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Octagar temperaments: Difference between revisions

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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]] [[regular temperament|temperaments]] [[tempering out]] the [[octagar comma]] ({{monzo|legend=1| 5 -4 3 -2 }}, [[ratio]]: 4000/3969).


Temperaments considered below are slendi and slithy. Discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
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* ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]
* ''[[Hamity]]'' (+2430/2401) → [[Amity family #Hamity|Amity family]]
* ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Pluto]]'' (+10976/10935) → [[Mirkwai clan #Pluto|Mirkwai clan]]
* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]
* ''[[Quintupole]]'' (+458752/455625) → [[Quintaleap family #Quintupole|Quintaleap family]]


== Slendi ==
Considered below are pluto, slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]].
Slendi may be described as the {{nowrap| 41 & 42 }} temperament. It has a generator of [[49/48]], seventeen of which give [[4/3]]. Its [[ploidacot]] is omega-17-cot.
 
== Pluto ==
{{Distinguish| Plutus }}


The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4000/3969, 84035/82944
[[Comma list]]: 4000/3969, 10976/10935


{{Mapping|legend=1| 1 2 3 3 | 0 -17 -28 -8 }}
{{Mapping|legend=1| 1 -2 -11 -10 | 0 7 26 25 }}
: mapping generators: ~2, ~49/48
: mapping generators: ~2, ~10/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.2515{{c}}, ~49/48 = 29.1870{{c}}
* [[WE]]: ~2 = 1199.4720{{c}}, ~10/7 = 614.5826{{c}}
: [[error map]]: {{val| +0.252 +2.368 -2.796 -1.568 }}
: [[error map]]: {{val| -0.528 +1.179 -1.358 +1.020 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 29.1715{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.8282{{c}}
: error map: {{val| 0.000 +2.129 -3.116 -2.198 }}
: error map: {{val| 0.000 +1.843 -0.780 +1.880 }}


{{Optimal ET sequence|legend=1| 1c, 40c, 41, 206 }}
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 283d, 404bd }}


[[Badness]] (Sintel): 3.62
[[Badness]] (Sintel): 1.46


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


Comma list: 100/99, 245/242, 46656/46585
Comma list: 540/539, 896/891, 1375/1372
 
Mapping: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
Mapping: {{mapping| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8027{{c}}, ~49/48 = 29.1751{{c}}
* WE: ~2 = 1199.7793{{c}}, ~10/7 = 614.6040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1875{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7074{{c}}


{{Optimal ET sequence|legend=0| 1ce, 40c, 41 }}
{{Optimal ET sequence|legend=0| 39dee, 41 }}


Badness (Sintel): 2.09
Badness (Sintel): 1.92


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 100/99, 144/143, 245/242, 343/338
Comma list: 100/99, 196/195, 245/242, 729/728


Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
Mapping: {{mapping| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.0915{{c}}, ~49/48 = 29.1033{{c}}
* WE: ~2 = 1199.0328{{c}}, ~10/7 = 614.2729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1576{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7277{{c}}


{{Optimal ET sequence|legend=0| 1ce, 41 }}
{{Optimal ET sequence|legend=0| 39deef, 41 }}


Badness (Sintel): 1.80
Badness (Sintel): 1.66


== Slithy ==
== 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
[[Subgroup]]: 2.3.5.7


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{{See also| 13th-octave temperaments #Tridecatonic }}
{{See also| 13th-octave 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
[[Subgroup]]: 2.3.5.7
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Badness (Sintel): 1.56
Badness (Sintel): 1.56
== Slendi ==
Slendi may be described as the {{nowrap| 41 & 42 }} temperament. It has a generator of [[49/48]], seventeen of which give [[4/3]]. Its [[ploidacot]] is omega-17-cot.
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 4000/3969, 84035/82944
{{Mapping|legend=1| 1 2 3 3 | 0 -17 -28 -8 }}
: mapping generators: ~2, ~49/48
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2515{{c}}, ~49/48 = 29.1870{{c}}
: [[error map]]: {{val| +0.252 +2.368 -2.796 -1.568 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 29.1715{{c}}
: error map: {{val| 0.000 +2.129 -3.116 -2.198 }}
{{Optimal ET sequence|legend=1| 1c, 40c, 41, 206 }}
[[Badness]] (Sintel): 3.62
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 100/99, 245/242, 46656/46585
Mapping: {{mapping| 1 2 3 3 4 | 0 -17 -28 -8 -22 }}
Optimal tunings:
* WE: ~2 = 1199.8027{{c}}, ~49/48 = 29.1751{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1875{{c}}
{{Optimal ET sequence|legend=0| 1ce, 40c, 41 }}
Badness (Sintel): 2.09
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 144/143, 245/242, 343/338
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -17 -28 -8 -22 -12 }}
Optimal tunings:
* WE: ~2 = 1199.0915{{c}}, ~49/48 = 29.1033{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 29.1576{{c}}
{{Optimal ET sequence|legend=0| 1ce, 41 }}
Badness (Sintel): 1.80


== Dhaivatic ==
== 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
[[Subgroup]]: 2.3.5.7


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Badness (Sintel): 1.55
Badness (Sintel): 1.55
== References ==


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