Cataharry comma: Difference between revisions
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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>. | 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]] {{nowrap| S26⋅S27<sup>2</sup> }}, meaning it's expressible as {{nowrap| ([[27/25]])/([[28/27]])<sup>2</sup>. }} | ||
== Temperaments == | == 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. | [[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. | ||