162/161: Difference between revisions
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'''162/161''', the '''minor kirnbergerisma''', is a [[small comma|small]] [[23-limit]] [[superparticular]] comma about 10.72 [[cent]]s in size. It forms the difference between [[23/18]] and [[9/7]]. | '''162/161''', the '''minor kirnbergerisma''', is a [[small comma|small]] [[23-limit]] [[superparticular]] comma about 10.72 [[cent]]s 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]]: <br><nowiki>[</nowiki>⟨1 0 0 0 0 0 0 0 1<nowiki>]</nowiki>, <br> ⟨0 1 0 0 0 0 0 0 4<nowiki>]</nowiki>, <br> ⟨0 0 1 0 0 0 0 0 0<nowiki>]</nowiki>, <br> ⟨0 0 0 1 0 0 0 0 -1<nowiki>]</nowiki>, <br> ⟨0 0 0 0 1 0 0 0 0<nowiki>]</nowiki>, <br> ⟨0 0 0 0 0 1 0 0 0<nowiki>]</nowiki>, <br> ⟨0 0 0 0 0 0 1 0 0<nowiki>]</nowiki>, <br> ⟨0 0 0 0 0 0 0 1 0<nowiki>]]</nowiki> | |||
: 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: <br><nowiki>[</nowiki>⟨1 0 0 1<nowiki>]</nowiki>, <br> ⟨0 1 0 4<nowiki>]</nowiki>, <br> ⟨0 0 1 -1<nowiki>]]</nowiki> | |||
: Mapping generators: ~2, ~3, ~7 | |||
Optimal tuning (POTE): ~2 = 1＼1, ~3/2 = 700.6501, ~7/4 = 970.4738 | |||
== Etymology == | == 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]]. The word ''kirnbergerisma'' was | 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 [[User:Xenllium|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 == | == See also == | ||
* [[Small comma]] | * [[Small comma]] | ||
* [[List of superparticular intervals]] | * [[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 | * [[161/120]] – a JI-approximated version of 1/2-comma-sharpened fourth | ||
* [[216/161]] – 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 | * [[240/161]] – a JI-approximated version of 1/2-comma-flattened fifth | ||
[[Category:Kirnbergerismic]] | [[Category:Kirnbergerismic]] | ||
[[Category:Minor kirnbergerismic]] | |||
[[Category:Commas named after music theorists]] | |||
[[Category:Commas named after composers]] | |||
Latest revision as of 01:04, 16 November 2024
| Interval information |
Twethuru comma
reduced
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