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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:  
{{Technical data page}}
* ''[[Quasisuper]]'', {64/63, 2430/2401} → [[Archytas clan #Quasisuper|Archytas clan]]
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:  
* ''[[Liese]]'', {81/80, 686/675} → [[Meantone family #Liese|Meantone family]]
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* [[Magic]], {225/224, 245/243} → [[Magic family #Magic|Magic family]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Guiron]]'', {1029/1024, 10976/10935} → [[Gamelismic clan #Guiron|Gamelismic clan]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[Echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* [[Hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[Dodecacot]]'', {3125/3087, 10976/10935} → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* ''[[Pluto]]'', {4000/3969, 10976/10935} → [[Mirkwai clan #Pluto|Mirkwai clan]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Yarman I]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


== 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 }}
: mapping generators: ~63/50, ~28/27


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


[[POTE generator]]: ~28/27 = 60.528
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
 
{{Val list|legend=1| 39d, 60, 99, 258, 357, 456 }}


[[Badness]]: 0.057499
[[Badness]]: 0.057499
Line 38: Line 39:
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 GPV sequence: {{Val list| 60e, 99e, 159, 258, 417d }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}


Badness: 0.050379
Badness: 0.050379
Line 51: Line 52:
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 GPV sequence: {{Val list| 99ef, 159, 258, 417d }}
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}


Badness: 0.046006
Badness: 0.046006
Line 64: Line 65:
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 GPV sequence: {{Val list| 99ef, 159, 258, 417dg }}
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}


Badness: 0.031678
Badness: 0.031678
Line 77: Line 78:
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 GPV sequence: {{Val list| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Badness: 0.043844
Badness: 0.043844
Line 90: Line 91:
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 GPV sequence: {{Val list| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


Badness: 0.030218
Badness: 0.030218
Line 103: Line 104:
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 GPV sequence: {{Val list| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


Badness: 0.049857
Badness: 0.049857
Line 116: Line 117:
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 GPV sequence: {{Val list| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


Badness: 0.031043
Badness: 0.031043
=== Hemichromat ===
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 10976/10935, 102487/102400
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}
Badness: 0.067173
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
Badness: 0.033420


== 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 }}


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
: mapping generators: ~1225/864, ~192/175


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


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


[[Badness]]: 0.065492
[[Badness]]: 0.065492
Line 146: Line 173:
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 GPV sequence: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}


Badness: 0.032080
Badness: 0.032080


== Commatic ==
== Bicommatic ==
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.
Used to be known simply as the ''commatic'' temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, 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 }}


{{Multival|legend=1| 10 38 36 37 29 -23 }}
: mapping generators: ~567/400, ~81/80


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


{{Val list|legend=1| 58, 118, 294, 412d, 530d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}


[[Badness]]: 0.084317
[[Badness]]: 0.084317
Line 176: Line 203:
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 GPV sequence: {{Val list| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


Badness: 0.030461
Badness: 0.030461
Line 189: Line 216:
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 GPV sequence: {{Val list| 58, 118, 176f }}
{{Optimal ET sequence|legend=1| 58, 118, 176f }}


Badness: 0.026336
Badness: 0.026336
Line 202: Line 229:
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 GPV sequence: {{Val list| 58, 118, 294ffg, 412dffgg }}
{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}


Badness: 0.022396
Badness: 0.022396


== Cotoneum ==
== Degrees ==
{{Main| Cotoneum }}
{{ See also | 20th-octave temperaments }}
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.
 
An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.


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.
By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.


Subgroup: 2.3.5.7
By looking at the mapping, we observe an 80-note [[mos scale]] is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].


[[Comma list]]: 10976/10935, 823543/819200
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 2 -18 -3}}, {{val|0 -1 49 14}}]
[[Comma list]]: 10976/10935, 390625/388962


{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}


[[POTE generator]]: ~3/2 = 702.317
: mapping generators: ~28/27, ~3


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)


[[Badness]]: 0.105632
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
 
[[Badness]]: 0.106471
 
Badness (Sintel): 2.694


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


Comma list: 441/440, 10976/10935, 16384/16335
Comma list: 1331/1323, 1375/1372, 2200/2187
 
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


Mapping: [{{val|1 2 -18 -3 13}}, {{val|0 -1 49 14 -23}}]
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)


POTE generator: ~3/2 = 702.303
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}


Optimal GPV sequence: {{Val list| 41, 135c, 176, 217 }}
Badness: 0.046770


Badness: 0.050966
Badness (Sintel): 1.546


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


Comma list: 364/363, 441/440, 3584/3575, 10976/10935
Comma list: 325/324, 352/351, 1001/1000, 1331/1323


Mapping: [{{val|1 2 -18 -3 13 29}}, {{val|0 -1 49 14 -23 -61}}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


POTE generator: ~3/2 = 702.306
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}


Badness: 0.036951
Badness: 0.032718
 
Badness (Sintel): 1.352


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000


Mapping: [{{val|1 2 -18 -3 13 29 41}}, {{val|0 -1 49 14 -23 -61 -89}}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


POTE generator: ~3/2 = 702.307
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}


Badness: 0.029495
Badness (Sintel): 1.171


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
 
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
 
Badness (Sintel): 1.273


Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~3/2 = 702.308
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


Badness: 0.021811
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)


== Degrees ==
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
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
Badness (Sintel): 1.209


[[Comma list]]: 10976/10935, 390625/388962
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29


[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405


{{Multival|legend=1| 20 40 60 17 39 27 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


[[POTE generator]]: ~3/2 = 703.015
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}


[[Badness]]: 0.106471
Badness (Sintel): 1.134


=== 11-limit ===
=== no-31's 37-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19.23.29.37


Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}


POTE generator: ~3/2 = 703.231
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)


Optimal GPV sequence: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}


Badness: 0.046770
Badness (Sintel): 1.127


=== 13-limit ===
=== no-31's 41-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41


Comma list: 325/324, 352/351, 1001/1000, 1331/1323
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870


Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 12 | 0 1 2 3 3 0 1 0 2 3 3 3 }}


POTE generator: ~3/2 = 703.080
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207


Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}


Badness: 0.032718
Badness (Sintel): 1.100


== 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


[[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


{{Val list|legend=1| 118, 593, 711, 829, 947 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
 
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}


[[Badness]]: 0.218314
[[Badness]]: 0.218314


=== 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 }}


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


{{Val list|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}


[[Badness]]: 0.132821
[[Badness]]: 0.132821
Line 359: Line 406:
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 GPV sequence: {{Val list| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


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:Pages with mostly numerical content]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"