Kirnberger's atom: Difference between revisions

'''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 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.

== Temperament ==

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]]).