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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, and bisupermajor, considered below, as well as the following discussed elsewhere:  
{{Technical data page}}
* ''[[Quasisuper]]'' → [[Archytas clan #Quasisuper|Archytas clan]] (+64/63)
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]]'' → [[Meantone family #Liese|Meantone family]] (+81/80)
 
* ''[[Unicorn]]'' → [[Unicorn family #Septimal unicorn|Unicorn family]] (+126/125)
Temperaments discussed elsewhere are:  
* [[Magic]] → [[Magic family #Magic|Magic family]] (+225/224 or 245/243)
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Guiron]]'' → [[Gamelismic clan #Guiron|Gamelismic clan]] (+1029/1024)
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Echidna]]'' → [[Diaschismic family #Echidna|Diaschismic family]] (+1728/1715 or 2048/2025)
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* [[Hemififths]] → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]] (+2401/2400 or 5120/5103)
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[Dodecacot]]'' → [[Tetracot family #Dodecacot|Tetracot family]] (+3125/3087)
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* [[Parakleismic]] → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]] (+3136/3125 or 4375/4374)
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[Pluto]]'' → [[Mirkwai clan #Pluto|Mirkwai clan]] (+4000/3969)
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Hendecatonic]]'' → [[Porwell temperaments #Hendecatonic|Porwell temperaments]] (+6144/6125)
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* ''[[Marfifths]]'' → [[Kleismic family #Marfifths|Kleismic family]] (+15625/15552)
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Cotoneum]]'' → [[Garischismic clan #Cotoneum|Garischismic clan]] (+33554432/33480783)
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Yarman I]]' → [[Turkish maqam music temperaments #Yarman I|Turkish maqam music temperaments]] (+244140625/243045684)
* ''[[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]]
 
Considered below are chromat, degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].


== Chromat ==
== 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.
The chromat temperament has a period of 1/3 octave and tempers out the hemimage (10976/10935) and the triwellisma (235298/234375). It is also described as an [[amity family|amity extension]] with third-octave period.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 235298/234375
[[Comma list]]: 10976/10935, 235298/234375


[[Mapping]]: [{{val| 3 4 5 6 }}, {{val| 0 5 13 16 }}]
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
: mapping generators: ~63/50, ~28/27


{{Multival|legend=1| 15 39 48 27 34 2 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~63/50 = 399.9549{{c}}, ~28/27 = 60.5216{{c}}
Mapping generators: ~63/50, ~28/27
: [[error map]]: {{val| -0.135 +0.473 +0.241 -0.751 }}
 
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~28/27 = 60.5162{{c}}
[[POTE generator]]: ~28/27 = 60.528
: error map: {{val| 0.000 +0.626 +0.397 -0.567 }}


{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}


[[Badness]]: 0.057499
[[Badness]] (Sintel): 1.46


=== 11-limit ===
=== 11-limit ===
Line 39: Line 46:
Comma list: 441/440, 4375/4356, 10976/10935
Comma list: 441/440, 4375/4356, 10976/10935


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


POTE generator: ~28/27 = 60.430
Optimal tunings:  
* WE: ~44/35 = 400.0359{{c}}, ~28/27 = 60.4357{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159, 258 }}


Badness: 0.050379
Badness (Sintel): 1.67


==== 13-limit ====
==== 13-limit ====
Line 52: Line 61:
Comma list: 364/363, 441/440, 625/624, 10976/10935
Comma list: 364/363, 441/440, 625/624, 10976/10935


Mapping: [{{val| 3 4 5 6 6 4 }}, {{val| 0 5 13 16 29 47 }}]
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


POTE generator: ~28/27 = 60.428
Optimal tunings:  
* WE: ~44/35 = 400.0382{{c}}, ~28/27 = 60.4342{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4331{{c}}


{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417d }}
{{Optimal ET sequence|legend=0| 60eff, 99ef, 159, 258, 417d }}


Badness: 0.046006
Badness (Sintel): 1.90


===== 17-limit =====
===== 17-limit =====
Line 65: Line 76:
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757


Mapping: [{{val| 3 4 5 6 6 4 10 }}, {{val| 0 5 13 16 29 47 15 }}]
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


