Aberschismic family: Difference between revisions

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=== Overview to 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~~. Thus~~ a ~~successful~~ mapping of 13 ~~can be found by fixing the~~ [[13/11]] at the minor third (~~C–Eb~~), tempering out [[~~352/351~~]], [[847/845]], and [[2080/2079]].

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:

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==== Subgroup extensions ====

A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].

== Pele ==

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== Kapo ==

Kapo tempers out 3025/3024, the [[lehmerisma]], as well as 16384/16335, the [[semiporwellisma]], thus splitting the [[~]][[5/3]] to two equal parts of ~[[128/99]] each. The only practical 13-limit extension maps [[13/11]] to the diatonic minor third, like the other temperaments in this family.

[[Subgroup]]: 2.3.5.7.11

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[[Badness]] (Sintel): 1.19

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 3025/3024

Mapping: {{mapping| 1 0 0 10 7 12 | 0 1 1 -5 -2 -5 | 0 0 2 2 -1 -1 }}

Optimal tunings:

* WE: ~2 = 1199.6256{{c}}, ~3/2 = 702.7250{{c}}, ~84/65 = 442.0740{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9733{{c}}, ~84/65 = 442.1684{{c}}

{{Optimal ET sequence|legend=0| 41, 46, 65d, 87, 111, 152f, 198, 350f, 437f, 635bcff }}

Badness (Sintel): 0.938

== Namaka ==

@@ Line 46: / Line 46: @@
 === Overview to 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. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third (C–Eb), tempering out [[352/351]], [[847/845]], and [[2080/2079]].
+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:
@@ Line 52: / Line 52: @@
 ==== Subgroup extensions ====
 A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]].
 == Pele ==
@@ Line 280: / Line 280: @@
 == Kapo ==
+Kapo tempers out 3025/3024, the [[lehmerisma]], as well as 16384/16335, the [[semiporwellisma]], thus splitting the [[~]][[5/3]] to two equal parts of ~[[128/99]] each. The only practical 13-limit extension maps [[13/11]] to the diatonic minor third, like the other temperaments in this family.
 [[Subgroup]]: 2.3.5.7.11
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 [[Badness]] (Sintel): 1.19
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 352/351, 847/845, 3025/3024
+Mapping: {{mapping| 1 0 0 10 7 12 | 0 1 1 -5 -2 -5 | 0 0 2 2 -1 -1 }}
+Optimal tunings:
+* WE: ~2 = 1199.6256{{c}}, ~3/2 = 702.7250{{c}}, ~84/65 = 442.0740{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9733{{c}}, ~84/65 = 442.1684{{c}}
+{{Optimal ET sequence|legend=0| 41, 46, 65d, 87, 111, 152f, 198, 350f, 437f, 635bcff }}
+Badness (Sintel): 0.938
 == Namaka ==