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Octagar temperaments: Difference between revisions

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Octagar temperaments temper out the octagar comma, {{monzo|5 -4 3 -2}} = 4000/3969.
{{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).


Temperaments discussed elsewhere include [[Garibaldi temperament|garibaldi]], [[Augmented family #Augene|augene]], [[Tetracot family #Octacot|octacot]], [[Orwellismic temperaments #Quartonic|quartonic]], [[superkleismic]], [[Meantone family #Injera|injera]], [[Diaschismic family #Bidia|bidia]], [[Kleismic family #Hemikleismic|hemikleismic]], [[Mirkwai clan #Pluto|pluto]], [[Porcupine family #Nautilus|nautilus]], [[Ripple family|ripple]], and [[Amity family #Hamity|hamity]].
Temperaments discussed elsewhere are:
* ''[[Sida]]'' (+25/24) → [[Dicot family #Sida|Dicot family]]
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]]
* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine 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]]
* [[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]]
* ''[[Hemikleismic]]'' (+6144/6125) → [[Kleismic family #Hemikleismic|Kleismic 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]].
Subgroup: 2.3.5.7
 
== 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.  


[[Comma list]]: 4000/3969, 84035/82944
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 2 3 3}}, {{val|0 -17 -28 -8}}]
[[Comma list]]: 4000/3969, 10976/10935


{{Multival|legend=1|17 28 8 5 -35 -60}}
{{Mapping|legend=1| 1 -2 -11 -10 | 0 7 26 25 }}
: mapping generators: ~2, ~10/7


[[POTE generator]]: ~49/48 = 29.181
[[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 }}


{{Val list|legend=1| 41, 206, 247bc, 288bcd, 329bcd }}
{{Optimal ET sequence|legend=1| 39d, 41, 80, 121, 283d, 404bd }}


[[Badness]]: 0.142903
[[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: [{{val|1 2 3 3 4}}, {{val|0 -17 -28 -8 -22}}]
Mapping: {{mapping| 1 -2 -11 -10 -16 | 0 7 26 25 38 }}


POTE generator: ~49/48 = 29.180
Optimal tunings:  
* WE: ~2 = 1199.7793{{c}}, ~10/7 = 614.6040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7074{{c}}


Vals: {{Val list| 41, 206e }}
{{Optimal ET sequence|legend=0| 39dee, 41 }}


Badness: 0.063217
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: [{{val|1 2 3 3 4 4}}, {{val|0 -17 -28 -8 -22 -1}}]
Mapping: {{mapping| 1 5 15 15 22 12 | 0 7 26 25 38 17 }}


POTE generator: ~49/48 = 29.125
Optimal tunings:  
* WE: ~2 = 1199.0328{{c}}, ~10/7 = 614.2729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.7277{{c}}


Vals: {{Val list| 41, 124ef, 165ef }}
{{Optimal ET sequence|legend=0| 39deef, 41 }}


Badness: 0.043644
Badness (Sintel): 1.66


== Slithy ==
== Slithy ==
Subgroup: 2.3.5.7
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


[[Comma list]]: 4000/3969, 420175/419904
[[Comma list]]: 4000/3969, 420175/419904


[[Mapping]]: [{{val|1 3 5 4}}, {{val|0 -19 -36 -16}}]
{{Mapping|legend=1| 1 -16 -31 -12 | 0 19 36 16 }}
 
: mapping generators: ~2, ~40/21
{{Multival|legend=1|19 36 16 13 -28 -64}}


[[POTE generator]]: ~21/20 = 89.282
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5464{{c}}, ~40/21 = 1110.2981{{c}}
: [[error map]]: {{val| -0.454 +0.996 -1.521 +1.387 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7026{{c}}
: error map: {{val| 0.000 +1.394 -1.021 +2.415 }}


{{Val list|legend=1| 27, 94, 121, 215, 336d }}
{{Optimal ET sequence|legend=1| 27, 67c, 94, 121, 215, 336d }}


