Octagar family: Difference between revisions

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The '''octagar family''' of ~~temperaments~~ are [[~~planar~~ temperament]]s tempering out the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.

The '''octagar family''' of [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7's 2.245 cents sharp, with just major thirds.

[[Linear temperament]]s that temper out the octagar comma can be found in [[octagar temperaments]].

== Octagar ==

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[4000/3969]]

~~[[Mapping]]: [~~{{~~val~~| 1 0 1 4 ~~}}, {{val~~| 0 1 0 -2 ~~}}, {{val~~| 0 0 2 3 }}]

~~Mapping~~ generators: ~2, ~3, ~63/40

: mapping generators: ~2, ~3, ~63/40

[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]

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[[Minimax tuning]]:

* [[7-odd-limit]]

: [{{monzo| 1 0 0 0 ~~}}, {{monzo~~| 5/6 1/3 1/2 -1/3 ~~}}, {{monzo~~| 0 0 1 0 ~~}}, {{monzo~~| 5/6 -2/3 1/2 2/3 }}]

: {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }}

: [[Eigenmonzo~~]]s~~ (unchanged-~~intervals~~): 2, 7/~~6, 5/4~~

: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7/3

* [[9-odd-limit]]

: [{{monzo| 1 0 0 0 ~~}}, {{monzo~~| 5/8 1/2 3/8 -1/4 ~~}}, {{monzo~~| 0 0 1 0 ~~}}, {{monzo~~| 5/4 -1 3/4 1/2 }}]

: {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }}

: [[Eigenmonzo~~]]s~~ (unchanged-~~intervals~~): 2, 5~~/4,~~ 9/7

: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7

{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}

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== Nakika ==

Subgroup: 2.3.5.7.11

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 100/99, 245/242

~~[[Mapping]]: [~~{{~~val~~| 1 0 1 4 4 ~~}}, {{val~~| 0 1 0 -2 -2 ~~}}, {{val~~| 0 0 2 3 4 }}]

Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]

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Comma list: 100/99, 105/104, 245/242

Mapping: [{{~~val~~| 1 0 1 4 4 2 ~~}}, {{val~~| 0 1 0 -2 -2 -1 ~~}}, {{val~~| 0 0 2 3 4 5 }}]

Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}

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[[Category:Temperament families]]

[[Category:Octagar family| ]]

[[Category:Octagar]]

[[Category:Octagar| ]]

[[Category:Rank 3]]

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-The '''octagar family''' of temperaments are [[planar temperament]]s tempering out the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.
+The '''octagar family''' of [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] the octagar comma, [[4000/3969]]. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7-odd-limit minimax tuning has fifths and 7's 2.245 cents sharp, with just major thirds.
 [[Linear temperament]]s that temper out the octagar comma can be found in [[octagar temperaments]].
 == Octagar ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: [[4000/3969]]
-[[Mapping]]: [{{val| 1 0 1 4 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 2 3 }}]
+{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}
-Mapping generators: ~2, ~3, ~63/40
+: mapping generators: ~2, ~3, ~63/40
 [[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
@@ Line 20: / Line 20: @@
 [[Minimax tuning]]:
 * [[7-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 5/6 1/3 1/2 -1/3 }}, {{monzo| 0 0 1 0 }}, {{monzo| 5/6 -2/3 1/2 2/3 }}]
+: {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }}
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/6, 5/4
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.7/3
 * [[9-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 5/8 1/2 3/8 -1/4 }}, {{monzo| 0 0 1 0 }}, {{monzo| 5/4 -1 3/4 1/2 }}]
+: {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }}
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 5/4, 9/7
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5.9/7
 {{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}
@@ Line 40: / Line 40: @@
 == Nakika ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 100/99, 245/242
-[[Mapping]]: [{{val| 1 0 1 4 4 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 2 3 4 }}]
+{{Mapping|legend=1| 1 0 1 4 4 | 0 1 0 -2 -2 | 0 0 2 3 4 }}
 Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]
@@ Line 67: / Line 67: @@
 Comma list: 100/99, 105/104, 245/242
-Mapping: [{{val| 1 0 1 4 4 2 }}, {{val| 0 1 0 -2 -2 -1 }}, {{val| 0 0 2 3 4 5 }}]
+Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}
 {{Optimal ET sequence|legend=1| 12f, 14cf, 15, 26, 29, 41 }}
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 [[Category:Temperament families]]
 [[Category:Octagar family| ]] <!-- main article -->
-[[Category:Octagar]]
+[[Category:Octagar| ]] <!-- key article -->
 [[Category:Rank 3]]