Hemimage temperaments: Difference between revisions

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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]]


== Commatic ==
== Chromat ==
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 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, 50421/50000
[[Comma list]]: 10976/10935, 235298/234375


[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}


{{Multival|legend=1| 10 38 36 37 29 -23 }}
: mapping generators: ~63/50, ~28/27


[[POTE generator]]: ~81/80 = 20.377
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528


{{Val list|legend=1| 58, 118, 294, 412d, 530d }}
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


[[Badness]]: 0.084317
[[Badness]]: 0.057499


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


Comma list: 441/440, 3388/3375, 8019/8000
Comma list: 441/440, 4375/4356, 10976/10935


Mapping: [{{val| 2 3 4 5 6 }}, {{val| 0 5 19 18 27 }}]
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


POTE generator: ~81/80 = 20.390
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430


Optimal GPV sequence: {{Val list| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}


Badness: 0.030461
Badness: 0.050379


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


Comma list: 196/195, 352/351, 729/728, 1001/1000
Comma list: 364/363, 441/440, 625/624, 10976/10935


Mapping: [{{val| 2 3 4 5 6 7 }}, {{val| 0 5 19 18 27 12 }}]
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


POTE generator: ~66/65 = 20.427
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.428


Optimal GPV sequence: {{Val list| 58, 118, 176f }}
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}


Badness: 0.026336
Badness: 0.046006


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


Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
 
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
 
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.438
 
{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
 
Badness: 0.031678
 
==== Catachrome ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
 
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.378


Mapping: [{{val| 2 3 4 5 6 7 8 }}, {{val| 0 5 19 18 27 12 5 }}]
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


POTE generator: ~66/65 = 20.378
Badness: 0.043844


Optimal GPV sequence: {{Val list| 58, 118, 294ffg, 412dffgg }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Badness: 0.022396
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913


== Chromat ==
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
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
Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.377


[[Comma list]]: 10976/10935, 235298/234375
{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}


[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
Badness: 0.030218


{{Multival|legend=1| 15 39 48 27 34 2 }}
==== Chromic ====
Subgroup: 2.3.5.7.11.13


[[POTE generator]]: ~28/27 = 60.528
Comma list: 196/195, 352/351, 729/728, 1875/1859


{{Val list|legend=1| 39d, 60, 99, 258, 357, 456 }}
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


[[Badness]]: 0.057499
Optimal tuning (POTE): ~44/35 = 1\3, ~27/26 = 60.456


== Degrees ==
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
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: 0.049857


[[Comma list]]: 10976/10935, 390625/388962
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


{{Multival|legend=1| 20 40 60 17 39 27 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


[[POTE generator]]: ~3/2 = 703.015
Optimal tuning (POTE): ~63/50 = 1\3, ~27/26 = 60.459


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}


[[Badness]]: 0.106471
Badness: 0.031043


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


Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 3025/3024, 10976/10935, 102487/102400


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


POTE generator: ~3/2 = 703.231
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511


Optimal GPV sequence: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.046770
Badness: 0.067173


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


Comma list: 325/324, 352/351, 1001/1000, 1331/1323
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935
 
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527


POTE generator: ~3/2 = 703.080
{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}


Optimal GPV sequence: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
Badness: 0.033420


Badness: 0.032718
== Bisupermajor ==
{{See also| Very high accuracy temperaments #Kwazy }}


== Subfourth ==
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10976/10935, 65536/64827
[[Comma list]]: 10976/10935, 65625/65536


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


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
: mapping generators: ~1225/864, ~192/175


[[POTE generator]]: ~21/16 = 475.991
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


[[Badness]]: 0.140722
[[Badness]]: 0.065492


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


Comma list: 540/539, 896/891, 12005/11979
Comma list: 385/384, 3388/3375, 9801/9800
 
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
 
Optimal tuning (POTE): ~99/70, ~11/10 = 162.773
 
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}
 
Badness: 0.032080
 
== Bicommatic ==
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
 
[[Comma list]]: 10976/10935, 50421/50000


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


POTE generator: ~21/16 = 475.995
: mapping generators: ~567/400, ~81/80


Optimal GPV sequence: {{Val list| 58, 121, 179e, 300bde }}
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377
 
