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]]
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).
* ''[[liese]]'', {81/80, 686/675} → [[Meantone family #Liese]]
* ''[[unicorn]]'', {126/125, 10976/10935} → [[Unicorn family #Septimal unicorn]]
* [[magic]], {225/224, 245/243} → [[Magic family #Magic]]
* ''[[guiron]]'', {1029/1024, 10976/10935} → [[Gamelismic clan #Guiron]]
* ''[[echidna]]'', {1728/1715, 2048/2025} → [[Diaschismic family #Echidna]]
* [[hemififths]], {2401/2400, 5120/5103} → [[Breedsmic temperaments #Hemififths]]
* ''[[dodecacot]]'', {3125/3087, 10976/10935} → [[Tetracot family #Dodecacot]]
* [[parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic]]
* ''[[pluto]]'', {4000/3969, 10976/10935} → [[Mirkwai clan #Pluto]]
* ''[[hendecatonic]]'', {6144/6125, 10976/10935} → [[Porwell temperaments #Hendecatonic]]
* ''[[marfifths]]'', {10976/10935, 15625/15552} → [[Kleismic family #Marfifths]]
* ''[[yarman]]'', {10976/10935, 244140625/243045684} → [[Turkish maqam music temperaments #Yarman]]


== Commatic ==
Temperaments discussed elsewhere are:
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.
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell 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]]


Subgroup: 2.3.5.7
Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].  


[[Comma list]]: 10976/10935, 50421/50000
== 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.
 
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]
[[Comma list]]: 10976/10935, 235298/234375


{{Multival|legend=1| 10 38 36 37 29 -23 }}
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27


[[POTE generator]]: ~81/80 = 20.377
[[Optimal tuning]]s:  
* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}
: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}
: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}


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


[[Badness]]: 0.084317
[[Badness]] (Sintel): 1.46


=== 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: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
 
Optimal tunings:
* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


Mapping: [{{val| 2 3 4 5 6 }}, {{val| 0 5 19 18 27 }}]
{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258 }}


POTE generator: ~81/80 = 20.390
Badness (Sintel): 1.67


Vals: {{Val list| 58, 118, 294, 412d }}
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.030461
Comma list: 364/363, 441/440, 625/624, 10976/10935


== Chromat ==
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
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 tunings:  
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}


[[Comma list]]: 10976/10935, 235298/234375
{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}


[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
Badness (Sintel): 1.90


{{Multival|legend=1| 15 39 48 27 34 2 }}
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~28/27 = 60.528
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757


{{Val list|legend=1| 39d, 60, 99, 258, 357, 456 }}
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


[[Badness]]: 0.057499
Optimal tunings:  
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


== Degrees ==
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}
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): 1.61


[[Comma list]]: 10976/10935, 390625/388962
==== Catachrome ====
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]
Comma list: 325/324, 441/440, 1001/1000, 10976/10935


{{Multival|legend=1| 20 40 60 17 39 27 }}
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


[[POTE generator]]: ~3/2 = 703.015
Optimal tunings:  
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


[[Badness]]: 0.106471
Badness (Sintel): 1.81


=== 11-limit ===
===== 17-limit =====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


POTE generator: ~3/2 = 703.231
Optimal tunings:  
* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}


Vals: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness: 0.046770
Badness (Sintel): 1.54


=== 13-limit ===
==== Chromic ====
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: 196/195, 352/351, 729/728, 1875/1859


Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


POTE generator: ~3/2 = 703.080
Optimal tunings:  
* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}


Vals: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


Badness: 0.032718
Badness (Sintel): 2.06


== Subfourth ==
===== 17-limit =====
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17
 
[[Comma list]]: 10976/10935, 65536/64827


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


[[POTE generator]]: ~21/16 = 475.991
Optimal tunings:  
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


[[Badness]]: 0.140722
Badness (Sintel): 1.58


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


Comma list: 540/539, 896/891, 12005/11979
Comma list: 3025/3024, 10976/10935, 102487/102400


Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


POTE generator: ~21/16 = 475.995
Optimal tunings:  
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}


Vals: {{Val list| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.045323
Badness (Sintel): 2.22


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


Comma list: 352/351, 364/363, 540/539, 676/675
Comma list: 676/675, 1001/1000, 3025/3024, 10976/10935


Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


POTE generator: ~21/16 = 475.996
Optimal tunings:  
* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}


Vals: {{Val list| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}


Badness: 0.023800
Badness (Sintel): 1.38


== Bisupermajor ==
== Bisupermajor ==
{{see also| Very high accuracy temperaments #Kwazy }}
: ''For the 5-limit version, see [[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
 
[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.0294{{c}}, ~192/175 = 162.8141{{c}}
: [[error map]]: {{val| +0.059 +0.587 -0.208 -0.957 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~192/175 = 162.8082{{c}}
: error map: {{val| 0.000 +0.510 -0.355 -1.087 }}
 
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
 
[[Badness]] (Sintel): 1.66
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 3388/3375, 9801/9800
 
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
 
Optimal tunings:
* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}
 
{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}
 
Badness (Sintel): 1.06
 
== 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


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
[[Comma list]]: 10976/10935, 50421/50000


[[POTE generator]]: ~192/175 = 162.8061
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
: mapping generators: ~567/400, ~81/80


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.0497{{c}}, ~81/80 = 20.3790{{c}}
: [[error map]]: {{val| +0.099 +0.089 +1.085 -1.756 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~81/80 = 20.3837{{c}}
: error map: {{val| 0.000 -0.037 +0.976 -1.920 }}


[[Badness]]: 0.065492
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}
 
[[Badness]] (Sintel): 2.13


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


Comma list: 385/384, 3388/3375, 9801/9800
Comma list: 441/440, 3388/3375, 8019/8000
 
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}
 
{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}
 
Badness (Sintel): 1.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 352/351, 729/728, 1001/1000
 
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}
 
{{Optimal ET sequence|legend=0| 58, 118, 176f }}
 
Badness (Sintel): 1.09
 
=== 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 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


POTE generators: ~11/10 = 162.7733
Optimal tunings:  
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}


Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=0| 58, 118 }}


Badness: 0.032080
Badness (Sintel): 1.14


== Cotoneum ==
== Degrees ==
{{Main| Cotoneum }}
{{About|the regular temperament|scale degrees|degree}}
{{See also| 20th-octave temperaments }}


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.
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
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.


[[Comma list]]: 10976/10935, 823543/819200
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.


[[Mapping]]: [{{val|1 2 -18 -3}}, {{val|0 -1 49 14}}]
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]].


{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
[[Subgroup]]: 2.3.5.7


[[POTE generator]]: ~3/2 = 702.317
[[Comma list]]: 10976/10935, 390625/388962


[[Minimax tuning]]:
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
* 7-odd-limit: ~3/2 = {{Monzo| 3/5 1/50 -1/50 }}
: mapping generators: ~28/27, ~3
: [{{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]]:
[[Optimal tuning]]s:  
* 7-odd-limit [[diamond monotone]]: ~3/2 = [701.5385, 702.8571] (38\65 to 41\70)
* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})
* 9-odd-limit diamond monotone: ~3/2 = [701.8868, 702.8571] (31\53 to 41\70)
: [[error map]]: {{val| -0.157 +0.812 -0.647 -0.220 }}
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.9550, 702.3575]
* [[CWE]]: ~28/27 = 60.0000{{c}}, ~3/2 = 702.9324{{c}} (~126/125 = 17.0676{{c}})
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.9550, 702.3575]
: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


[[Badness]]: 0.105632
[[Badness]] (Sintel): 2.69


=== 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 }}
 
Optimal tunings:
* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})
 
