161/160

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Interval information
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 [math]\text{P1}^{7,23}_{5}[/math]
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, [math]\sqrt{nd}[/math])
~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.

See also