# 161/160

 Ratio 161/160 Subgroup monzo 2.5.7.23 [-5 -1 1 1⟩ Size in cents 10.78654¢ Name major kirnbergerisma Color name 23ozg2, twethozogu 2nd Twethozogu comma FJS name $\text{P1}^{7,23}_{5}$ Special properties superparticular,reduced Tenney height (log2 nd) 14.6528 Weil height (log2 max(n, d)) 14.6618 Wilson height (sopfr (nd)) 45 Harmonic entropy(Shannon, $\sqrt{nd}$) ~2.2106 bits Comma size small S-expression S21 × S22 × S23 open this interval in xen-calc

161/160, the major kirnbergerisma, is a small 23-limit superparticular comma about 10.79 cents in size. It forms the difference between 23/16 and 10/7.

## Temperaments

Tempering out this comma in the 23-limit leads to the rank-8 major kirnbergerismic temperament. In the 2.5.7.23 subgroup, tempering it out leads to the rank-3 major kirnbergic temperament.

### Major kirnbergerismic

Subgroup: full 23-limit

Comma list: 161/160

Mapping:
[⟨1 0 0 0 0 0 0 0  5],
⟨0 1 0 0 0 0 0 0  0],
⟨0 0 1 0 0 0 0 0  1],
⟨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 = 700.5007, ~5/4 = 385.1726, ~7/4 = 964.8038, ~11/8 = 548.1438, ~13/8 = 837.1324, ~17/16 = 101.2050, ~19/16 = 293.6154

### Major kirnbergic

Subgroup: 2.5.7.23

Comma list: 161/160

Mapping:
[⟨1 0 0  5],
⟨0 1 0  1],
⟨0 0 1 -1]]

Mapping generators: ~2, ~5, ~7

Optimal tuning (POTE): ~2 = 1＼1, ~5/4 = 385.1726, ~7/4 = 964.8038

## Etymology

The name major kirnbergerisma is contrast to the minor kirnbergerisma, 162/161, measuring about 10.72¢, which is the difference between 23/18 and 9/7. 161/160 is larger than 162/161 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.