Hemimage temperaments: Difference between revisions

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This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[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}}
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).  
 
Temperaments discussed elsewhere are:  
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
Line 9: Line 12:
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]
* ''[[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
Line 23: Line 29:


{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
{{Mapping|legend=1| 3 4 5 6 | 0 5 13 16 }}
{{Multival|legend=1| 15 39 48 27 34 2 }}
: mapping generators: ~63/50, ~28/27
: mapping generators: ~63/50, ~28/27


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528
[[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 }}


{{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 41: Line 48:
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~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 54: Line 63:
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}
Mapping: {{mapping| 3 4 5 6 6 4 | 0 5 13 16 29 47 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~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 67: Line 78:
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}
Mapping: {{mapping| 3 4 5 6 6 4 10 | 0 5 13 16 29 47 15 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~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 80: Line 93:
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}
Mapping: {{mapping| 3 4 5 6 6 12 | 0 5 13 16 29 -6 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~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 93: Line 108:
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}
Mapping: {{mapping| 3 4 5 6 6 12 10 | 0 5 13 16 29 -6 15 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~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 106: Line 123:
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}
Mapping: {{mapping| 3 4 5 6 6 9 | 0 5 13 16 29 14 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~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 119: Line 138:
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}
Mapping: {{mapping| 3 4 5 6 6 9 10 | 0 5 13 16 29 14 15 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~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 ===
=== Hemichromat ===
Line 132: Line 153:
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}
Mapping: {{mapping| 3 4 5 6 10 | 0 10 26 32 5 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
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=1| 39d, 120cd, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=0| 39d, 120cd, 159, 198, 357, 912b }}


Badness: 0.067173
Badness (Sintel): 2.22


==== 13-limit ====
==== 13-limit ====
Line 145: Line 168:
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}
Mapping: {{mapping| 3 4 5 6 10 8 | 0 10 26 32 5 41 }}


Optimal tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
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=1| 39df, 120cdff, 159, 198, 357, 912b }}
{{Optimal ET sequence|legend=0| 39df, 120cdff, 159, 198, 357, 912b }}


Badness: 0.033420
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
Line 159: Line 184:


{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
: mapping generators: ~1225/864, ~192/175
: mapping generators: ~1225/864, ~192/175


{{Multival|legend=1| 16 -10 34 -53 9 107 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~1225/864 = 600.0294{{c}}, ~192/175 = 162.8141{{c}}
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806
: [[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 177: Line 203:
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}
Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}


Optimal tuning (POTE): ~99/70, ~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


== Bicommatic ==
== Bicommatic ==
Line 191: Line 219:


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


[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~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 209: Line 238:
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}
Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~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 222: Line 253:
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}
Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~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 235: Line 268:
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}
Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~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 ==
{{ See also | 20th-octave temperaments }}
{{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.  
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.  


Line 256: Line 293:


{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
: mapping generators: ~28/27, ~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}})
[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)
: [[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}})
{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}
: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}


[[Badness]]: 0.106471
{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}


Badness (Dirichlet): 2.694
[[Badness]] (Sintel): 2.69


=== 11-limit ===
=== 11-limit ===
Line 276: Line 312:
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}


Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)
Optimal tunings:
 
* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})
{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})


Badness: 0.046770
{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}


Badness (Dirichlet): 1.546
Badness (Sintel): 1.55


=== 13-limit ===
=== 13-limit ===
Line 291: Line 327:
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}


Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)
Optimal tunings:
 
* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})
{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})


Badness: 0.032718
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Dirichlet): 1.352
Badness (Sintel): 1.35


=== 17-limit ===
=== 17-limit ===
Line 306: Line 342:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}


Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
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=1| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Dirichlet): 1.171
Badness (Sintel): 1.17


=== 19-limit ===
=== 19-limit ===
Line 319: Line 357:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}


Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)
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=1| 20cde, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Dirichlet): 1.273
Badness (Sintel): 1.27


=== 23-limit ===
=== 23-limit ===
Line 332: Line 372:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}


Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)
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=1| 20cdei, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Dirichlet): 1.209
Badness (Sintel): 1.21


=== 29-limit ===
=== 29-limit ===
Line 345: Line 387:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}


Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)
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=1| 20cdeij, 60e, 80, 140 }}
{{Optimal ET sequence|legend=0| 60e, 80, 140 }}


Badness (Dirichlet): 1.134
Badness (Sintel): 1.13


=== no-31's 37-limit ===
=== 2.3.5.7.11.13.17.19.23.29.37 subgroup ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.37
Subgroup: 2.3.5.7.11.13.17.19.23.29.37


Line 358: Line 402:
Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 9 | 0 1 2 3 3 0 1 0 2 3 3 }}
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 tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222 (~100/99 = 16.778)
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=1| 20cdeijl, 60el, 80, 140 }}
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}


Badness (Dirichlet): 1.127
Badness (Sintel): 1.13


=== no-31's 41-limit ===
=== 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
Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41


Line 371: Line 417:
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 }}
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 }}


Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.207
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}})


{{Optimal ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
{{Optimal ET sequence|legend=0| 60el, 80, 140 }}


Badness (Dirichlet): 1.100
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)]].''
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 61 4 -29 }}
 
{{Mapping|legend=1| 1 -8 1 | 0 29 4 }}
 
: mapping generators: ~2, ~98304/78125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621
 
{{Optimal ET sequence|legend=1| 118, 593, 711, 829, 947 }}


[[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


Line 401: Line 436:
{{Mapping|legend=1| 1 -8 1 -20 | 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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 416: Line 453:
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}
Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 (Sintel): 1.26


Badness: 0.038186
== 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.  


== Subfourth ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 10976/10935, 65536/64827
[[Comma list]]: 10976/10935, 51200/50421
 
{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}
 
: mapping generators: ~2, ~21/16


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}
Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~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: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}
Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 temperaments| ]] <!-- main article -->
[[Category:Hemimage temperaments| ]] <!-- main article -->
[[Category:Hemimage| ]] <!-- key article -->
[[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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