162/161

 Ratio 162/161 Subgroup monzo 2.3.7.23 [1 4 -1 -1⟩ Size in cents 10.71975¢ Name minor kirnbergerisma Color name 23ur-2, twethuru negative 2nd Twethuru comma FJS name $\text{P1}_{7,23}$ Special properties superparticular,reduced Tenney height (log2 nd) 14.6708 Weil height (log2 max(n, d)) 14.6797 Wilson height (sopfr (nd)) 44 Harmonic entropy(Shannon, $\sqrt{nd}$) ~2.19918 bits Comma size small open this interval in xen-calc

162/161, the minor kirnbergerisma, is a small 23-limit superparticular comma about 10.72 cents in size. It forms the difference between 23/18 and 9/7.

Temperaments

Tempering out this comma in the 23-limit leads to the rank-8 minor kirnbergerismic temperament. In the 2.3.7.23 subgroup, tempering it out leads to the rank-3 minor kirnbergic temperament.

Minor kirnbergerismic

Subgroup: full 23-limit

Comma list: 162/161

Mapping:
[⟨1 0 0 0 0 0 0 0  1],
⟨0 1 0 0 0 0 0 0  4],
⟨0 0 1 0 0 0 0 0  0],
⟨0 0 0 1 0 0 0 0 -1],
⟨0 0 0 0 1 0 0 0  0],
⟨0 0 0 0 0 1 0 0  0],
⟨0 0 0 0 0 0 1 0  0],
⟨0 0 0 0 0 0 0 1  0]]

Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13, ~17, ~19

Optimal tuning (POTE): ~2 = 1＼1, ~3/2 = 700.6501, ~5/4 = 386.6717, ~7/4 = 970.4738, ~11/8 = 551.8513, ~13/8 = 841.0982, ~17/16 = 105.5856, ~19/16 = 298.1680

Minor kirnbergic

Subgroup: 2.3.7.23

Comma list: 162/161

Mapping:
[⟨1 0 0  1],
⟨0 1 0  4],
⟨0 0 1 -1]]

Mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~2 = 1＼1, ~3/2 = 700.6501, ~7/4 = 970.4738

Etymology

The name minor kirnbergerisma is contrast to the major kirnbergerisma, 161/160, measuring about 10.79¢, which is the difference between 23/16 and 10/7. 162/161 is smaller than 161/160 by a 23-limit unnoticeable comma, 25921/25920. The word kirnbergerisma was introduced by Xenllium in 2023, for the following reason - 161/160 or 162/161 may be found in JI-approximated version of Kirnberger II as the difference between 1/2-comma-flattened fifth and pure fifth.