Aberschismic family: Difference between revisions
→Kapo: the only practical extension for 13 is to map 13/11 to m3, like the other temps, even tho it's not supported by the OPV here |
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=== Overview to extensions === | === Overview to extensions === | ||
==== 11- and 13-limit extensions ==== | ==== 11- and 13-limit extensions ==== | ||
Strong extensions of aberschismic are [[#Pele|pele]], [[#Laka|laka]], [[#Akea|akea]], and [[#Lono|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (C–vG♭); laka, up augmented third (C–^E♯); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F♯). All these extensions follow the trend of tuning the fifth a little sharp | Strong extensions of aberschismic are [[#Pele|pele]], [[#Laka|laka]], [[#Akea|akea]], and [[#Lono|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (C–vG♭); laka, up augmented third (C–^E♯); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F♯). All these extensions follow the trend of tuning the fifth a little sharp, so a mapping for 13 arises from identifying [[13/11]] with the minor third (C–E♭), which is further facilitated by the fact that the minor third in aberschismic is the exact mean of [[6/5]] and [[7/6]], thus tempering out the [[semiparticular]] [[847/845]] ([[S-expression|S11/S13]]) and by extension [[352/351]] and [[2080/2079]]. | ||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
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==== Subgroup extensions ==== | ==== Subgroup extensions ==== | ||
A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]]. | A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]]. | ||
== Pele == | == Pele == | ||
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Comma list: 196/195, 352/351, 364/363 | Comma list: 196/195, 352/351, 364/363 | ||
{{Mapping|legend=0| 1 0 0 10 17 22 | 0 1 0 -6 -10 -13 | 0 0 1 1 1 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 196/195, 256/255, 352/351, 364/363 | Comma list: 196/195, 256/255, 352/351, 364/363 | ||
{{Mapping|legend=0| 1 0 0 10 17 22 8 | 0 1 0 -6 -10 -13 -1 | 0 0 1 1 1 1 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 352/351, 540/539, 729/728 | Comma list: 352/351, 540/539, 729/728 | ||
{{Mapping|legend=0| 1 0 0 10 -18 -13 | 0 1 0 -6 15 12 | 0 0 1 1 -1 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 352/351, 400/399, 456/455, 495/494 | Comma list: 352/351, 400/399, 456/455, 495/494 | ||
{{Mapping|legend=0| 1 0 0 10 -18 -13 -6 | 0 1 0 -6 15 12 5 | 0 0 1 1 -1 -1 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 325/324, 352/351, 385/384 | Comma list: 325/324, 352/351, 385/384 | ||
{{Mapping|legend=0| 1 0 0 10 -3 2 | 0 1 0 -6 7 4 | 0 0 1 1 -2 -2 }} | |||
Lattice basis: | Lattice basis: | ||
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Comma list: 176/175, 351/350, 847/845 | Comma list: 176/175, 351/350, 847/845 | ||
{{Mapping|legend=0| 1 0 0 10 6 11 | 0 1 0 -6 -6 -9 | 0 0 1 1 3 3 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 352/351, 847/845, 3025/3024 | Comma list: 352/351, 847/845, 3025/3024 | ||
{{Mapping|legend=0| 1 0 0 10 7 12 | 0 1 1 -5 -2 -5 | 0 0 2 2 -1 -1 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 352/351, 676/675, 847/845 | Comma list: 352/351, 676/675, 847/845 | ||
{{Mapping|legend=0| 1 0 0 10 -6 -1 | 0 2 0 -12 9 3 | 0 0 1 1 1 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
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Comma list: 400/399, 1216/1215 | Comma list: 400/399, 1216/1215 | ||
{{Mapping|legend=0| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }} | |||
Optimal tunings: | Optimal tunings: | ||
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== References == | == References == | ||
[[Category:Aberschismic family| ]] <!-- main article --> | |||
[[Category:Aberschismic| ]] <!-- key article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-3 temperaments]] | ||