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}
 
Badness (Sintel): 1.55
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 1001/1000, 1331/1323
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
 
Optimal tunings:
* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})
 
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
 
Badness (Sintel): 1.35
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000
 
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


Mapping: [{{val|1 2 -18 -3 13}}, {{val|0 -1 49 14 -23}}]
Optimal tunings:  
* WE: ~28/27 = 60.0058{{c}}, ~3/2 = 703.0364{{c}} (~100/99 = 17.0335{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0061{{c}} (~100/99 = 16.9939{{c}})


POTE generator: ~3/2 = 702.303
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Minimax tuning:
Badness (Sintel): 1.17
* 11-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}
: Eigenmonzos (unchanged intervals): 2, 11/10


Tuning ranges:
=== 19-limit ===
* 11-odd-limit diamond monotone: ~3/2 = [702.1277, 702.4390] (55\94 to 24\41)
Subgroup: 2.3.5.7.11.13.17.19
* 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 }}
Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475


Badness: 0.050966
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}


=== 13-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13
* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})
 
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
 
Badness (Sintel): 1.27
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 364/363, 441/440, 3584/3575, 10976/10935
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399


Mapping: [{{val|1 2 -18 -3 13 29}}, {{val|0 -1 49 14 -23 -61}}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


POTE generator: ~3/2 = 702.306
Optimal tunings:  
* WE: ~28/27 = 59.9990{{c}}, ~3/2 = 703.1804{{c}} (~100/99 = 16.8074{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1870{{c}} (~100/99 = 16.8130{{c}})


Minimax tuning:
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
* 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:
Badness (Sintel): 1.21
* 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 }}
=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29


Badness: 0.036951
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405


=== 17-limit ===
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
Optimal tunings:  
* WE: ~29/28 = 59.9990{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})


Mapping: [{{val|1 2 -18 -3 13 29 41}}, {{val|0 -1 49 14 -23 -61 -89}}]
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


POTE generator: ~3/2 = 702.307
Badness (Sintel): 1.13


Minimax tuning:
=== 2.3.5.7.11.13.17.19.23.29.37 subgroup ===
* 17-odd-limit: ~3/2 = {{Monzo| 42/71 -1/71 -1/71 0 1/71 }}
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
: Eigenmonzos (unchanged intervals): 2, 15/11


Tuning ranges:
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405, 481/480
* 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 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}


Badness: 0.029495
Optimal tunings:  
* WE: ~29/28 = 60.0001{{c}}, ~3/2 = 703.2183{{c}} (~100/99 = 16.7827{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2178{{c}} (~100/99 = 16.7822{{c}})


=== 19-limit ===
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
Badness (Sintel): 1.13


Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]
=== 2.3.5.7.11.13.17.19.23.29.37.41 subgroup ===
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


Minimax tuning:  
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 }}
* 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:
Optimal tunings:  
* 19- and 21-odd-limit diamond monotone: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
* WE: ~29/28 = 59.9998{{c}}, ~3/2 = 703.2088{{c}} (~100/99 = 16.7882{{c}})
* 19- and 21-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3771]
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2104{{c}} (~100/99 = 16.7896{{c}})
* 19- and 21-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3771]


Vals: {{Val list| 41, 176, 217 }}
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}


Badness: 0.021811
Badness (Sintel): 1.10


== 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.
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''
 
Squarschimidt may be described as {{nowrap| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]].  