[[Badness]]: 0.107482
[[Badness]] (Sintel): 2.72


=== 11-limit ===
=== 11-limit ===
Line 64: Line 172:
Comma list: 540/539, 2200/2187, 4375/4356
Comma list: 540/539, 2200/2187, 4375/4356


Mapping: [{{val|1 3 5 4 8}}, {{val|0 -19 -36 -16 -61}}]
Mapping: {{mapping| 1 -16 -31 -12 -53 | 0 19 36 16 61 }}


POTE generator: ~21/20 = 89.291
Optimal tunings:  
* WE: ~2 = 1199.5884{{c}}, ~40/21 = 1110.3276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.6969{{c}}


Vals: {{Val list| 94, 121, 215, 336de, 551bde }}
{{Optimal ET sequence|legend=0| 27e, 67ce, 94, 121, 215, 336de }}


Badness: 0.045895
Badness (Sintel): 1.52


=== 13-limit ===
=== 13-limit ===
Line 77: Line 187:
Comma list: 325/324, 352/351, 540/539, 1575/1573
Comma list: 325/324, 352/351, 540/539, 1575/1573


Mapping: [{{val|1 3 5 4 8 4}}, {{val|0 -19 -36 -16 -61 -4}}]
Mapping: {{mapping| 1 -16 -31 -12 -53 0 | 0 19 36 16 61 4 }}
 
Optimal tunings:
* WE: ~2 = 1199.5032{{c}}, ~40/21 = 1110.2545{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7037{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 67ce, 94, 121, 215, 336def }}
 
Badness (Sintel): 1.18
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1199.5051{{c}}, ~40/21 = 1110.2555{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/21 = 1110.7033{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 67ceg, 94, 121, 215, 336defg }}
 
Badness (Sintel): 1.01
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1199.6184{{c}}, ~19/10 = 1110.3583{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.7036{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 67cegh, 94, 121, 215 }}
 
Badness (Sintel): 0.998
 
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483


POTE generator: ~21/20 = 89.286
Mapping: {{mapping| 1 -16 -31 -12 -53 0 -57 -30 -76 | 0 19 36 16 61 4 66 37 87 }}


Vals: {{Val list| 94, 121, 215, 336def }}
Optimal tunings:  
* WE: ~2 = 1199.6651{{c}}, ~19/10 = 1110.3928{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/10 = 1110.6965{{c}}


Badness: 0.028492
{{Optimal ET sequence|legend=0| 27egi, 94, 121i, 215 }}


== Quintupole ==
Badness (Sintel): 1.07
The ''quintupole'' temperament tempers out the octagar comma and the mistisma (458752/455625) in the 7-limit; 896/891 (pentacircle), 1375/1372 (moctdel), and 4375/4356 (fantares, luluzoquadyo) in the 11-limit. The word "quintupole" means ''five poles'', but also a play on the words "quintuple" and "polypyth". It is so named because the generator is 1/5 of the [[Porwell temperaments #Polypyth|polypyth]] fourth (~4/3, about 495.8 cents).  


Subgroup: 2.3.5.7
== Tridecatonic ==
{{See also| 13th-octave temperaments #Tridecatonic }}


[[Comma list]]: 4000/3969, 458752/455625
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.


[[Mapping]]: [{{val|1 2 1 0}}, {{val|0 -5 16 34}}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 5 -16 -34 -37 -68 -34 }}
[[Comma list]]: 4000/3969, 8748/8575


[[POTE generator]]: ~135/128 = 99.175
{{Mapping|legend=1| 13 0 -11 16 | 0 1 2 1 }}
: mapping generators: ~21/20, ~3


{{Vals|legend=1| 12, 97, 109, 121 }}
[[Optimal tuning]]s:
* [[WE]]: ~21/20 = 92.2197{{c}}, ~3/2 = 701.2915{{c}}
: [[error map]]: {{val| -1.144 -1.807 -0.435 +6.837 }}
* [[CWE]]: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2352{{c}}
: error map: {{val| 0.000 -0.720 +0.772 +9.332 }}


