Porwell temperaments: Difference between revisions
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This | This page collects [[rank-2 temperament|rank-2]] '''porwell temperaments''', which [[tempering out|temper out]] the [[porwell comma]] ({{monzo|legend=1| 11 1 -3 -2 }}, [[ratio]]: [[6144/6125]]). For the family of [[rank-3 temperament]]s, see [[Porwell family]]. | ||
Temperaments discussed elsewhere are: | Temperaments discussed elsewhere are: | ||
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* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]] | * [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]] | ||
* ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]] | * ''[[Freivald]]'' (+6272/6075) → [[Passion family #Freivald|Passion family]] | ||
* ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]] | * ''[[Hemikleismic]]'' (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]] | ||
* ''[[Twothirdtonic]]'' (+686/675) → [[Sengic temperaments #Twothirdtonic|Sengic temperaments]] | * ''[[Twothirdtonic]]'' (+686/675) → [[Sengic temperaments #Twothirdtonic|Sengic temperaments]] | ||
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* ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]] | * ''[[Hemimaquila]]'' (+{{monzo| -5 10 5 -8 }}) → [[Maquila family #Hemimaquila|Maquila family]] | ||
Considered below are hendecatonic, nessafof, grendel, countermiracle, aufo, absurdity, polypyth, whoops, dodifo, and | Considered below are hendecatonic, nessafof, grendel, countermiracle, aufo, absurdity, polypyth, whoops, dodifo, icositritonic, and decimaleap, in the order of increasing [[badness]]. | ||
== Hendecatonic == | == Hendecatonic == | ||
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Badness (Sintel): 1.27 | Badness (Sintel): 1.27 | ||
== Decimaleap == | |||
: ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Quintaleap]].'' | |||
Decimaleap may be described as the 121 & 145 temperament. It divides the [[4/3|perfect fourth]] into ten [[~]][[36/35]] quartertone generators; its [[ploidacot]] is thus omega-decacot. Three generator steps reach an accurate ~[[12/11]], from which we can derive an 11-limit extension where [[896/891]] and [[14700/14641]] are tempered out, and a 13-limit extension that follows from [[pentacircle clan #Parapyth|parapyth]]. | |||
Decimaleap was named by [[Xenllium]] in 2021 as a portmanteau of ''decima'' (Latin for "one tenth") and ''[[leapday]]'', because the generator is 1/10 of a perfect fourth which can serve as a tuning for leapday. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 6144/6125, 393379840/387420489 | |||
{{Mapping|legend=1| 1 2 1 5 | 0 -10 32 -53 }} | |||
: mapping generators: ~2, ~36/35 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.4302{{c}}, ~36/35 = 49.5975{{c}} | |||
: [[error map]]: {{val| -0.570 +0.930 +0.237 -0.343 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 49.6234{{c}} | |||
: error map: {{val| 0.000 +1.811 +1.635 +1.134 }} | |||
{{Optimal ET sequence|legend=1| 24, 97d, 121, 145, 266 }} | |||
[[Badness]] (Sintel): 7.95 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 896/891, 6144/6125, 14700/14641 | |||
{{Mapping|legend=0| 1 2 1 5 4 | 0 -10 32 -53 -13 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.3373{{c}}, ~36/35 = 49.5943{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.6247{{c}} | |||
{{Optimal ET sequence|legend=0| 24, 97d, 121, 145, 266e }} | |||
Badness (Sintel): 2.80 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 352/351, 364/363, 676/675, 6144/6125 | |||
{{Mapping|legend=0| 1 2 1 5 4 3 | 0 -10 32 -53 -13 17 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.3078{{c}}, ~36/35 = 49.5914{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 49.6217{{c}} | |||
{{Optimal ET sequence|legend=0| 24, 97d, 121, 266ef, 653bbcdeeefff }} | |||
Badness (Sintel): 1.73 | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 256/255, 352/351, 364/363, 676/675, 1156/1155 | |||
{{Mapping|legend=0| 1 2 1 5 4 3 5 | 0 -10 32 -53 -13 17 -22 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.2893{{c}}, ~35/34 = 49.5913{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/34 = 49.6230{{c}} | |||
{{Optimal ET sequence|legend=0| 24, 97dg, 121, 266efg }} | |||
Badness (Sintel): 1.30 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | |||
Comma list: 256/255, 352/351, 361/360, 364/363, 456/455, 665/663 | |||
{{Mapping|legend=0| 1 2 1 5 4 3 5 4 | 0 -10 32 -53 -13 17 -22 6 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.3776{{c}}, ~34/33 = 49.5984{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~34/33 = 49.6257{{c}} | |||
{{Optimal ET sequence|legend=0| 24, 97dg, 121, 145, 266efg }} | |||
Badness (Sintel): 1.29 | |||
== References == | == References == | ||