Kirnberger's atom

Kirnberger's atom ([161 -84 -12⟩), is an unnoticeable 5-limit comma, 0.01536093 cents in size. It is the difference between:

• 81/80 and 11 schismas,
• A pythagorean comma and 12 schismas
• 12 81/80's and 11 pythagorean commas
• The raider and the pirate comma.

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 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.

Temperaments

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 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

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).