161/160
| Interval information |
Twethozogu comma
reduced
(Shannon, [math]\sqrt{nd}[/math])
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
- Small comma
- List of superparticular intervals
- 161/108 – a JI-approximated version of 1/2-comma-flattened fifth
- 161/120 – a JI-approximated version of 1/2-comma-sharpened fourth
- 216/161 – a JI-approximated version of 1/2-comma-sharpened fourth
- 240/161 – a JI-approximated version of 1/2-comma-flattened fifth