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'''Hemimage temperaments''' temper out the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935.  
{{Technical data page}}
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).  


Discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* [[Archytas clan #quasisuper|Quasisuper]]
* [[Quasisuper]] (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* [[Meantone family #Liese|Liese]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* [[Unicorn family #Alicorn|Alicorn]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* [[Magic family #magic|Magic]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* [[Schismatic family #Guiron|Guiron]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* [[Diaschismic family #Echidna|Echidna]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* [[Breedsmic temperaments #hemififths|Hemififths]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* [[Ragismic microtemperaments #Parakleismic|Parakleismic]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* [[Mirkwai clan #Pluto|Pluto]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* [[Porwell temperaments #Hendecatonic|Hendecatonic]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* [[Turkish maqam music temperaments|Yarman]]
* ''[[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]]


= Commatic =
Considered below are degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].
Subgroup: 2.3.5.7
 
== Bisupermajor ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 65625/65536
 
{{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


[[Comma list]]: 10976/10935, 50421/50000
[[Comma list]]: 10976/10935, 50421/50000


[[Mapping]]: [{{val| 2 3 4 5 }}, {{val| 0 5 19 18 }}]
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
 
: mapping generators: ~567/400, ~81/80
{{Multival|legend=1| 10 38 36 37 29 -23 }}


[[POTE generator]]: ~81/80 = 20.377
[[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| 58, 118, 294, 412d, 530d }}
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


[[Badness]]: 0.0843
[[Badness]] (Sintel): 2.13


== 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, 3388/3375, 8019/8000


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


POTE generator: ~81/80 = 20.390
Comma list: 196/195, 352/351, 729/728, 1001/1000


Vals: {{Val list| 58, 118, 294, 412d }}
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


Badness: 0.0305
Optimal tunings:  
* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}


= Chromat =
{{Optimal ET sequence|legend=0| 58, 118, 176f }}
{{see also|Amity family}}


Subgroup: 2.3.5.7
Badness (Sintel): 1.09


[[Comma list]]: 10976/10935, 235298/234375
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560


{{Multival|legend=1| 15 39 48 27 34 2 }}
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


[[POTE generator]]: ~28/27 = 60.528
Optimal tunings:  
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}


{{Val list|legend=1| 60, 99, 258, 357, 456 }}
{{Optimal ET sequence|legend=0| 58, 118 }}


[[Badness]]: 0.0575
Badness (Sintel): 1.14


= Degrees =
== Degrees ==
Subgroup: 2.3.5.7
{{About|the regular temperament|scale degrees|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]] ({{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.
 
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]].
 
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
[[Comma list]]: 10976/10935, 390625/388962


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


{{Multival|legend=1| 20 40 60 17 39 27 }}
[[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 }}


[[POTE generator]]: ~3/2 = 703.015
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


{{Val list|legend=1| 60, 80, 140, 640b, 780b, 920b }}
[[Badness]] (Sintel): 2.69


[[Badness]]: 0.1065
=== 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: 1331/1323, 1375/1372, 2200/2187


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


POTE generator: ~3/2 = 703.231
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| 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}


Badness: 0.0468
Badness (Sintel): 1.55


== 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: 325/324, 352/351, 1001/1000, 1331/1323


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


POTE generator: ~3/2 = 703.080
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}})


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


Badness: 0.0327
Badness (Sintel): 1.35


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


[[Comma list]]: 10976/10935, 65536/64827
Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


[[POTE generator]]: ~21/16 = 475.991
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}})


{{Val list|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


[[Badness]]: 0.1407
Badness (Sintel): 1.17


== 11-limit ==
=== 19-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17.19
 
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 tunings:
* 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: 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 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}})
 
{{Optimal ET sequence|legend=0| 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: {{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{{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}})
 
{{Optimal ET sequence|legend=0| 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 {{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]].


Comma list: 540/539, 896/891, 12005/11979
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
[[Comma list]]: 10976/10935, 29360128/29296875


POTE generator: ~21/16 = 475.995
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}


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


Badness: 0.0453
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}


== 13-limit ==
[[Badness]] (Sintel): 3.36
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
Comma list: 3025/3024, 5632/5625, 10976/10935


POTE generator: ~21/16 = 475.996
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


Vals: {{Val list| 58, 121, 179ef, 300bdef }}
Optimal tunings:  
* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}


Badness: 0.0238
{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}


= Bisupermajor =
Badness (Sintel): 1.26
{{see also| Very high accuracy temperaments #Kwazy }}


Subgroup: 2.3.5.7
== 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 list]]: 10976/10935, 65625/65536
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 2 1 6 1 }}, {{val| 0 8 -5 17 }}]
[[Comma list]]: 10976/10935, 51200/50421


[[POTE generator]]: ~192/175 = 162.8061
{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}


{{Val list|legend=1| 22, 74d, 96d, 118, 140, 258, 398 }}
[[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]]: 0.0655
{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}


== 11-limit ==
[[Badness]] (Sintel): 4.79


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


Comma list: 385/384, 3388/3375, 9801/9800
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| 2 1 6 1 8 }}, {{val| 0 8 -5 17 -4 }}]
Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}


POTE generators: ~11/10 = 162.7733
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| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}


Badness: 0.0321
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]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"