POTE generator: ~28/27 = 60.438
Optimal tunings:  
* WE: ~44/35 = 399.9982{{c}}, ~28/27 = 60.4374{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4375{{c}}


{{Optimal ET sequence|legend=1| 99ef, 159, 258, 417dg }}
{{Optimal ET sequence|legend=0| 99ef, 159, 258, 417dg }}


Badness: 0.031678
Badness (Sintel): 1.61


==== Catachrome ====
==== Catachrome ====
Line 78: Line 91:
Comma list: 325/324, 441/440, 1001/1000, 10976/10935
Comma list: 325/324, 441/440, 1001/1000, 10976/10935


Mapping: [{{val| 3 4 5 6 6 12 }}, {{val| 0 5 13 16 29 -6 }}]
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


POTE generator: ~28/27 = 60.378
Optimal tunings:  
* WE: ~44/35 = 400.1386{{c}}, ~28/27 = 60.3986{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3929{{c}}


{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness: 0.043844
Badness (Sintel): 1.81


===== 17-limit =====
===== 17-limit =====
Line 91: Line 106:
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913


Mapping: [{{val| 3 4 5 6 6 12 10 }}, {{val| 0 5 13 16 29 -6 15 }}]
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


POTE generator: ~28/27 = 60.377
Optimal tunings:  
* WE: ~44/35 = 400.1115{{c}}, ~28/27 = 60.3935{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.3893{{c}}


{{Optimal ET sequence|legend=1| 60e, 99e, 159 }}
{{Optimal ET sequence|legend=0| 60e, 99e, 159 }}


Badness: 0.030218
Badness (Sintel): 1.54


==== Chromic ====
==== Chromic ====
Line 104: Line 121:
Comma list: 196/195, 352/351, 729/728, 1875/1859
Comma list: 196/195, 352/351, 729/728, 1875/1859


Mapping: [{{val| 3 4 5 6 6 9 }}, {{val| 0 5 13 16 29 14 }}]
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


POTE generator: ~27/26 = 60.456
Optimal tunings:  
* WE: ~44/35 = 399.9082{{c}}, ~28/27 = 60.4425{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4380{{c}}


{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


Badness: 0.049857
Badness (Sintel): 2.06


===== 17-limit =====
===== 17-limit =====
Line 117: Line 136:
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
Comma list: 170/169, 196/195, 352/351, 375/374, 595/594


Mapping: [{{val| 3 4 5 6 6 9 10 }}, {{val| 0 5 13 16 29 14 15 }}]
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


POTE generator: ~27/26 = 60.459
Optimal tunings:  
* WE: ~44/35 = 399.8948{{c}}, ~28/27 = 60.4435{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~28/27 = 60.4385{{c}}


{{Optimal ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
{{Optimal ET sequence|legend=0| 60e, 99ef, 159f }}


Badness: 0.031043
Badness (Sintel): 1.58
 
=== Hemichromat ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 10976/10935, 102487/102400
 
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9750{{c}}, ~55/54 = 30.2568{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2561{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
 
Optimal tunings:
* WE: ~63/50 = 399.9741{{c}}, ~55/54 = 30.2584{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~55/54 = 30.2577{{c}}
 
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}
 
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
{{Multival|legend=1| 16 -10 34 -53 9 107 }}


[[POTE generator]]: ~192/175 = 162.806
[[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 }}
{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}


[[Badness]]: 0.065492
[[Badness]] (Sintel): 1.66


=== 11-limit ===
=== 11-limit ===
Line 147: Line 201:
Comma list: 385/384, 3388/3375, 9801/9800
Comma list: 385/384, 3388/3375, 9801/9800


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


POTE generators: ~11/10 = 162.773
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=1| 22, 74d, 96d, 118, 258e, 376de }}
{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}


Badness: 0.032080
Badness (Sintel): 1.06


== Commatic ==
== Bicommatic ==
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.
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, 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 }}
[[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 }}


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


[[Badness]]: 0.084317
[[Badness]] (Sintel): 2.13


=== 11-limit ===
=== 11-limit ===
Line 177: Line 236:
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 }}


POTE generator: ~81/80 = 20.390
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=1| 58, 118, 294, 412d }}
{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}


