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]].

== 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]]: <nowiki>[</nowiki>⟨1 0 0 0 0 0 0 0  1<nowiki>]</nowiki>,  ⟨0 1 0 0 0 0 0 0  4<nowiki>]</nowiki>,  ⟨0 0 1 0 0 0 0 0  0<nowiki>]</nowiki>,  ⟨0 0 0 1 0 0 0 0 -1<nowiki>]</nowiki>,  ⟨0 0 0 0 1 0 0 0  0<nowiki>]</nowiki>,  ⟨0 0 0 0 0 1 0 0  0<nowiki>]</nowiki>,  ⟨0 0 0 0 0 0 1 0  0<nowiki>]</nowiki>,  ⟨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: <nowiki>[</nowiki>⟨1 0 0  1<nowiki>]</nowiki>,  ⟨0 1 0  4<nowiki>]</nowiki>,  ⟨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 ==

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[[Category:Kirnbergerismic]]

[[Category:Minor kirnbergerismic]]

[[Category:Commas named after music theorists]]

[[Category:Commas named after composers]]

@@ Line 6: / Line 6: @@
 '''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 &nbsp;1<nowiki>]</nowiki>, <br>&nbsp;⟨0 1 0 0 0 0 0 0 &nbsp;4<nowiki>]</nowiki>, <br>&nbsp;⟨0 0 1 0 0 0 0 0 &nbsp;0<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 = 700.6501, ~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 &nbsp;1<nowiki>]</nowiki>, <br>&nbsp;⟨0 1 0 &nbsp;4<nowiki>]</nowiki>, <br>&nbsp;⟨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 ==
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 [[Category:Kirnbergerismic]]
+[[Category:Minor kirnbergerismic]]
+[[Category:Commas named after music theorists]]
+[[Category:Commas named after composers]]