[[Badness]]: 0.111620
{{Optimal ET sequence|legend=1| 26, 39d, 65d }}
 
[[Badness]] (Sintel): 3.52


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


Comma list: 896/891, 1375/1372, 4375/4356
Comma list: 99/98, 1944/1925, 2560/2541


Mapping: [{{val|1 2 1 0 -1}}, {{val|0 -5 16 34 54}}]
Mapping: {{mapping| 13 0 -11 16 45 | 0 1 2 1 0 }}


POTE generator: ~132/125 = 99.156
Optimal tunings:  
* WE: ~21/20 = 92.2333{{c}}, ~3/2 = 701.1371{{c}}
* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 701.2751{{c}}


Vals: {{Val list| 12, 109, 121, 351bde, 472bdee }}
{{Optimal ET sequence|legend=0| 26, 39d, 65d }}


Badness: 0.056501
Badness (Sintel): 2.09


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


Comma list: 352/351, 364/363, 625/624, 2704/2695
Comma list: 99/98, 144/143, 275/273, 1040/1029
 
Mapping: {{mapping| 13 0 -11 16 45 7 | 0 1 2 1 0 2 }}
 
Optimal tunings:
* WE: ~21/20 = 92.2248{{c}}, ~3/2 = 700.7272{{c}}
* CWE: ~21/20 = 92.3077{{c}}, ~3/2 = 700.8621{{c}}
 
{{Optimal ET sequence|legend=0| 26, 39df, 65d }}
 
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


Mapping: [{{val|1 2 1 0 -1 -2}}, {{val|0 -5 16 34 54 69}}]
[[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 }}


POTE generator: ~55/52 = 99.165
{{Optimal ET sequence|legend=1| 1c, 40c, 41, 206 }}


Vals: {{Val list| 12f, 109, 121 }}
[[Badness]] (Sintel): 3.62


Badness: 0.038431
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 17-limit ===
Comma list: 100/99, 245/242, 46656/46585
Subgroup: 2.3.5.7.11.13.17
 
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 ==
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
 
[[Comma list]]: 4000/3969, 2100875/2097152
 
{{Mapping|legend=1| 1 -4 -3 6 | 0 21 20 -12 }}
: mapping generators: ~2, ~6/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9902{{c}}, ~6/5 = 319.1955{{c}}
: [[error map]]: {{val| -0.010 +1.190 -2.374 +0.769 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 319.1980{{c}}
: error map: {{val| 0.000 +1.202 -2.354 +0.798 }}
 
{{Optimal ET sequence|legend=1| 15, 64c, 79, 94, 203, 297c }}
 
[[Badness]] (Sintel): 4.26
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 4000/3969


Comma list: 256/255, 352/351, 364/363, 375/374, 442/441
Mapping: {{Mapping| 1 -4 -3 6 0 | 0 21 20 -12 13 }}


Mapping: [{{val|1 2 1 0 -1 -2 5}}, {{val|0 -5 16 34 54 69 -11}}]
Optimal tunings:  
* WE: ~2 = 1200.0983{{c}}, ~6/5 = 319.2229{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1985{{c}}


POTE generator: ~18/17 = 99.172
{{Optimal ET sequence|legend=0| 15, 64c, 79, 94, 203, 297c }}


Vals: {{Val list| 12f, 109, 121 }}
Badness (Sintel): 1.69


Badness: 0.028721
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 19-limit ===
Comma list: 275/273, 325/324, 385/384, 1331/1323
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374
Mapping: {{Mapping| 1 -4 -3 6 0 -8 | 0 21 20 -12 13 44 }}


Mapping: [{{val|1 2 1 0 -1 -2 5 4}}, {{val|0 -5 16 34 54 69 -11 3}}]
Optimal tunings:  
* WE: ~2 = 1200.0732{{c}}, ~6/5 = 319.1840{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 319.1658{{c}}


POTE generator: ~18/17 = 99.164
{{Optimal ET sequence|legend=0| 15, 79, 94 }}


Vals: {{Val list| 12f, 109, 121 }}
Badness (Sintel): 1.55


Badness: 0.023818
== References ==


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