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma]]: {{monzo| 61 4 -29 }}
[[Comma list]]: 10976/10935, 29360128/29296875


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


[[POTE generator]]: ~98304/78125 = 396.6208
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9006{{c}}, ~1125/896 = 396.6104{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1125/896 = 396.6417{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}


{{Val list|legend=1| 118, 593, 711, 829, 947 }}
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


[[Badness]]: 0.218314
[[Badness]] (Sintel): 3.36


=== 7-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11


[[Comma list]]: 10976/10935, 29360128/29296875
Comma list: 3025/3024, 5632/5625, 10976/10935
 
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
 
Optimal tunings:
* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}
 
{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}
 
Badness (Sintel): 1.26
 
== Leapmonth ==
Leapmonth may be described as the {{nowrap| 63 & 80 }} temperament, generated by a [[3/2|perfect fifth]] and being a strong extension of [[leapfrog]]. It was named by [[Flora Canou]] in 2025 following the pattern demonstrated by ''leapday'' and ''leapweek'', the two simpler extensions of leapfrog.
 
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 -8 1 -20 }}, {{val| 0 29 4 69 }}]
[[Comma list]]: 10976/10935, 51200/50421


{{Multival|legend=1| 29 4 69 -61 28 149 }}
{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}


[[POTE generator]]: ~1125/896 = 396.643
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8005{{c}}, ~3/2 = 704.2543{{c}}
: [[error map]]: {{val| -1.200 +1.100 -0.659 +2.186 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}
: error map: {{val| 0.000 +2.977 +1.093 +5.150 }}


{{Val list|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}


[[Badness]]: 0.132821
[[Badness]] (Sintel): 4.79


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


Comma list: 3025/3024, 5632/5625, 10976/10935
Comma list: 540/539, 896/891, 1331/1323
 
Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}
 
Optimal tunings:
* WE: ~2 = 1198.8679{{c}}, ~3/2 = 704.2911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}
 
Badness (Sintel): 1.88
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 352/351, 364/363, 540/539


Mapping: [{{val| 1 -8 1 -20 -21 }}, {{val| 0 29 4 69 74 }}]
Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}


POTE generator: ~44/35 = 396.644
Optimal tunings:  
* WE: ~2 = 1199.1781{{c}}, ~3/2 = 704.4551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}}


Vals: {{Val list| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}


Badness: 0.038186
Badness (Sintel): 1.53


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 18:12, 29 April 2026

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).

Temperaments discussed elsewhere are:

Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing badness.

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 tunings:

  • WE: ~63/50 = 399.9549 ¢, ~28/27 = 60.5216 ¢
error map: ⟨-0.135 +0.473 +0.241 -0.751]
  • CWE: ~63/50 = 400.0000 ¢, ~28/27 = 60.5162 ¢
error map: ⟨0.000 +0.626 +0.397 -0.567]

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

Badness (Sintel): 1.46

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 tunings:

  • WE: ~44/35 = 400.0359 ¢, ~28/27 = 60.4357 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢

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

Badness (Sintel): 1.67

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 tunings:

  • WE: ~44/35 = 400.0382 ¢, ~28/27 = 60.4342 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4331 ¢

Optimal ET sequence: 60eff, 99ef, 159, 258, 417d

Badness (Sintel): 1.90

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 tunings:

  • WE: ~44/35 = 399.9982 ¢, ~28/27 = 60.4374 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4375 ¢

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

Badness (Sintel): 1.61

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 tunings:

  • WE: ~44/35 = 400.1386 ¢, ~28/27 = 60.3986 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3929 ¢

Optimal ET sequence: 60e, 99e, 159

Badness (Sintel): 1.81

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 tunings:

  • WE: ~44/35 = 400.1115 ¢, ~28/27 = 60.3935 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.3893 ¢

Optimal ET sequence: 60e, 99e, 159

Badness (Sintel): 1.54

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 tunings:

  • WE: ~44/35 = 399.9082 ¢, ~28/27 = 60.4425 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4380 ¢

Optimal ET sequence: 60e, 99ef, 159f

Badness (Sintel): 2.06

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 tunings:

  • WE: ~44/35 = 399.8948 ¢, ~28/27 = 60.4435 ¢
  • CWE: ~44/35 = 400.0000 ¢, ~28/27 = 60.4385 ¢

Optimal ET sequence: 60e, 99ef, 159f

Badness (Sintel): 1.58

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 tunings:

  • WE: ~63/50 = 399.9750 ¢, ~55/54 = 30.2568 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2561 ¢

Optimal ET sequence: 39d, 120cd, 159, 198, 357, 912b

Badness (Sintel): 2.22

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 tunings:

  • WE: ~63/50 = 399.9741 ¢, ~55/54 = 30.2584 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~55/54 = 30.2577 ¢

Optimal ET sequence: 39df, 120cdff, 159, 198, 357, 912b

Badness (Sintel): 1.38

Bisupermajor

For the 5-limit version, see 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 tunings:

  • WE: ~1225/864 = 600.0294 ¢, ~192/175 = 162.8141 ¢
error map: ⟨+0.059 +0.587 -0.208 -0.957]
  • CWE: ~1225/864 = 600.0000 ¢, ~192/175 = 162.8082 ¢
error map: ⟨0.000 +0.510 -0.355 -1.087]

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

Badness (Sintel): 1.66

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 tunings:

  • WE: ~99/70 = 600.1224 ¢, ~11/10 = 162.8065 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~11/10 = 162.7788 ¢

Optimal ET sequence: 22, 74d, 96d, 118, 258e, 376de, 634dee

Badness (Sintel): 1.06

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 tunings:

  • WE: ~567/400 = 600.0497 ¢, ~81/80 = 20.3790 ¢
error map: ⟨+0.099 +0.089 +1.085 -1.756]
  • CWE: ~567/400 = 600.0000 ¢, ~81/80 = 20.3837 ¢
error map: ⟨0.000 -0.037 +0.976 -1.920]

Optimal ET sequence: 58, 118, 294, 412d

Badness (Sintel): 2.13

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 tunings:

  • WE: ~99/70 = 600.0401 ¢, ~81/80 = 20.3913 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~81/80 = 20.3948 ¢

Optimal ET sequence: 58, 118, 294, 412d

Badness (Sintel): 1.01

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 tunings:

  • WE: ~99/70 = 599.8514 ¢, ~66/65 = 20.4215 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~66/65 = 20.4093 ¢

Optimal ET sequence: 58, 118, 176f

Badness (Sintel): 1.09

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 tunings:

  • WE: ~17/12 = 600.0257 ¢, ~66/65 = 20.3789 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~66/65 = 20.3804 ¢

Optimal ET sequence: 58, 118

Badness (Sintel): 1.14

Degrees

This page is about the regular temperament. For scale degrees, see degree.
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 tunings:

  • WE: ~28/27 = 59.9922 ¢, ~3/2 = 702.9233 ¢ (~126/125 = 16.9828 ¢)
error map: ⟨-0.157 +0.812 -0.647 -0.220]
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 702.9324 ¢ (~126/125 = 17.0676 ¢)
error map: ⟨0.000 +0.977 -0.449 -0.029]

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

Badness (Sintel): 2.69

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 tunings:

  • WE: ~28/27 = 59.9929 ¢, ~3/2 = 703.1478 ¢ (~100/99 = 16.7666 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1556 ¢ (~100/99 = 16.8444 ¢)

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

Badness (Sintel): 1.55

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 tunings:

  • WE: ~28/27 = 59.9996 ¢, ~3/2 = 703.0749 ¢ (~100/99 = 16.9197 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.0770 ¢ (~100/99 = 16.9230 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.35

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 tunings:

  • WE: ~28/27 = 60.0058 ¢, ~3/2 = 703.0364 ¢ (~100/99 = 17.0335 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.0061 ¢ (~100/99 = 16.9939 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.17

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 tunings:

  • WE: ~28/27 = 59.9961 ¢, ~3/2 = 703.1523 ¢ (~100/99 = 16.8015 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1777 ¢ (~100/99 = 16.8223 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.27

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 tunings:

  • WE: ~28/27 = 59.9990 ¢, ~3/2 = 703.1804 ¢ (~100/99 = 16.8074 ¢)
  • CWE: ~28/27 = 60.0000 ¢, ~3/2 = 703.1870 ¢ (~100/99 = 16.8130 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.21

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 tunings:

  • WE: ~29/28 = 59.9990 ¢, ~3/2 = 703.1829 ¢ (~100/99 = 16.8055 ¢)
  • CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.1891 ¢ (~100/99 = 16.8109 ¢)

Optimal ET sequence: 60e, 80, 140

Badness (Sintel): 1.13

2.3.5.7.11.13.17.19.23.29.37 subgroup

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 tunings:

  • WE: ~29/28 = 60.0001 ¢, ~3/2 = 703.2183 ¢ (~100/99 = 16.7827 ¢)
  • CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.2178 ¢ (~100/99 = 16.7822 ¢)

Optimal ET sequence: 60el, 80, 140

Badness (Sintel): 1.13

2.3.5.7.11.13.17.19.23.29.37.41 subgroup

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 tunings:

  • WE: ~29/28 = 59.9998 ¢, ~3/2 = 703.2088 ¢ (~100/99 = 16.7882 ¢)
  • CWE: ~29/28 = 60.0000 ¢, ~3/2 = 703.2104 ¢ (~100/99 = 16.7896 ¢)

Optimal ET sequence: 60el, 80, 140

Badness (Sintel): 1.10

Squarschmidt

For the 5-limit version, see Father–3 equivalence continuum #Squarschmidt (5-limit).

Squarschimidt may be described as 118 & 121 temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875. In the 11-limit, it tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 29360128/29296875

Mapping: [⟨1 -8 1 -20], ⟨0 29 4 69]]

Optimal tunings:

  • WE: ~2 = 1199.9006 ¢, ~1125/896 = 396.6104 ¢
error map: ⟨-0.099 +0.543 +0.029 -0.719]
  • CWE: ~2 = 1200.0000 ¢, ~1125/896 = 396.6417 ¢
error map: ⟨0.000 +0.653 +0.253 -0.552]

Optimal ET sequence: 118, 239, 357, 596

Badness (Sintel): 3.36

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 tunings:

  • WE: ~2 = 1199.9005 ¢, ~44/35 = 396.6107 ¢
  • CWE: ~2 = 1200.0000 ¢, ~44/35 = 396.6419 ¢

Optimal ET sequence: 118, 239, 357, 596

Badness (Sintel): 1.26

Leapmonth

Leapmonth may be described as the 63 & 80 temperament, generated by a perfect fifth and being a strong extension of leapfrog. It was named by Flora Canou in 2025 following the pattern demonstrated by leapday and leapweek, the two simpler extensions of leapfrog.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 51200/50421

Mapping: [⟨1 0 -58 -21], ⟨0 1 38 15]]

Optimal tunings:

  • WE: ~2 = 1198.8005 ¢, ~3/2 = 704.2543 ¢
error map: ⟨-1.200 +1.100 -0.659 +2.186]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9318 ¢
error map: ⟨0.000 +2.977 +1.093 +5.150]

Optimal ET sequence: 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd

Badness (Sintel): 4.79

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 1331/1323

Mapping: [⟨1 0 -58 -21 -14], ⟨0 1 38 15 11]]

Optimal tunings:

  • WE: ~2 = 1198.8679 ¢, ~3/2 = 704.2911 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9318 ¢

Optimal ET sequence: 17c, 46c, 63, 80, 223bde, 303bdde

Badness (Sintel): 1.88

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 352/351, 364/363, 540/539

Mapping: [⟨1 0 -58 -21 -14 -1], ⟨0 1 38 15 11 8]]

Optimal tunings:

  • WE: ~2 = 1199.1781 ¢, ~3/2 = 704.4551 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9218 ¢

Optimal ET sequence: 17c, 46c, 63, 80, 143d

Badness (Sintel): 1.53

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