Badness: 0.030461
Badness (Sintel): 1.01


=== 13-limit ===
=== 13-limit ===
Line 190: Line 251:
Comma list: 196/195, 352/351, 729/728, 1001/1000
Comma list: 196/195, 352/351, 729/728, 1001/1000


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


POTE generator: ~66/65 = 20.427
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=1| 58, 118, 176f }}
{{Optimal ET sequence|legend=0| 58, 118, 176f }}


Badness: 0.026336
Badness (Sintel): 1.09


=== 17-limit ===
=== 17-limit ===
Line 203: Line 266:
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560
Comma list: 170/169, 196/195, 289/288, 352/351, 561/560


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


POTE generator: ~66/65 = 20.378
Optimal tunings:  
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}


{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}
{{Optimal ET sequence|legend=0| 58, 118 }}


Badness: 0.022396
Badness (Sintel): 1.14


== Degrees ==
== 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.
{{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 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.
 
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
[[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 }}


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


{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b, 920b }}
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


[[Badness]]: 0.106471
[[Badness]] (Sintel): 2.69


=== 11-limit ===
=== 11-limit ===
Line 233: Line 310:
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}})


{{Optimal ET sequence|legend=1| 60e, 80, 140, 360, 500be, 860bde }}
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}


Badness: 0.046770
Badness (Sintel): 1.55


=== 13-limit ===
=== 13-limit ===
Line 246: Line 325:
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 }}
 
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


POTE generator: ~3/2 = 703.080
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


{{Optimal ET sequence|legend=1| 60e, 80, 140, 500be, 640be, 780be }}
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: 0.032718
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


== Squarschmidt ==
Badness (Sintel): 1.17
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.
 
=== 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
 
=== 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: {{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{{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}})
 
{{Optimal ET sequence|legend=0| 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


Subgroup: 2.3.5
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 }}


[[Comma]]: {{monzo| 61 4 -29 }}
Optimal tunings:  
* WE: ~29/28 = 59.9998{{c}}, ~3/2 = 703.2088{{c}} (~100/99 = 16.7882{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.2104{{c}} (~100/99 = 16.7896{{c}})


[[Mapping]]: [{{val| 1 -8 1 }}, {{val| 0 29 4 }}]
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}


[[POTE generator]]: ~98304/78125 = 396.621
Badness (Sintel): 1.10


{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}
== Squarschmidt ==
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''


[[Badness]]: 0.218314
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]].  


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 10976/10935, 29360128/29296875
[[Comma list]]: 10976/10935, 29360128/29296875


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


{{Multival|legend=1| 29 4 69 -61 28 149 }}
[[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 }}


[[POTE generator]]: ~1125/896 = 396.643
{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}
 
{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}


[[Badness]]: 0.132821
[[Badness]] (Sintel): 3.36


=== 11-limit ===
=== 11-limit ===
Line 289: Line 451:
Comma list: 3025/3024, 5632/5625, 10976/10935
Comma list: 3025/3024, 5632/5625, 10976/10935


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


POTE generator: ~44/35 = 396.644
Optimal 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=1| 118, 239, 357, 596 }}
{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}


Badness: 0.038186
Badness (Sintel): 1.26


== Subfourth ==
== Leapmonth ==
Subgroup: 2.3.5.7
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, 65536/64827
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 0 17 4 }}, {{val| 0 4 -37 -3 }}]
[[Comma list]]: 10976/10935, 51200/50421


{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}
{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}


[[POTE generator]]: ~21/16 = 475.991
[[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| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}


[[Badness]]: 0.140722
[[Badness]] (Sintel): 4.79


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


Comma list: 540/539, 896/891, 12005/11979
Comma list: 540/539, 896/891, 1331/1323


Mapping: [{{val| 1 0 17 4 11 }}, {{val| 0 4 -37 -3 -19 }}]
Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}


POTE generator: ~21/16 = 475.995
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=1| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}


Badness: 0.045323
Badness (Sintel): 1.88


=== 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: 169/168, 352/351, 364/363, 540/539


Mapping: [{{val| 1 0 17 4 11 16 }}, {{val| 0 4 -37 -3 -19 -31 }}]
Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}


POTE generator: ~21/16 = 475.996
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=1| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}


Badness: 0.023800
Badness (Sintel): 1.53


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Hemimage]]
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimage_temperaments"