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d, 530d }}
 
[[Badness]]: 0.084317


Badness: 0.045323
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 13-limit ===
Comma list: 441/440, 3388/3375, 8019/8000
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 364/363, 540/539, 676/675
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}


Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390


POTE generator: ~21/16 = 475.996
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


Optimal GPV sequence: {{Val list| 58, 121, 179ef, 300bdef }}
Badness: 0.030461


Badness: 0.023800
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


== Bisupermajor ==
Comma list: 196/195, 352/351, 729/728, 1001/1000
{{see also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


[[Comma list]]: 10976/10935, 65625/65536
Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427


[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
{{Optimal ET sequence|legend=1| 58, 118, 176f }}


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
Badness: 0.026336


[[POTE generator]]: ~192/175 = 162.806
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560


[[Badness]]: 0.065492
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


=== 11-limit ===
Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378
Subgroup: 2.3.5.7.11


Comma list: 385/384, 3388/3375, 9801/9800
{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}


Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
Badness: 0.022396


POTE generators: ~11/10 = 162.773
== Degrees ==
{{ 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.  


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


Badness: 0.032080
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.


== Cotoneum ==
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]].
{{Main| Cotoneum }}


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


[[Comma list]]: 10976/10935, 823543/819200
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}


[[Mapping]]: [{{val|1 2 -18 -3}}, {{val|0 -1 49 14}}]
: mapping generators: ~28/27, ~3


{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)


[[POTE generator]]: ~3/2 = 702.317
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
[[Badness]]: 0.106471


[[Badness]]: 0.105632
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: [{{val|1 2 -18 -3 13}}, {{val|0 -1 49 14 -23}}]
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


POTE generator: ~3/2 = 702.303
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)


Optimal GPV sequence: {{Val list| 41, 135c, 176, 217 }}
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}


Badness: 0.050966
Badness: 0.046770
 
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: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


Mapping: [{{val|1 2 -18 -3 13 29}}, {{val|0 -1 49 14 -23 -61}}]
Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)


POTE generator: ~3/2 = 702.306
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
Badness: 0.032718


Badness: 0.036951
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
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}
 
Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)
 
{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}
 
Badness (Sintel): 1.209
 
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)
 
{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
 
Badness (Sintel): 1.134
 
=== no-31's 37-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
 
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}
 
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)
 
{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
 
Badness (Sintel): 1.127


Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]
=== no-31's 41-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41


POTE generator: ~3/2 = 702.308
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 451/450, 476/475, 481/480, 2871/2870


Optimal GPV sequence: {{Val list| 41, 176, 217 }}
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 }}


Badness: 0.021811
Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
 
{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
 
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 307: 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


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimage]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:29, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 temperaments tempering out the hemimage comma (monzo: [5 -7 -1 3⟩, ratio: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:

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]]

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

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

Optimal ET sequence: 39d, 60, 99, 258, 357, 456

Badness: 0.057499

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 10976/10935

Mapping: [⟨3 4 5 6 6], ⟨0 5 13 16 29]]

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

Optimal ET sequence: 60e, 99e, 159, 258, 417d

Badness: 0.050379

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 10976/10935

Mapping: [⟨3 4 5 6 6 4], ⟨0 5 13 16 29 47]]

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

Optimal ET sequence: 99ef, 159, 258, 417d

Badness: 0.046006

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757

Mapping: [⟨3 4 5 6 6 4 10], ⟨0 5 13 16 29 47 15]]

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

Optimal ET sequence: 99ef, 159, 258, 417dg

Badness: 0.031678

Catachrome

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440, 1001/1000, 10976/10935

Mapping: [⟨3 4 5 6 6 12], ⟨0 5 13 16 29 -6]]

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

Optimal ET sequence: 60e, 99e, 159

Badness: 0.043844

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913

Mapping: [⟨3 4 5 6 6 12 10], ⟨0 5 13 16 29 -6 15]]

