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'''161/160''', the '''major kirnbergerisma''', is a [[small comma|small]] [[23-limit]] [[superparticular]] comma about 10.79 [[cent]]s in size. It forms the difference between [[23/16]] and [[10/7]].
'''161/160''', the '''major kirnbergerisma''', is a [[small comma|small]] [[23-limit]] [[superparticular]] comma about 10.79 [[cent]]s 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]]: <br><nowiki>[</nowiki>⟨1 0 0 0 0 0 0 0 &nbsp;5<nowiki>]</nowiki>, <br>&nbsp;⟨0 1 0 0 0 0 0 0 &nbsp;0<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 1 0 0 0 0 0 &nbsp;1<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 0 1 0 0 0 0 -1<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 0 0 1 0 0 0 &nbsp;0<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 0 0 0 1 0 0 &nbsp;0<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 0 0 0 0 1 0 &nbsp;0<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 0 0 0 0 0 1  &nbsp;0<nowiki>]]</nowiki>
: 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: <br><nowiki>[</nowiki>⟨1 0 0 &nbsp;5<nowiki>]</nowiki>, <br>&nbsp;⟨0 1 0 &nbsp;1<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 1 -1<nowiki>]]</nowiki>
: Mapping generators: ~2, ~5, ~7
Optimal tuning (POTE): ~2 = 1＼1, ~5/4 = 385.1726, ~7/4 = 964.8038


== Etymology ==
== 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]]. The word ''kirnbergerisma'' was named 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, and 161/160 is larger than 162/161 by a [[23-limit]] [[unnoticeable comma]], 25921/25920.
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 [[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
* [[161/108]] – a JI-approximated version of 1/2-comma-flattened fifth
* [[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:Major kirnbergerismic]]
[[Category:Commas named after music theorists]]
[[Category:Commas named after composers]]

Latest revision as of 01:04, 16 November 2024

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]\displaystyle{ \text{P1}^{7,23}_{5} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 14.6528
Weil norm (log2 max(n, d)) 14.6618
Wilson norm (sopfr(nd)) 45
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

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