTE generator]]: ~192/175 = 162.806


Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


Badness: 0.032718
[[Badness]]: 0.065492


== Subfourth ==
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 10976/10935, 65536/64827
Comma list: 385/384, 3388/3375, 9801/9800


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
POTE generators: ~11/10 = 162.773


[[POTE generator]]: ~21/16 = 475.991
Optimal GPV sequence: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
Badness: 0.032080


[[Badness]]: 0.140722
== 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.


=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


Comma list: 540/539, 896/891, 12005/11979
[[Comma list]]: 10976/10935, 50421/50000


Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]


POTE generator: ~21/16 = 475.995
{{Multival|legend=1| 10 38 36 37 29 -23 }}


Optimal GPV sequence: {{Val list| 58, 121, 179e, 300bde }}
[[POTE generator]]: ~81/80 = 20.377


Badness: 0.045323
{{Val list|legend=1| 58, 118, 294, 412d, 530d }}


=== 13-limit ===
[[Badness]]: 0.084317
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 364/363, 540/539, 676/675
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
Comma list: 441/440, 3388/3375, 8019/8000


POTE generator: ~21/16 = 475.996
Mapping: [{{val| 2 3 4 5 6 }}, {{val| 0 5 19 18 27 }}]


Optimal GPV sequence: {{Val list| 58, 121, 179ef, 300bdef }}
POTE generator: ~81/80 = 20.390


Badness: 0.023800
Optimal GPV sequence: {{Val list| 58, 118, 294, 412d }}


== Bisupermajor ==
Badness: 0.030461
{{see also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 10976/10935, 65625/65536
Comma list: 196/195, 352/351, 729/728, 1001/1000


[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
Mapping: [{{val| 2 3 4 5 6 7 }}, {{val| 0 5 19 18 27 12 }}]


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
POTE generator: ~66/65 = 20.427


[[POTE generator]]: ~192/175 = 162.806
Optimal GPV sequence: {{Val list| 58, 118, 176f }}


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
Badness: 0.026336


[[Badness]]: 0.065492
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


=== 11-limit ===
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
Subgroup: 2.3.5.7.11


Comma list: 385/384, 3388/3375, 9801/9800
Mapping: [{{val| 2 3 4 5 6 7 8 }}, {{val| 0 5 19 18 27 12 5 }}]


Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
POTE generator: ~66/65 = 20.378


POTE generators: ~11/10 = 162.773
Optimal GPV sequence: {{Val list| 58, 118, 294ffg, 412dffgg }}


Optimal GPV sequence: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
Badness: 0.022396
 
Badness: 0.032080


== Cotoneum ==
== Cotoneum ==
Line 336: Line 280:


Badness: 0.021811
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]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
{{Multival|legend=1| 20 40 60 17 39 27 }}
[[POTE generator]]: ~3/2 = 703.015
{{Val list|legend=1| 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: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
POTE generator: ~3/2 = 703.231
Optimal GPV sequence: {{Val list| 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: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
POTE generator: ~3/2 = 703.080
Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
Badness: 0.032718


== Squarschmidt ==
== Squarschmidt ==
Line 379: Line 366:


Badness: 0.038186
Badness: 0.038186
== Subfourth ==
Subgroup: 2.3.5.7
[[Comma list]]: 10976/10935, 65536/64827
[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
[[POTE generator]]: ~21/16 = 475.991
{{Val list|legend=1| 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: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
POTE generator: ~21/16 = 475.995
Optimal GPV sequence: {{Val list| 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: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
POTE generator: ~21/16 = 475.996
Optimal GPV sequence: {{Val list| 58, 121, 179ef, 300bdef }}
Badness: 0.023800


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemimage]]
[[Category:Hemimage]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"