325/324: Difference between revisions

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The '''marveltwin comma''', '''325/324''', is a [[13-limit]] (also 2.3.5.13 [[subgroup]]) [[comma]] measuring about 5.3{{cent}}. Its significance is in that it marks the difference between:

The '''marveltwin comma''', '''325/324''', is a [[small comma|small]] [[13-limit]] (also 2.3.5.13 [[subgroup]]) [[superparticular ratio|superparticular]] [[comma]] measuring about 5.3 [[cent]]s. Its significance is in that it marks the difference between [[16/13]] and a stack of two [[10/9]]'s, and [[13/9]] and a stack of two [[6/5]]'s. It can also be expressed as the difference between [[1053/1024]] and a stack of two [[81/80]]'s.

* [[16/13]] and two [[10/9]]'s

In terms of other superparticular ratios, it is the difference between:

* [[13/8]] and [[81/50]] ([[8/5]] * [[81/80]])

* [[13/9]] and two [[6/5]]'s

* [[1053/1024]] and two [[81/80]]'s

* [[25/24]] and [[27/26]]

* [[100/99]] and [[144/143]]

Being a [[triangle-particular|triangle superparticular]], it factors neatly into ([[625/624]])([[676/675]]). Other factorizations are:

Being a [[triangle-particular|triangle superparticular]], it factors neatly into ([[625/624]])⋅([[676/675]]). Other factorizations are:

* ([[385/384]])⋅([[2080/2079]])

* ([[364/363]])⋅([[3025/3024]])

* ([[352/351]])⋅([[4225/4224]])

* ([[351/350]])⋅([[4375/4374]])

* ([[385/384]])([[2080/2079]])

In the 17-limit it additionally factors into ([[595/594]])⋅([[715/714]]).

* ([[364/363]])([[3025/3024]])

* ([[352/351]])([[4225/4224]])

* ([[351/350]])([[4375/4374]])

* ([[595/594]])([[715/714]]) ~~In the 17-limit~~.

== Temperaments ==

Tempering out this comma leads to the rank-5 [[Marveltwin|marveltwin temperament]], which identifies [[16/13]] by two [[10/9]]'s stacked and enables [[marveltwin chords]]. The most significant further temperament is to temper out 625/624 and/or 676/675, equating 26/25 with 25/24~27/26. This in the 2.3.5.13 [[subgroup]] is the rank-2 [[cata]] temperament.

[[Tempering out]] this comma leads to the rank-5 [[Marveltwin|marveltwin temperament]], which identifies [[16/13]] by two [[10/9]]'s stacked and enables [[marveltwin chords]]. The most significant further temperament is to temper out 625/624 and/or 676/675, equating 26/25 with 25/24~27/26. This in the 2.3.5.13 [[subgroup]] is the rank-2 [[cata]] temperament.

== See also ==

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 }}
-The '''marveltwin comma''', '''325/324''', is a [[13-limit]] (also 2.3.5.13 [[subgroup]]) [[comma]] measuring about 5.3{{cent}}. Its significance is in that it marks the difference between:
+The '''marveltwin comma''', '''325/324''', is a [[small comma|small]] [[13-limit]] (also 2.3.5.13 [[subgroup]]) [[superparticular ratio|superparticular]] [[comma]] measuring about 5.3 [[cent]]s. Its significance is in that it marks the difference between [[16/13]] and a stack of two [[10/9]]'s, and [[13/9]] and a stack of two [[6/5]]'s. It can also be expressed as the difference between [[1053/1024]] and a stack of two [[81/80]]'s.
-* [[16/13]] and two [[10/9]]'s
+In terms of other superparticular ratios, it is the difference between:
-* [[13/8]] and [[81/50]] ([[8/5]] * [[81/80]])
-* [[13/9]] and two [[6/5]]'s
-* [[1053/1024]] and two [[81/80]]'s
 * [[25/24]] and [[27/26]]
 * [[100/99]] and [[144/143]]
-Being a [[triangle-particular|triangle superparticular]], it factors neatly into ([[625/624]])([[676/675]]). Other factorizations are:
+Being a [[triangle-particular|triangle superparticular]], it factors neatly into ([[625/624]])⋅([[676/675]]). Other factorizations are:
+* ([[385/384]])⋅([[2080/2079]])
+* ([[364/363]])⋅([[3025/3024]])
+* ([[352/351]])⋅([[4225/4224]])
+* ([[351/350]])⋅([[4375/4374]])
-* ([[385/384]])([[2080/2079]])
+In the 17-limit it additionally factors into ([[595/594]])⋅([[715/714]]).
-* ([[364/363]])([[3025/3024]])
-* ([[352/351]])([[4225/4224]])
-* ([[351/350]])([[4375/4374]])
-* ([[595/594]])([[715/714]]) In the 17-limit.
 == Temperaments ==
-Tempering out this comma leads to the rank-5 [[Marveltwin|marveltwin temperament]], which identifies [[16/13]] by two [[10/9]]'s stacked and enables [[marveltwin chords]]. The most significant further temperament is to temper out 625/624 and/or 676/675, equating 26/25 with 25/24~27/26. This in the 2.3.5.13 [[subgroup]] is the rank-2 [[cata]] temperament.
+[[Tempering out]] this comma leads to the rank-5 [[Marveltwin|marveltwin temperament]], which identifies [[16/13]] by two [[10/9]]'s stacked and enables [[marveltwin chords]]. The most significant further temperament is to temper out 625/624 and/or 676/675, equating 26/25 with 25/24~27/26. This in the 2.3.5.13 [[subgroup]] is the rank-2 [[cata]] temperament.
 == See also ==