Aberschismic family: Difference between revisions

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~~{{Interwiki~~

~~| en = Aberschismic family~~

~~| de = Aberrschismisch~~

~~| es =~~

~~| ja =~~

~~| ro =~~

~~| ko = 헤미패미티 (음률)~~

}}

The '''aberschismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: [[5120/5103]]).

== Aberschismic ==

Aberschismic (formerly ''hemifamity'') divides an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] with the augmented fourth (~~C–F#~~) and [[50/49]] with the [[Pythagorean comma]].

Aberschismic (formerly ''hemifamity'') divides an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] with the augmented fourth (C–F♯) and [[50/49]] with the [[Pythagorean comma]] (C–B♯′).

Aberschismic can be further tempered to [[garibaldi]], which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, aberschismic can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.

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[[Projection pair]]s: 7 5120/729

~~; Music~~

* [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]

* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]

=== 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–vGb~~); 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. 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]].

Temperaments discussed elsewhere include:

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-{{Interwiki
-| en = Aberschismic family
-| de = Aberrschismisch
-| es =
-| ja =
-| ro =
-| ko = 헤미패미티 (음률)
-}}
 {{Technical data page}}
 The '''aberschismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: [[5120/5103]]).
 == Aberschismic ==
-Aberschismic (formerly ''hemifamity'') divides an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] with the augmented fourth (C–F#) and [[50/49]] with the [[Pythagorean comma]].
+{{Main| Aberschismic }}
+Aberschismic (formerly ''hemifamity'') divides an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] with the augmented fourth (C–F♯) and [[50/49]] with the [[Pythagorean comma]] (C–B♯′).
 Aberschismic can be further tempered to [[garibaldi]], which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, aberschismic can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths.
@@ Line 49: / Line 43: @@
 [[Projection pair]]s: 7 5120/729
-; Music
-* [http://www.archive.org/details/Choraled ''Choraled''] [http://www.archive.org/download/Choraled/Genewardsmith-Choraled.mp3 play] by [[Gene Ward Smith]]
-* [http://clones.soonlabel.com/public/micro/hemifamity27/hemifamity27-IF-20100917.mp3 ''Hemifamity27''] by [[Chris Vaisvil]]
 === 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–vGb); 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. 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]].
 Temperaments discussed elsewhere include: