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 {{Technical data page}}
-This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).
+This is a collection of [[rank-2 temperament|rank-2]] '''octagar temperaments''', which [[tempering out|temper out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).
 Temperaments discussed elsewhere are:
+* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
+* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
+* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
 * ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
+* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
 * ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
+* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
 * ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]]
-* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
+* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
-* [[Augene]] (+64/63 or 126/125) → [[Augmented family #Septimal augmented (augene)|Augmented family]]
-* [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
 * ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
-* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]]
 * [[Superkleismic]] (+875/864 or 1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
-* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family]]
-* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]]
 * ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]
+* ''[[Quartonic]]'' (+1728/1715) → [[Quartonic family #Septimal quartonic|Quartonic family]]
 * ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]
-* ''[[Pluto]]'' (+10976/10935) → [[Mirkwai clan #Pluto|Mirkwai clan]]
+* ''[[Trisedodge]]'' (+110592/109375) → [[Trisedodge family #Septimal trisedodge|Trisedodge family]]
-* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
+Considered below are pluto, slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
+== Pluto ==
+{{Distinguish| Plutus }}
+Pluto, named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18465.html Yahoo! Tuning Group | ''19-limit Pluto temperament'']</ref>, can be described as the {{nowrap| 41 & 80 }} temperament. It is generated by a flattened [[~]][[10/7]] [[tritone]], seven of which give the [[interval class]] of [[3/1|3]], and the [[ploidacot]] for this temperament is gamma-heptacot. [[121edo|59\121]] is about perfect as a tuning.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 4000/3969, 10976/10935
+{{Mapping|legend=1| 1 -2 -11 -10 | 0 7 26 25 }}
+: mapping generators: ~2, ~10/7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.4720{{c}}, ~10/7 = 614.5826{{c}}
+: [[error map]]: {{val| -0.528 +1.179 -1.358 +1.020 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.8282{{c}}
+: error map: {{val| 0.000 +1.843 -0.780 +1.880 }}
+{{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 283d, 404bd }}
+[[Badness]] (Sintel): 1.46
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 540/539, 896/891, 1375/1372
+{{Mapping|legend=0| 1 -2 -11 -10 5 | 0 7 26 25 -3 }}
+Optimal tunings:
+* WE: ~2 = 1200.1835{{c}}, ~10/7 = 614.4677{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8616{{c}}
+{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
+Badness (Sintel): 0.987
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 325/324, 352/351, 364/363, 540/539
+{{Mapping|legend=0| 1 -2 -11 -10 5 16 | 0 7 26 25 -3 -24 }}
+Optimal tunings:
+* WE: ~2 = 1199.2469{{c}}, ~10/7 = 614.4908{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8684{{c}}
+{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
+Badness (Sintel): 1.06
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 256/255, 325/324, 352/351, 364/363, 540/539
+{{Mapping|legend=0| 1 -2 -11 -10 5 16 21 | 0 7 26 25 -3 -24 -33 }}
+Optimal tunings:
+* WE: ~2 = 1199.2009{{c}}, ~10/7 = 614.4746{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8813{{c}}
+{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
+Badness (Sintel): 1.09
+===== 19-limit =====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594
+{{Mapping|legend=0| 1 -2 -11 -10 5 16 21 -6 | 0 7 26 25 -3 -24 -33 20 }}
+Optimal tunings:
+* WE: ~2 = 1199.2833{{c}}, ~10/7 = 614.5237{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8876{{c}}
+{{Optimal ET sequence|legend=0| 39d, 41, 80, 121 }}
+Badness (Sintel): 1.07
+==== Orcus ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 144/143, 196/195, 275/273, 896/891
+{{Mapping|legend=0| 1 -2 -11 -10 5 -5 | 0 7 26 25 -3 17 }}
+Optimal tunings:
+* WE: ~2 = 1198.7372{{c}}, ~10/7 = 614.2418{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.8502{{c}}
+{{Optimal ET sequence|legend=0| 39df, 41, 80f, 121ff }}
+Badness (Sintel): 1.38
+=== Plutino ===
+Subgroup: 2.3.5.7.11
+Comma list: 100/99, 245/242, 10976/10935
+{{Mapping|legend=0| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}
+Optimal tunings:
+* WE: ~2 = 1199.7793{{c}}, ~10/7 = 614.6040{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7074{{c}}
+{{Optimal ET sequence|legend=0| 39dee, 41 }}
+Badness (Sintel): 1.92
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 100/99, 196/195, 245/242, 729/728
+{{Mapping|legend=0| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}
+Optimal tunings:
+* WE: ~2 = 1199.0328{{c}}, ~10/7 = 614.2729{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7277{{c}}
+{{Optimal ET sequence|legend=0| 39deef, 41 }}
-Considered below are slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
+Badness (Sintel): 1.66
 == Slithy ==
+Slithy tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 27 & 94 }} temperament with a [[ploidacot]] signature of 17-sheared 19-cot. [[94edo]], [[121edo]] and [[215edo]] can all be used as tunings.
