5120/5103: Difference between revisions

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'''5120/5103''', the '''aberschisma''' (formerly ''hemifamity comma''), is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 5.76 [[cent]]s. It is the difference between [[7/5]] (small septimal tritone) and [[1024/729]] (Pythagorean diminished fifth), or between [[10/7]] (large septimal tritone) and [[729/512]] (Pythagorean augmented fourth). It is also the difference between [[15/14]] and [[2187/2048]], between [[21/20]] and [[256/243]], between [[80/63]] and [[81/64]], etc., hence, between the septimal comma of [[64/63]] and the syntonic comma of [[81/80]], ~~as well as~~ between [[~~225~~/~~224~~]], the ~~marvel comma,~~ and [[~~32805/32768~~]], the schisma.

'''5120/5103''', the '''aberschisma''' (formerly ''hemifamity comma''), is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 5.76 [[cent]]s. It is the difference between [[7/5]] (small septimal tritone) and [[1024/729]] (Pythagorean diminished fifth), or between [[10/7]] (large septimal tritone) and [[729/512]] (Pythagorean augmented fourth). It is also the difference between [[15/14]] and [[2187/2048]], between [[21/20]] and [[256/243]], between [[80/63]] and [[81/64]], etc., hence, between the septimal comma of [[64/63]] and the syntonic comma of [[81/80]].

In terms of smaller commas, it is the difference between the [[marvel comma]] and the [[32805/32768|schisma]], as well as the sum of the schisma and the [[septischisma]]. In terms of [[S-expression]]s, this can be seen through the fact that one way of expressing the schisma is {{nowrap| S15/(S8/S9) }} while the septischisma can be expressed as {{nowrap| (S8/S9)<sup>2</sup>/S15 }}, that is, the reciprocal of the septischisma is an aberschisma below the schisma.

== Temperaments ==

[[Tempering out]] this comma in the 7-limit leads to the rank-3 [[aberschismic]] temperament, which splits the septimal quartertone [[36/35]] into two equal parts, each representing 81/80~64/63. Rank-2 temperaments that temper it out include [[hemififths]], [[amity]], and [[garibaldi]] to name three. [[Edo]]s that temper it out include {{EDOs| 12, 29, 34, 41, 46, 53, 58, 87, 94, 99, 111, 140, 145, 152, 239, 292, 391 }}, etc.

See [[Aberschismic family]] for the ~~rank-3~~ [[family]] where it is tempered out. See [[Aberschismic temperaments]] for a collection of rank-2 temperaments where it is tempered out.

See [[Aberschismic family]] for the [[family]] of rank-3 temperaments where it is tempered out. See [[Aberschismic temperaments]] for a collection of rank-2 temperaments where it is tempered out.

== Sagittal notation ==

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[[Category:Aberschismic]]

[[Category:Commas named for their regular temperament properties]]

[[Category:Commas named by translating something into another language]]

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-'''5120/5103''', the '''aberschisma''' (formerly ''hemifamity comma''), is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 5.76 [[cent]]s. It is the difference between [[7/5]] (small septimal tritone) and [[1024/729]] (Pythagorean diminished fifth), or between [[10/7]] (large septimal tritone) and [[729/512]] (Pythagorean augmented fourth). It is also the difference between [[15/14]] and [[2187/2048]], between [[21/20]] and [[256/243]], between [[80/63]] and [[81/64]], etc., hence, between the septimal comma of [[64/63]] and the syntonic comma of [[81/80]], as well as between [[225/224]], the marvel comma, and [[32805/32768]], the schisma.
+'''5120/5103''', the '''aberschisma''' (formerly ''hemifamity comma''), is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 5.76 [[cent]]s. It is the difference between [[7/5]] (small septimal tritone) and [[1024/729]] (Pythagorean diminished fifth), or between [[10/7]] (large septimal tritone) and [[729/512]] (Pythagorean augmented fourth). It is also the difference between [[15/14]] and [[2187/2048]], between [[21/20]] and [[256/243]], between [[80/63]] and [[81/64]], etc., hence, between the septimal comma of [[64/63]] and the syntonic comma of [[81/80]].
+In terms of smaller commas, it is the difference between the [[marvel comma]] and the [[32805/32768|schisma]], as well as the sum of the schisma and the [[septischisma]]. In terms of [[S-expression]]s, this can be seen through the fact that one way of expressing the schisma is {{nowrap| S15/(S8/S9) }} while the septischisma can be expressed as {{nowrap| (S8/S9)<sup>2</sup>/S15 }}, that is, the reciprocal of the septischisma is an aberschisma below the schisma.
 == Temperaments ==
 [[Tempering out]] this comma in the 7-limit leads to the rank-3 [[aberschismic]] temperament, which splits the septimal quartertone [[36/35]] into two equal parts, each representing 81/80~64/63. Rank-2 temperaments that temper it out include [[hemififths]], [[amity]], and [[garibaldi]] to name three. [[Edo]]s that temper it out include {{EDOs| 12, 29, 34, 41, 46, 53, 58, 87, 94, 99, 111, 140, 145, 152, 239, 292, 391 }}, etc.
-See [[Aberschismic family]] for the rank-3 [[family]] where it is tempered out. See [[Aberschismic temperaments]] for a collection of rank-2 temperaments where it is tempered out.
+See [[Aberschismic family]] for the [[family]] of rank-3 temperaments where it is tempered out. See [[Aberschismic temperaments]] for a collection of rank-2 temperaments where it is tempered out.
 == Sagittal notation ==
@@ Line 24: / Line 26: @@
 [[Category:Aberschismic]]
+[[Category:Commas named for their regular temperament properties]]
+[[Category:Commas named by translating something into another language]]