Kirnberger's atom
| Factorization | 2161 × 3-84 × 5-12 |
| Monzo | [161 -84 -12⟩ |
| Size in cents | 0.015360929¢ |
| Name | Kirnberger's atom |
| Color name | s14g1212, sepbisa-quadtrigu 12th |
| FJS name | [math]\text{19d12}_{5,5,5,5,5,5,5,5,5,5,5,5}[/math] |
| Special properties | reduced |
| Tenney height (log2 nd) | 322 |
| Weil height (log2 max(n, d)) | 322 |
| Wilson height (sopfr (nd)) | 634 |
| Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.3983 bits |
| Comma size | unnoticeable |
| open this interval in xen-calc | |
Kirnberger's atom, is a 5-limit unnoticeable comma. It is the difference between the syntonic comma and a stack of eleven schismas; [161 -84 -12⟩ in monzo and 0.01536093 cents 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 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 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 mapping and 3 steps by patent val).