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

Optimal ET sequence: 60e, 99e, 159

Badness: 0.030218

Chromic

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 4 5 6 6 9], ⟨0 5 13 16 29 14]]

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

Optimal ET sequence: 60e, 99ef, 159f, 258ff

Badness: 0.049857

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 352/351, 375/374, 595/594

Mapping: [⟨3 4 5 6 6 9 10], ⟨0 5 13 16 29 14 15]]

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

Optimal ET sequence: 60e, 99ef, 159f, 258ff

Badness: 0.031043

Hemichromat

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 10976/10935, 102487/102400

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: 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: [⟨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: 39df, 120cdff, 159, 198, 357, 912b

Badness: 0.033420

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]]

mapping generators: ~1225/864, ~192/175

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

Optimal ET sequence: 22, 74d, 96d, 118, 140, 258, 398, 656d

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]]

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

Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de

Badness: 0.032080

Bicommatic

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

Comma list: 10976/10935, 50421/50000

Mapping: [⟨2 3 4 5], ⟨0 5 19 18]]

mapping generators: ~567/400, ~81/80

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

Optimal ET sequence: 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: [⟨2 3 4 5 6], ⟨0 5 19 18 27]]

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

Optimal ET sequence: 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: [⟨2 3 4 5 6 7], ⟨0 5 19 18 27 12]]

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

Optimal ET sequence: 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: [⟨2 3 4 5 6 7 8], ⟨0 5 19 18 27 12 5]]

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

Optimal ET sequence: 58, 118, 294ffg, 412dffgg

Badness: 0.022396

Degrees

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 ¢, we get a uniquely elegant extension to the 29-limit which tempers out (33/25)/(29/22) = 726/725, S28 = 784/783 and S29 = 841/840. An edo as large as 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.

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.

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.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 390625/388962

Mapping: [⟨20 0 -17 -39], ⟨0 1 2 3]]

mapping generators: ~28/27, ~3

Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)

Optimal ET sequence: 20cd, 60, 80, 140, 640b, 780b

Badness: 0.106471

Badness (Sintel): 2.694

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]]

Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)

Optimal ET sequence: 20cd, 60e, 80, 140, 360

Badness: 0.046770

Badness (Sintel): 1.546

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]]

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

Optimal ET sequence: 20cde, 60e, 80, 140

Badness: 0.032718

Badness (Sintel): 1.352

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000

Mapping: [⟨20 0 -17 -39 -26 74 50], ⟨0 1 2 3 3 0 1]]

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

Optimal ET sequence: 20cde, 60e, 80, 140

Badness (Sintel): 1.171

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475

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: 20cde, 60e, 80, 140

Badness (Sintel): 1.273

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27], ⟨0 1 2 3 3 0 1 0 2]]

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

Optimal ET sequence: 20cdei, 60e, 80, 140

Badness (Sintel): 1.209

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2], ⟨0 1 2 3 3 0 1 0 2 3]]

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

Optimal ET sequence: 20cdeij, 60e, 80, 140

Badness (Sintel): 1.134

no-31's 37-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480

Mapping: [⟨20 0 -17 -39 -26 74 50 85 27 2 9], ⟨0 1 2 3 3 0 1 0 2 3 3]]

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

Optimal ET sequence: 20cdeijl, 60el, 80, 140

Badness (Sintel): 1.127

no-31's 41-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41

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: [⟨20 0 -17 -39 -26 74 50 85 27 2 9 12], ⟨0 1 2 3 3 0 1 0 2 3 3 3]]

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

Optimal ET sequence: 20cdeijlm, 60el, 80, 140

Badness (Sintel): 1.100

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 list: [61 4 -29⟩

Mapping: [⟨1 -8 1], ⟨0 29 4]]

mapping generators: ~2, ~98304/78125

Optimal tuning (POTE): ~2 = 1\1, ~98304/78125 = 396.621

Optimal ET sequence: 118, 593, 711, 829, 947

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]]

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

Optimal ET sequence: 118, 239, 357, 596, 1549bd

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]]

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

Optimal ET sequence: 118, 239, 357, 596

Badness: 0.038186

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