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]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat|Amity family]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]


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


[[Comma list]]: 10976/10935, 50421/50000
== Bisupermajor ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 10 38 36 37 29 -23 }}
[[Comma list]]: 10976/10935, 65625/65536


[[POTE generator]]: ~81/80 = 20.377
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
: mapping generators: ~1225/864, ~192/175


{{Val list|legend=1| 58, 118, 294, 412d, 530d }}
[[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 }}


[[Badness]]: 0.084317
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
 
[[Badness]] (Sintel): 1.66


=== 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: 385/384, 3388/3375, 9801/9800


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


POTE generator: ~81/80 = 20.390
Optimal tunings:  
* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}


Vals: {{Val list| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}


Badness: 0.030461
Badness (Sintel): 1.06


== Chromat ==
== Bicommatic ==
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.
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, 235298/234375
[[Comma list]]: 10976/10935, 50421/50000
 
[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
 
{{Multival|legend=1| 15 39 48 27 34 2 }}
 
[[POTE generator]]: ~28/27 = 60.528
 
{{Val list|legend=1| 39d, 60, 99, 258, 357, 456 }}
 
[[Badness]]: 0.057499
 
== Degrees ==
Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 10976/10935, 390625/388962
 
[[Mapping]]: [{{val| 20 0 -17 -39 }}, {{val| 0 1 2 3 }}]


{{Multival|legend=1| 20 40 60 17 39 27 }}
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
: mapping generators: ~567/400, ~81/80


[[POTE generator]]: ~3/2 = 703.015
[[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 }}


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


[[Badness]]: 0.106471
[[Badness]] (Sintel): 2.13


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


Comma list: 1331/1323, 1375/1372, 2200/2187
Comma list: 441/440, 3388/3375, 8019/8000


Mapping: [{{val| 20 0 -17 -39 -26 }}, {{val| 0 1 2 3 3 }}]
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}


POTE generator: ~3/2 = 703.231
Optimal tunings:  
* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}


Vals: {{Val list| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}


Badness: 0.046770
Badness (Sintel): 1.01


=== 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: 196/195, 352/351, 729/728, 1001/1000


Mapping: [{{val| 20 0 -17 -39 -26 74 }}, {{val| 0 1 2 3 3 0 }}]
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


POTE generator: ~3/2 = 703.080
Optimal tunings:  
* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}


Vals: {{Val list| 60e, 80, 140, 500be, 640be, 780be }}
{{Optimal ET sequence|legend=0| 58, 118, 176f }}


Badness: 0.032718
Badness (Sintel): 1.09


== Subfourth ==
=== 17-limit ===
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7.11.13.17


[[Comma list]]: 10976/10935, 65536/64827
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
Optimal tunings:
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}


[[POTE generator]]: ~21/16 = 475.991
{{Optimal ET sequence|legend=0| 58, 118 }}


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
Badness (Sintel): 1.14


[[Badness]]: 0.140722
== Degrees ==
{{About|the regular temperament|scale degrees|degree}}
{{See also| 20th-octave temperaments }}


=== 11-limit ===
Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.  
Subgroup: 2.3.5.7.11


Comma list: 540/539, 896/891, 12005/11979
An obvious extension to the 23-limit exists by mapping [[23/20]] to 4\20 (1\5), [[69/56]] to 6\20 (3\10), and [[23/18]] to 7\20. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by mapping [[29/22]] to 2\5, we get a uniquely elegant extension to the 29-limit which tempers out [[726/725]], which is the difference between [[33/25]] and [[29/22]], as well as [[784/783]] ({{S|28}}) and [[841/840]] ({{S|29}}). 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.


Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
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 for prime [[37/1|37]] agreeing with many [[semiconvergent]]s, tempering out [[481/480]]. By mapping [[60/41]] and [[41/28]] to 11\20 or equivalently [[56/41]] and [[41/30]] to 9\20 and by mapping [[44/41]] to 1\10 (among many other equivalences), there is a very efficient extension for prime [[41/1|41]] tempering out [[451/450]].


