@@ Line 1: / Line 1: @@
-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}}
-* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
+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]]
+* ''[[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]]
-* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn 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]]
-* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
 * ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
-* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
+* ''[[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]]
-* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]
+* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat|Amity family]]
-* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
 * ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
-* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
 * ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]
-== Chromat ==
+Considered below are degrees, bicommatic, bisupermajor, squarschmidt, and leapmonth, in the order of increasing [[badness]].
-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
-[[Comma list]]: 10976/10935, 235298/234375
-{{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
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528
-{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}
-[[Badness]]: 0.057499
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 441/440, 4375/4356, 10976/10935
-Mapping: {{mapping| 3 4 5 6 6 | 0 5 13 16 29 }}
-Optimal tuning (POTE): ~44/35 = 1\3, ~28/27 = 60.430
-{{Optimal ET sequence|legend=1| 60e, 99e, 159, 258, 417d }}
-Badness: 0.050379
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 364/363, 441/440, 625/624, 10976/10935
-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 ET sequence|legend=1| 99ef, 159, 258, 417d }}
-Badness: 0.046006
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 364/363, 375/374, 441/440, 595/594, 3773/3757
-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 ET sequence|legend=1| 99ef, 159, 258, 417dg }}
-Badness: 0.031678
-==== Catachrome ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 325/324, 441/440, 1001/1000, 10976/10935
-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 ET sequence|legend=1| 60e, 99e, 159 }}
-Badness: 0.043844
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 273/272, 325/324, 375/374, 441/440, 4928/4913
-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 ET sequence|legend=1| 60e, 99e, 159 }}
-Badness: 0.030218
-==== Chromic ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 196/195, 352/351, 729/728, 1875/1859
-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 ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
-Badness: 0.049857
-===== 17-limit =====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 170/169, 196/195, 352/351, 375/374, 595/594
-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 ET sequence|legend=1| 60e, 99ef, 159f, 258ff }}
-Badness: 0.031043
-=== 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 tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2511
-{{Optimal ET sequence|legend=1| 39d, 120cd, 159, 198, 357, 912b }}
-Badness: 0.067173
-==== 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 tuning (CTE): ~63/50 = 1\3, ~55/54 = 30.2527
-{{Optimal ET sequence|legend=1| 39df, 120cdff, 159, 198, 357, 912b }}
-Badness: 0.033420
 == Bisupermajor ==
-{{See also| Very high accuracy temperaments #Kwazy }}
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 159: / Line 30: @@
 {{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 }}
+[[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 }}
-[[Badness]]: 0.065492
+[[Badness]] (Sintel): 1.66
 === 11-limit ===
@@ Line 177: / Line 49: @@
 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
-== 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
@@ Line 191: / Line 65: @@
 {{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 }}
-[[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 ===
@@ Line 209: / Line 84: @@
 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 ===
@@ Line 222: / Line 99: @@
 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 ===
@@ Line 235: / Line 114: @@
 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 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 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
@@ Line 249: / Line 139: @@
 {{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}})
-[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015
+: [[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 ===
@@ Line 269: / Line 158: @@
 Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}
-Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231
+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 ===
@@ Line 284: / Line 173: @@
 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
+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 ===
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288
+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 tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107
+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 ===
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399
+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 tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107
+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 ===
-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.
 Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459
+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 tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169
+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 ===
-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 ([[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 doesn't appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
 Subgroup: 2.3.5.7.11.13.17.19.23.29
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459, 726/725
+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 tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171
+Optimal tunings:
+* WE: ~29/28 = 59.9990{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})
-{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}
+* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})
-Badness (Dirichlet): 1.134
-=== no-31's 37-limit ===
-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.
-Subgroup: 2.3.5.7.11.13.17.19.23.29.37
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459, 726/725, 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 tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.222
-{{Optimal ET sequence|legend=1| 20cdeijl, 60el, 80, 140 }}
-Badness (Dirichlet): 1.127
-=== no-31's 41-limit ===
-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:
-:44:54:56:58:60:65:69:74:82:85
-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 Degrees. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
-Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41
-Comma list: 325/324, 352/351, 1001/1000, 1331/1323, 289/288, 400/399, 460/459, 726/725, 481/480
-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 ET sequence|legend=1| 20cdeijlm, 60el, 80, 140 }}
+{{Optimal ET sequence|legend=0| 60e, 80, 140 }}
-Badness (Dirichlet): 1.100
+Badness (Sintel): 1.13
 == 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
@@ Line 412: / Line 252: @@
 {{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 ===
@@ Line 427: / Line 269: @@
 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=0| 118, 239, 357, 596 }}
-{{Optimal ET sequence|legend=1| 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
-[[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 ===
 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 ===
 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:Hemimage temperaments| ]] <!-- main article -->
-[[Category:Hemimage| ]] <!-- key article -->
 [[Category:Rank 2]]

Hemimage temperaments: Difference between revisions