Kirnberger's atom

Interval information

Factorization	2161 × 3-84 × 5-12
Monzo	[161 -84 -12⟩
Size in cents	0.01536093¢
Name	Kirnberger's atom
Color name	s14g1212, sepbisa-quadtrigu 12th
FJS name	[math]\displaystyle{ \text{19d12}_{5,5,5,5,5,5,5,5,5,5,5,5} }[/math]
Special properties	reduced, reduced subharmonic
Tenney norm (log2 nd)	322
Weil norm (log2 max(n, d))	322
Wilson norm (sopfr(nd))	634
Comma size	unnoticeable
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Kirnberger's atom (monzo: [161 -84 -12⟩), or simply the atom, is an unnoticeable 5-limit comma, 0.01536093 cents in size. It is the difference between a syntonic comma and a stack of eleven 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 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 octaves, (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 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 edos 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).