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 ~~the~~ [[81/80~~|syntonic comma~~]] and ~~a stack of eleven~~ [[32805/32768|schismas]], ~~between the~~ [[~~Pythagorean~~ comma]] and ~~a stack of twelve~~ schismas~~, or equivalently, between twelve syntonic~~ commas and ~~eleven Pythagorean commas~~.

'''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:

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

* 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>.

== Temperaments ==

Kirnberger's atom is [[tempering out|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 Kirnberger's atom.

Kirnberger's atom is [[tempering out|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 a multiple of 12 cannot temper out Kirnberger's atom.

== Approximation ==

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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 the [[81/80|syntonic comma]] and a stack of eleven [[32805/32768|schismas]], between the [[Pythagorean comma]] and a stack of twelve schismas, or equivalently, between twelve syntonic commas and eleven Pythagorean commas.
+'''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:
+* [[81/80]] and 11 [[32805/32768|schismas]],
+* 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>.
 == Temperaments ==
-Kirnberger's atom is [[tempering out|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 Kirnberger's atom.
+Kirnberger's atom is [[tempering out|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 a multiple of 12 cannot temper out Kirnberger's atom.
 == Approximation ==