Kirnberger's atom: Difference between revisions

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'''Kirnberger's atom''' ({{monzo|~~legned~~=1| 161 -84 -12 }}), is an [[unnoticeable comma|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between:

'''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.

* [[81/80]] ~~and 11~~ [[~~32805~~/~~32768|schismas~~]],

[[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.

* A [[~~pythagorean comma~~]] ~~and 12 schismas~~

* 12 81/80's and 11 pythagorean commas

* The [[~~raider~~]] ~~and~~ the [[~~pirate~~]] ~~comma~~.

~~[[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~~.

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

== Temperaments ==

[[Tempering out]] Kirnberger's atom leads 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]]. Many notable [[~~EDO|edos~~]] temper out Kirnberger's atom, such as ~~{{EDOs| 12, 612, 624, 1236, 1848, 2460, 3072, 3684, 4296, 7980, 12276, 16572, 20868, 25164, and 46032 }}~~. 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.

[[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. [[~~78123edo~~]]~~'s step size is even closer~~, but ~~Kirnberger's atom~~ is not consistently represented (1 step via [[direct approximation]] and 3 steps by [[patent val]]).

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]]).

[[Category:Atomic]]

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-'''Kirnberger's atom''' ({{monzo|legned=1| 161 -84 -12 }}), is an [[unnoticeable comma|unnoticeable]] [[5-limit]] [[comma]], 0.01536093 [[cent]]s in size. It is the difference between:
+'''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.
-* [[81/80]] and 11 [[32805/32768|schismas]],
+[[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>.
-* A [[pythagorean comma]] and 12 schismas
-* 12 81/80's and 11 pythagorean commas
-* The [[raider]] and the [[pirate]] comma.
-[[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>.
+It may also be expressed as the difference between the [[raider comma]] and the [[pirate comma]].
 == Temperaments ==
-[[Tempering out]] Kirnberger's atom leads 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]]. Many notable [[EDO|edos]] temper out Kirnberger's atom, such as {{EDOs| 12, 612, 624, 1236, 1848, 2460, 3072, 3684, 4296, 7980, 12276, 16572, 20868, 25164, and 46032 }}. 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.
+[[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. [[78123edo]]'s step size is even closer, but Kirnberger's atom is not consistently represented (1 step via [[direct approximation]] and 3 steps by [[patent val]]).
+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]]).
 [[Category:Atomic]]