POTE generator: ~21/16 = 475.995
The 80-note generator chain is ideal, so [[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]].


Vals: {{Val list| 58, 121, 179e, 300bde }}
[[Subgroup]]: 2.3.5.7


Badness: 0.045323
[[Comma list]]: 10976/10935, 390625/388962
 
=== 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
 
Vals: {{Val list| 58, 121, 179ef, 300bdef }}
 
Badness: 0.023800
 
== Bisupermajor ==
{{see also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
: mapping generators: ~28/27, ~3


[[Comma list]]: 10976/10935, 65625/65536
[[Optimal tuning]]s:  
* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})
: [[error map]]: {{val| -0.157 +0.812 -0.647 -0.220 }}
* [[CWE]]: ~28/27 = 60.0000{{c}}, ~3/2 = 702.9324{{c}} (~126/125 = 17.0676{{c}})
: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}


[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
[[Badness]] (Sintel): 2.69
 
[[POTE generator]]: ~192/175 = 162.8061
 
{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
 
[[Badness]]: 0.065492


=== 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: 1331/1323, 1375/1372, 2200/2187


Mapping: [{{val| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


POTE generators: ~11/10 = 162.7733
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}})


Vals: {{Val list| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}


Badness: 0.032080
Badness (Sintel): 1.55


== Cotoneum ==
=== 13-limit ===
{{Main| Cotoneum }}
Subgroup: 2.3.5.7.11.13


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.
Comma list: 325/324, 352/351, 1001/1000, 1331/1323


Subgroup: 2.3.5.7
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


[[Comma list]]: 10976/10935, 823543/819200
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}})


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


{{Multival|legend=1| 1 -49 -14 -80 -25 105 }}
Badness (Sintel): 1.35


[[POTE generator]]: ~3/2 = 702.317
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Minimax tuning]]:  
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000
* 7-odd-limit: ~3/2 = {{Monzo| 3/5 1/50 -1/50 }}
: [{{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]]:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}
* 7-odd-limit [[diamond monotone]]: ~3/2 = [701.5385, 702.8571] (38\65 to 41\70)
* 9-odd-limit diamond monotone: ~3/2 = [701.8868, 702.8571] (31\53 to 41\70)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.9550, 702.3575]
* 7- and 9-odd-limit diamond monotone and tradeoff: ~3/2 = [701.9550, 702.3575]


{{Val list|legend=1| 41, 135c, 176, 217, 258, 475 }}
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}})


[[Badness]]: 0.105632
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


=== 11-limit ===
Badness (Sintel): 1.17
Subgroup: 2.3.5.7.11


Comma list: 441/440, 10976/10935, 16384/16335
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val|1 2 -18 -3 13}}, {{val|0 -1 49 14 -23}}]
Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475


POTE generator: ~3/2 = 702.303
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}


Minimax tuning:  
Optimal tunings:  
* 11-odd-limit: ~3/2 = {{Monzo| 41/72 0 -1/72 0 1/72 }}
* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})
: Eigenmonzos (unchanged intervals): 2, 11/10
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})


Tuning ranges:
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
* 11-odd-limit diamond monotone: ~3/2 = [702.1277, 702.4390] (55\94 to 24\41)
* 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 }}
Badness (Sintel): 1.27


Badness: 0.050966
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


=== 13-limit ===
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399
Subgroup: 2.3.5.7.11.13


Comma list: 364/363, 441/440, 3584/3575, 10976/10935
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


Mapping: [{{val|1 2 -18 -3 13 29}}, {{val|0 -1 49 14 -23 -61}}]
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}})


POTE generator: ~3/2 = 702.306
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Minimax tuning:
Badness (Sintel): 1.21
* 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:
=== 29-limit ===
* 13- and 15-odd-limit diamond monotone: ~3/2 = [702.2222, 702.4390] (79\135 to 24\41)
Subgroup: 2.3.5.7.11.13.17.19.23.29
* 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 }}
Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405


