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. | ||
Latest revision as of 14:21, 2 September 2026
| Interval information |
The cataharry comma (ratio: 19683/19600, monzo: [-4 9 -2 -2⟩) is a small 7-limit comma measuring about 6.1 cents. It is the difference between the 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 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, meaning it's expressible as (27/25)/(28/27)2.
Temperaments
Tempering out this comma leads to the rank-3 cataharry temperament, which splits the 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[1].