Kirnberger's atom: Difference between revisions

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'''Kirnberger's atom''', is a [[5-limit]] [[~~unnoticeable~~ comma]]. It is the difference between [[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~~.

{{Infobox Interval

| Monzo = 161 -84 -12

| Name = Kirnberger's atom

| Color name = s14g1212, sepbisa-quadtrigu 12th

| Comma = yes

}}

'''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 fifth, which is ~~flattened~~ the perfect fifth of [[3/2]] 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)<~~font style="vertical-align: super;font-size: 0.8em;"~~>12</~~font~~> / 2<~~font style="vertical-align: super;font-size: 0.8em;"~~>7</~~font~~> = 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.

Kirnberger's atom is tempered out in such notable EDOs as 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|temperament]] in which eleven schismas make up a syntonic comma and twelve schismas make up a [[~~531441/524288|~~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.

== 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.

[[~~Category:~~5-limit]]

== Approximation ==

[[Category:~~Unnoticeable comma~~]]

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]]).

[[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 [[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.
+{{Infobox Interval
+| Monzo = 161 -84 -12
+| Name = Kirnberger's atom
+| Color name = s<sup>14</sup>g<sup>12</sup>12, sepbisa-quadtrigu 12th
+| Comma = yes
+}}
+'''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 fifth, which is flattened the perfect fifth of [[3/2]] 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)<font style="vertical-align: super;font-size: 0.8em;">12</font> / 2<font style="vertical-align: super;font-size: 0.8em;">7</font> = 2<font style="vertical-align: super;font-size: 0.8em;">161</font> 3<font style="vertical-align: super;font-size: 0.8em;">-84</font> 5<font style="vertical-align: super;font-size: 0.8em;">-12</font>, 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>.
-Kirnberger's atom is tempered out in such notable EDOs as 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|temperament]] in which eleven schismas make up a syntonic comma and twelve schismas make up a [[531441/524288|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.
+== 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.
-[[Category:5-limit]]
+== Approximation ==
-[[Category:Unnoticeable comma]]
+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]]).
+[[Category:Atomic]]
+[[Category:Kirnberger]]
+[[Category:Commas named after composers]]
+[[Category:Commas named after music theorists]]