Cataharry comma: Difference between revisions

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The '''cataharry comma''' ([[ratio]]: 19683/19600, {{monzo|legend=1| -4 9 -2 -2 }}) is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 6.1 [[cent]]s. It is the difference between the [[81/70|septimal ultramajor second (81/70)]] and its [[fourth complement]].

The '''cataharry comma''' ([[ratio]]: 19683/19600, {{monzo|legend=1| -4 9 -2 -2 }}) is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 6.1 [[cent]]s. It is the difference between the [[81/70|septimal ultramajor second (81/70)]] and its [[fourth complement]]. In terms of commas, it is equal to ([[81/80]])/([[245/243]]) and to ([[81/80]])2/([[49/48]]). The latter equivalence implies that if the cataharry comma is tempered out and if neither [[81/80]] nor [[49/48]] are tempered out, there will be an interval ~[[81/80]] above [[8/7]] and ~[[81/80]] below [[7/6]], and analogously for [[12/7]] and [[7/4]].

In terms of commas, it is equal to ([[81/80]])/([[245/243]]) and to ([[81/80]])2/([[49/48]]). The latter equivalence implies that if the cataharry comma is tempered out and if neither [[81/80]] nor [[49/48]] are tempered out, there will be an interval ~[[81/80]] above [[8/7]] and ~[[81/80]] below [[7/6]], and analogously for [[12/7]] and [[7/4]].

In the 13-limit, this comma factors into ([[351/350]])⋅([[729/728]]), which equals ([[676/675]])⋅(729/728)2, making it a [[lopsided comma]] with [[S-expression]] S26⋅S272.

== Temperaments ==

Tempering out this comma leads to the rank-3 [[cataharry]] temperament, which splits the [[4/3|just perfect fourth (4/3)]] into two exact hemifourths, each representing 81/70~280/243. See [[Cataharry family]] for the rank-3 family where it is tempered out. See [[Cataharry temperaments]] for a collection of rank-2 temperaments where it is tempered out.

[[Tempering out]] this comma leads to the rank-3 [[cataharry]] temperament, which splits the [[4/3|just perfect fourth (4/3)]] into two exact hemifourths, each representing 81/70~280/243.

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

== Etymology ==

The name ''cataharry'' was given by [[Gene Ward Smith]] in 2005 as a contraction of ''[[catakleismic]]'' and ''[[harry]]''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12900.html Yahoo! Tuning Group | ''Seven limit comma names from pairs of temperament names'']</ref>.

== ~~Notes~~ ==

== References ==

[[Category:Cataharry]]

[[Category:Commas named by combining multiple temperament names]]

[[Category:Commas named after ~~individuals~~]]

[[Category:Commas named after composers]]

[[Category:Commas named after music theorists]]

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+The '''cataharry comma''' ([[ratio]]: 19683/19600, {{monzo|legend=1| -4 9 -2 -2 }}) is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 6.1 [[cent]]s. It is the difference between the [[81/70|septimal ultramajor second (81/70)]] and its [[fourth complement]].
-The '''cataharry comma''' ([[ratio]]: 19683/19600, {{monzo|legend=1| -4 9 -2 -2 }}) is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 6.1 [[cent]]s. It is the difference between the [[81/70|septimal ultramajor second (81/70)]] and its [[fourth complement]]. In terms of commas, it is equal to ([[81/80]])/([[245/243]]) and to ([[81/80]])<sup>2</sup>/([[49/48]]). The latter equivalence implies that if the cataharry comma is tempered out and if neither [[81/80]] nor [[49/48]] are tempered out, there will be an interval ~[[81/80]] above [[8/7]] and ~[[81/80]] below [[7/6]], and analogously for [[12/7]] and [[7/4]].
+In terms of commas, it is equal to ([[81/80]])/([[245/243]]) and to ([[81/80]])<sup>2</sup>/([[49/48]]). The latter equivalence implies that if the cataharry comma is tempered out and if neither [[81/80]] nor [[49/48]] are tempered out, there will be an interval ~[[81/80]] above [[8/7]] and ~[[81/80]] below [[7/6]], and analogously for [[12/7]] and [[7/4]].
+In the 13-limit, this comma factors into ([[351/350]])⋅([[729/728]]), which equals ([[676/675]])⋅(729/728)<sup>2</sup>, making it a [[lopsided comma]] with [[S-expression]] S26⋅S27<sup>2</sup>.
 == Temperaments ==
-Tempering out this comma leads to the rank-3 [[cataharry]] temperament, which splits the [[4/3|just perfect fourth (4/3)]] into two exact hemifourths, each representing 81/70~280/243. See [[Cataharry family]] for the rank-3 family where it is tempered out. See [[Cataharry temperaments]] for a collection of rank-2 temperaments where it is tempered out.
+[[Tempering out]] this comma leads to the rank-3 [[cataharry]] temperament, which splits the [[4/3|just perfect fourth (4/3)]] into two exact hemifourths, each representing 81/70~280/243.
+See [[Cataharry family]] for the rank-3 family where it is tempered out. See [[Cataharry temperaments]] for a collection of rank-2 temperaments where it is tempered out.
 == Etymology ==
 The name ''cataharry'' was given by [[Gene Ward Smith]] in 2005 as a contraction of ''[[catakleismic]]'' and ''[[harry]]''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12900.html Yahoo! Tuning Group | ''Seven limit comma names from pairs of temperament names'']</ref>.
-== Notes ==
+== References ==
 [[Category:Cataharry]]
 [[Category:Commas named by combining multiple temperament names]]
-[[Category:Commas named after individuals]]
+[[Category:Commas named after composers]]
+[[Category:Commas named after music theorists]]