Badness: 0.036951
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


=== 17-limit ===
Optimal tunings:
Subgroup: 2.3.5.7.11.13.17
* 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}})


Comma list: 364/363, 441/440, 595/594, 3584/3575, 8281/8262
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Mapping: [{{val|1 2 -18 -3 13 29 41}}, {{val|0 -1 49 14 -23 -61 -89}}]
Badness (Sintel): 1.13


POTE generator: ~3/2 = 702.307
== Squarschmidt ==
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''


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


Tuning ranges:
[[Subgroup]]: 2.3.5.7
* 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 }}
[[Comma list]]: 10976/10935, 29360128/29296875


Badness: 0.029495
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}


=== 19-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13.17.19
* [[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 }}


Comma list: 343/342, 364/363, 441/440, 595/594, 1216/1215, 1729/1728
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


Mapping: [{{val|1 2 -18 -3 13 29 41 -14}}, {{val|0 -1 49 14 -23 -61 -89 44}}]
[[Badness]] (Sintel): 3.36


POTE generator: ~3/2 = 702.308
=== 11-limit ===
Subgroup: 2.3.5.7.11


Minimax tuning:  
Comma list: 3025/3024, 5632/5625, 10976/10935
* 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:
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
* 19- and 21-odd-limit diamond monotone: ~3/2 = [702.2727, 702.4390] (103\176 to 24\41)
* 19- and 21-odd-limit diamond tradeoff: ~3/2 = [701.9550, 702.3771]
* 19- and 21-odd-limit diamond monotone and tradeoff: ~3/2 = [702.2727, 702.3771]


Vals: {{Val list| 41, 176, 217 }}
Optimal tunings:  
* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}


Badness: 0.021811
{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}


== Squarschmidt ==
Badness (Sintel): 1.26
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
== 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.  


[[Comma]]: {{monzo| 61 4 -29 }}
[[Subgroup]]: 2.3.5.7


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


[[POTE generator]]: ~98304/78125 = 396.6208
{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}


{{Val list|legend=1| 118, 593, 711, 829, 947 }}
[[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 }}


[[Badness]]: 0.218314
{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}


=== 7-limit ===
[[Badness]] (Sintel): 4.79
Subgroup: 2.3.5.7


[[Comma list]]: 10976/10935, 29360128/29296875
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[Mapping]]: [{{val| 1 -8 1 -20 }}, {{val| 0 29 4 69 }}]
Comma list: 540/539, 896/891, 1331/1323


{{Multival|legend=1| 29 4 69 -61 28 149 }}
Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}


[[POTE generator]]: ~1125/896 = 396.643
Optimal tunings:  
* WE: ~2 = 1198.8679{{c}}, ~3/2 = 704.2911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}


{{Val list|legend=1| 118, 239, 357, 596, 1549bd }}
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}


[[Badness]]: 0.132821
Badness (Sintel): 1.88


=== 11-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 3025/3024, 5632/5625, 10976/10935
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]]

Latest revision as of 15:12, 3 June 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 degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing badness.

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 mapping 23/20 to 4\20 (1\5), 69/56 to 6\20 (3\10), and 23/18 to 7\20. By observing that 1\20 works as 30/29~29/28~28/27, with 29/28 being especially accurate, and by mapping 29/22 to 2\5, we get a uniquely elegant extension to the 29-limit which tempers out 726/725, which is the difference between 33/25 and 29/22, as well as 784/783 (S28) and 841/840 (S29). 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 for prime 37 agreeing with many semiconvergents, tempering out 481/480. By mapping 60/41 and 41/28 to 11\20 or equivalently 56/41 and 41/30 to 9\20 and by mapping 44/41 to 1\10 (among many other equivalences), there is a very efficient extension for prime 41 tempering out 451/450.

The 80-note generator chain is ideal, so 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

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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