Kirnberger's atom: Difference between revisions

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'''Kirnberger's atom''', is a [[5-limit]] [[~~unnoticeable~~ comma]]. It is the difference between ~~the~~ [[~~81/80|~~syntonic comma]] and a stack of eleven [[~~32805/32768|schismas~~]]~~; {{monzo|161 -84 -12}} in~~ [[~~monzo~~]] and ~~0.01536093 [[cent]]s in size~~.

'''Kirnberger's atom''' ({{monzo|legend=1| 161 -84 -12 }}), or simply the '''atom''', is an [[unnoticeable comma|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between a [[syntonic comma]] and a stack of eleven [[schisma]]s, between the [[Pythagorean comma]] and a stack of twelve schismas, or equivalently, between twelve syntonic commas and eleven Pythagorean commas.

Kirnberger's fifth, which is the perfect fifth of [[3/2]] flattened by a [[schisma]] is practically identical to seven steps of [[12edo]], which realizes a rational intonation version of the equal temperament. ~~Twelve~~ of Kirnberger's fifths of [[~~16384/10935~~]] ~~exceed seven octaves by the tiny interval of~~ (16384/10935)12 / 27 ~~= 2161 3-84 5-12, Kirnberger's atom~~.

[[16384/10935|Kirnberger's fifth]], which is the perfect fifth of [[3/2]] flattened by a [[schisma]], is practically identical to seven steps of [[12edo]], which realizes a rational intonation version of the equal temperament. Kirnberger's atom arises as the tiny interval by which twelve of Kirnberger's fifths exceed seven [[octave]]s, (16384/10935)12/27.

~~== Temperament ==~~

It may also be expressed as the difference between the [[raider comma]] and the [[pirate comma]].

~~Kirnberger's atom is tempered out in such notable EDOs~~ as ~~{{EDOs| 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 3684, 4296, 4308, 4908, 7980, 12276, 16572, 20868, 25164, 29460, 33756, and 46032 }}, leading to~~ the [[~~Very high accuracy temperaments #Atomic|atomic temperament~~]]~~, in which eleven schismas make up a syntonic comma~~ and ~~twelve schismas make up a~~ [[~~Pythagorean~~ comma]]~~; any tuning system ([[41edo]], for example) which the number of divisions of the octave is not multiple of 12 cannot be tempering out the Kirnberger's atom~~.

~~However, if one wants to accurately represent the interval without tempering it out, there are very large EDOs that do this.~~ [[~~78005edo~~]] ~~not only has a step size that's very close to~~ Kirnberger's atom ~~and consistently represents it, but it's also one of, if not~~ the ~~most accurate~~ 5-limit ~~EDO for its size~~. [[~~78123edo~~]]'s ~~step size is even closer, but~~ Kirnberger's atom ~~is not consistently represented (1 step via~~ [[~~Direct approximation|direct mapping~~]] ~~and 3 steps by~~ [[~~patent val~~]])

== Temperaments ==

[[Tempering out]] Kirnberger's atom leads to the 5-limit version of [[atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a Pythagorean comma. Many notable [[edo]]s temper out Kirnberger's atom, such as [[612edo]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.

== Approximation ==

However, if one wants to accurately represent the interval without tempering it out, there are very large edos that do this. [[78005edo]] not only has a step size that is very close to Kirnberger's atom and consistently represents it, but it is also one of, if not the most accurate 5-limit edo for its size. The edo with the closest step to Kirnberger's atom is [[78120edo]], but it is not consistently represented (1 step via [[direct approximation]] and 24 steps by [[patent val]]).

== ~~See also~~ ==

* [[~~Unnoticeable comma~~]]

[[Category:Atomic]]

[[Category:Kirnberger]]

[[Category:Commas named after composers]]

[[Category:Commas named after music theorists]]

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-'''Kirnberger's atom''', is a [[5-limit]] [[unnoticeable comma]]. It is the difference between the [[81/80|syntonic comma]] and a stack of eleven [[32805/32768|schismas]]; {{monzo|161 -84 -12}} in [[monzo]] and 0.01536093 [[cent]]s in size.
+'''Kirnberger's atom''' ({{monzo|legend=1| 161 -84 -12 }}), or simply the '''atom''', is an [[unnoticeable comma|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between a [[syntonic comma]] and a stack of eleven [[schisma]]s, between the [[Pythagorean comma]] and a stack of twelve schismas, or equivalently, between twelve syntonic commas and eleven Pythagorean commas.
-Kirnberger's fifth, which is the perfect fifth of [[3/2]] flattened by a [[schisma]] is practically identical to seven steps of [[12edo]], which realizes a rational intonation version of the equal temperament. Twelve of Kirnberger's fifths of [[16384/10935]] exceed seven octaves by the tiny interval of (16384/10935)<sup>12</sup> / 2<sup>7</sup> = 2<sup>161</sup> 3<sup>-84</sup> 5<sup>-12</sup>, Kirnberger's atom.
+[[16384/10935|Kirnberger's fifth]], which is the perfect fifth of [[3/2]] flattened by a [[schisma]], is practically identical to seven steps of [[12edo]], which realizes a rational intonation version of the equal temperament. Kirnberger's atom arises as the tiny interval by which twelve of Kirnberger's fifths exceed seven [[octave]]s, (16384/10935)<sup>12</sup>/2<sup>7</sup>.
-== Temperament ==
+It may also be expressed as the difference between the [[raider comma]] and the [[pirate comma]].
-Kirnberger's atom is tempered out in such notable EDOs as {{EDOs| 12, 612, 624, 1236, 1848, 2460, 3072, 3084, 3684, 4296, 4308, 4908, 7980, 12276, 16572, 20868, 25164, 29460, 33756, and 46032 }}, leading to the [[Very high accuracy temperaments #Atomic|atomic temperament]], in which eleven schismas make up a syntonic comma and twelve schismas make up a [[Pythagorean comma]]; any tuning system ([[41edo]], for example) which the number of divisions of the octave is not multiple of 12 cannot be tempering out the Kirnberger's atom.
-However, if one wants to accurately represent the interval without tempering it out, there are very large EDOs that do this. [[78005edo]] not only has a step size that's very close to Kirnberger's atom and consistently represents it, but it's also one of, if not the most accurate 5-limit EDO for its size. [[78123edo]]'s step size is even closer, but Kirnberger's atom is not consistently represented (1 step via [[Direct approximation|direct mapping]] and 3 steps by [[patent val]])
+== Temperaments ==
+[[Tempering out]] Kirnberger's atom leads to the 5-limit version of [[atomic]] temperament, in which eleven schismas make up a syntonic comma, and twelve schismas make up a Pythagorean comma. Many notable [[edo]]s temper out Kirnberger's atom, such as [[612edo]]. Any tuning system (such as [[41edo]]) for which the number of divisions of the octave is not divisible by 12 cannot temper out Kirnberger's atom.
+== Approximation ==
+However, if one wants to accurately represent the interval without tempering it out, there are very large edos that do this. [[78005edo]] not only has a step size that is very close to Kirnberger's atom and consistently represents it, but it is also one of, if not the most accurate 5-limit edo for its size. The edo with the closest step to Kirnberger's atom is [[78120edo]], but it is not consistently represented (1 step via [[direct approximation]] and 24 steps by [[patent val]]).
-== See also ==
-* [[Unnoticeable comma]]
 [[Category:Atomic]]
 [[Category:Kirnberger]]
+[[Category:Commas named after composers]]
+[[Category:Commas named after music theorists]]