Octagar temperaments: Difference between revisions
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* ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]] | * ''[[Bidia]]'' (+2048/2025 or 3136/3125) → [[Diaschismic family #Bidia|Diaschismic family]] | ||
* ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]] | * ''[[Rip]]'' (+36/35) → [[Ripple family #Rip|Ripple family]] | ||
* ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]] | * ''[[Nautilus]]'' (+49/48) → [[Porcupine family #Nautilus|Porcupine family]] | ||
* ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]] | * ''[[Roman]]'' (+525/512) → [[Wesley family #Roman|Wesley family]] | ||
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* ''[[Trisedodge]]'' (+110592/109375) → [[Trisedodge family #Septimal trisedodge|Trisedodge family]] | * ''[[Trisedodge]]'' (+110592/109375) → [[Trisedodge family #Septimal trisedodge|Trisedodge family]] | ||
Considered below are pluto, slithy, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]]. | Considered below are pluto, slithy, quintupole, tridecatonic, slendi and dhaivatic, in the order of increasing [[badness]]. | ||
== Pluto == | == Pluto == | ||
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Badness (Sintel): 1.07 | Badness (Sintel): 1.07 | ||
== Quintupole == | |||
{{Main| Quintupole }} | |||
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Quintaleap]].'' | |||
Quintupole may be described as the 12 & 121 temperament with a [[ploidacot]] signature of omega-pentacot. It was named by [[Xenllium]] in 2021; the word ''quintupole'' means "five poles", but also a play on the words ''quintuple'' and ''[[porwell temperaments #Polypyth|polypyth]]'', so named because the generator is 1/5 of a [[4/3|perfect fourth]] of about 495.8 cents, which can serve as a tuning for polypyth. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 4000/3969, 458752/455625 | |||
{{Mapping|legend=1| 1 2 1 0 | 0 -5 16 34 }} | |||
: mapping generators: ~2, ~135/128 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.3243{{c}}, ~135/128 = 99.1196{{c}} | |||
: [[error map]]: {{val| -0.676 +1.095 -1.075 +1.241 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~135/128 = 99.1615{{c}} | |||
: error map: {{val| 0.000 +2.237 +0.271 +2.666 }} | |||
{{Optimal ET sequence|legend=1| 12, 85d, 97, 109, 121 }} | |||
[[Badness]] (Sintel): 2.82 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 896/891, 1375/1372, 4375/4356 | |||
{{Mapping|legend=0| 1 2 1 0 -1 | 0 -5 16 34 54 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.3424{{c}}, ~35/33 = 99.1015{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.1436{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 97e, 109, 121, 351bde }} | |||
Badness (Sintel): 1.87 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 352/351, 364/363, 625/624, 2704/2695 | |||
{{Mapping|legend=0| 1 2 1 0 -1 -2 | 0 -5 16 34 54 69 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.3223{{c}}, ~35/33 = 99.1087{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.1536{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 97ef, 109, 121 }} | |||
Badness (Sintel): 1.59 | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 256/255, 352/351, 364/363, 375/374, 442/441 | |||
{{Mapping|legend=0| 1 2 1 0 -1 -2 5 | 0 -5 16 34 54 69 -11 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.2234{{c}}, ~18/17 = 99.1076{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.1629{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 109, 121 }} | |||
Badness (Sintel): 1.46 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374 | |||
{{Mapping|legend=0| 1 2 1 0 -1 -2 5 4 | 0 -5 16 34 54 69 -11 3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.3678{{c}}, ~18/17 = 99.1115{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.1576{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 109, 121 }} | |||
Badness (Sintel): 1.45 | |||
== Tridecatonic == | == Tridecatonic == | ||