 [[Subgroup]]: 2.3.5.7
@@ Line 44: / Line 173: @@
 Comma list: 540/539, 2200/2187, 4375/4356
-Mapping: {{mapping| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}
 Optimal tunings:
@@ Line 59: / Line 188: @@
 Comma list: 325/324, 352/351, 540/539, 1575/1573
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}
 Optimal tunings:
@@ Line 74: / Line 203: @@
 Comma list: 325/324, 352/351, 375/374, 540/539, 595/594
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 | 0 19 36 16 61 4 66 }}
 Optimal tunings:
@@ Line 89: / Line 218: @@
 Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 -30 | 0 19 36 16 61 4 66 37 }}
 Optimal tunings:
@@ Line 104: / Line 233: @@
 Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483
-Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}
+{{Mapping|legend=0| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}
 Optimal tunings:
@@ Line 115: / Line 244: @@
 == Tridecatonic ==
-{{See also| 13th-octave temperaments #Tridecatonic }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tridecatonic]]
-The tridecatonic temperament ({{nowrap| 26 & 39 }}) has a period of 1/13 octave. It tempers out the tridecalimma, {{monzo| -11 26 -13 }} in the 5-limit. There are some extensions for 7-limit, including tridecatonic ({{nowrap| 26 & 39d }}, tempering out 4000/3969), [[ragismic microtemperaments #Trideci|trideci]] ({{nowrap| 26 & 39 }}, tempering out 4375/4374), and [[26th-octave temperaments|bosonic]] ({{nowrap| 26 & 130 }}, tempering out 321489/320000).
+Tridecatonic has a period of 1/13 octave, and may be described as {{nowrap| 26 & 39 }}. It tempers out [[devil's tridecalimma]], {{monzo| -11 26 -13 }} in the 5-limit. Its [[ploidacot]] is 13-ploid monocot.
 [[Subgroup]]: 2.3.5.7
@@ Line 141: / Line 270: @@
 Comma list: 99/98, 1944/1925, 2560/2541
-Mapping: {{mapping| 13 0 -11 16 45 | 0 1 2 1 0 }}
+{{Mapping|legend=0| 13 0 -11 16 45 | 0 1 2 1 0 }}
 Optimal tunings:
@@ Line 156: / Line 285: @@
 Comma list: 99/98, 144/143, 275/273, 1040/1029
-Mapping: {{mapping| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}
+{{Mapping|legend=0| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}
 Optimal tunings:
@@ Line 193: / Line 322: @@
 Comma list: 100/99, 245/242, 46656/46585
-Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
+{{Mapping|legend=0| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
 Optimal tunings:
@@ Line 208: / Line 337: @@
 Comma list: 100/99, 144/143, 245/242, 343/338
-Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
+{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
 Optimal tunings:
@@ Line 219: / Line 348: @@
 == Dhaivatic ==
+Named by [[Xenllium]] in 2023, dhaivatic tempers out the [[rainy comma]] and may be described as the {{nowrap| 15 & 94 }} temperament. Its [[ploidacot]] is epsilon-21-cot.
 [[Subgroup]]: 2.3.5.7
@@ Line 266: / Line 397: @@
 Badness (Sintel): 1.55
+== References ==
+[[Category:Octagar temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Octagar temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]

Octagar temperaments: Difference between revisions

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