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Hemimage temperaments: Difference between revisions

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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 [[Archytas clan #quasisuper|quasisuper]], [[Meantone family #Liese|liese]], [[Marvel temperaments #Pluto|pluto]], [[Magic family #magic|magic]], [[Diaschismic family #Echidna|echidna]], [[Breedsmic temperaments #hemififth|hemififth]], [[Ragismic microtemperaments #Parakleismic|parakleismic]], [[Porwell temperaments #Hendecatonic|hendecatonic]], [[Schismatic family #Guiron|guiron]], and [[Unicorn family #Alicorn|alicorn]].
Temperaments discussed elsewhere are:
* [[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]]


= Commatic =
Considered below are degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].
Commas: 10976/10935, 50421/50000


POTE generator: ~81/80 = 20.377
== Bisupermajor ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


Map: [<2 3 4 5|, <0 5 19 18|]
[[Subgroup]]: 2.3.5.7


Wedgie: <<10 38 36 37 29 -23||
[[Comma list]]: 10976/10935, 65625/65536


EDOs: 58, 118, 294, 412d, 530d
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
: mapping generators: ~1225/864, ~192/175


Badness: 0.0843
[[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}
Commas: 441/440, 3388/3375, 8019/8000


POTE generator: ~81/80 = 20.390
[[Badness]] (Sintel): 1.66


Map: [<2 3 4 5 6|, <0 5 19 18 27|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 58, 118, 294, 412d
Comma list: 385/384, 3388/3375, 9801/9800


Badness: 0.0305
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}


= Chromat =
Optimal tunings:
{{see also|Amity family}}
* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}


Commas: 10976/10935, 235298/234375
{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}


POTE generator: ~28/27 = 60.528
Badness (Sintel): 1.06


Map: [<3 4 5 6|, <0 5 13 16|]
== 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.


Wedgie: <<15 39 48 27 34 2||
[[Subgroup]]: 2.3.5.7


EDOs: 60, 99, 258, 357, 456
[[Comma list]]: 10976/10935, 50421/50000


Badness: 0.0575
{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
: mapping generators: ~567/400, ~81/80


= Degrees =
[[Optimal tuning]]s:
Commas: 10976/10935, 390625/388962
* [[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 }}


POTE generator: ~3/2 = 703.015
{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}


Map: [<20 0 -17 -39|, <0 1 2 3|]
[[Badness]] (Sintel): 2.13


Wedgie: <<20 40 60 17 39 27||
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 60, 80, 140, 640b, 780b, 920b
Comma list: 441/440, 3388/3375, 8019/8000


Badness: 0.1065
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}


== 11-limit ==
Optimal tunings:
Commas: 1331/1323, 1375/1372, 2200/2187
* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}


POTE generator: ~3/2 = 703.231
{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}


Map: [<20 0 -17 -39 -26|, <0 1 2 3 3|]
Badness (Sintel): 1.01


EDOs: 80, 140, 360, 500be, 860bde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0468
Comma list: 196/195, 352/351, 729/728, 1001/1000


== 13-limit ==
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
Commas: 325/324, 352/351, 1001/1000, 1331/1323


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}}


Map: [<20 0 -17 -39 -26 74|, <0 1 2 3 3 0|]
{{Optimal ET sequence|legend=0| 58, 118, 176f }}


EDOs: 60e, 80, 140, 500be, 640be, 780be
Badness (Sintel): 1.09


Badness: 0.0327
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


= Subfourth =
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
Commas: 10976/10935, 65536/64827


POTE generator: ~21/16 = 475.991
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


Map: [<1 0 17 4|, <0 4 -37 -3|]
Optimal tunings:  
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}


EDOs: 58, 121, 179, 300bd, 479bcd
{{Optimal ET sequence|legend=0| 58, 118 }}


Badness: 0.1407
Badness (Sintel): 1.14


== 11-limit ==
== Degrees ==
Commas: 540/539, 896/891, 12005/11979
{{About|the regular temperament|scale degrees|degree}}
{{See also| 20th-octave temperaments }}


POTE generator: ~21/16 = 475.995
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.  


Map: [<1 0 17 4 11|, <0 4 -37 -3 -19|]
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.


EDOs: 58, 121, 179e, 300bde
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]].


Badness: 0.0453
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]].


== 13-limit ==
[[Subgroup]]: 2.3.5.7
Commas: 352/351, 364/363, 540/539, 676/675


POTE generator: ~21/16 = 475.996
[[Comma list]]: 10976/10935, 390625/388962


Map: [<1 0 17 4 11 16|, <0 4 -37 -3 -19 -31|]
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
: mapping generators: ~28/27, ~3


EDOs: 58, 121, 179ef, 300bdef
[[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 }}


Badness: 0.0238
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


= Bisupermajor =
[[Badness]] (Sintel): 2.69
{{see also| Very high accuracy temperaments #Kwazy }}


Commas: 10976/10935, 65625/65536
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~192/175 = 162.8061
Comma list: 1331/1323, 1375/1372, 2200/2187


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


EDOs: 22, 74d, 96d, 118, 140, 258, 398
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}})


Badness: 0.0655
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}


== 11-limit ==
Badness (Sintel): 1.55
Commas: 385/384, 3388/3375, 9801/9800


POTE generators: ~11/10 = 162.7733
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [<2 1 6 1 8|, <0 8 -5 17 -4|]
Comma list: 325/324, 352/351, 1001/1000, 1331/1323


EDOs: 22, 74d, 96d, 118, 258e, 376de
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


Badness: 0.0321
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}})


[[Category:Theory]]
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
[[Category:Temperament]]
 
[[Category:Hemimage]]
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 }}
 
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}})
 
{{Optimal ET sequence|legend=0| 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: {{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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 10976/10935, 29360128/29296875
 
{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{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
 
[[Comma list]]: 10976/10935, 51200/50421
 
{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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: {{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: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}
 
Optimal tunings:
* WE: ~2 = 1199.1781{{c}}, ~3/2 = 704.4551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}
 
Badness (Sintel): 1.53
 
[[Category:Temperament collections]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"