162/161
| Interval information |
Twethuru comma
reduced
(Shannon, [math]\sqrt{nd}[/math])
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.
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