Octagar family: Difference between revisions

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-The '''octagar family''' of temperaments are [[planar temperament]]s tempering out [[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 temperaments are [[planar temperament]]s tempering out [[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.
-== Octagar  ==
+== Octagar ==
-[[Comma]] c = 4000/3969
+[[Comma]]: 4000/3969
--limit minimax: 3 and 7 1/6c sharp, 5 just
+[[Mapping]]: [{{val|1 0 1 4}}, {{val|0 1 0 -2}}, {{val|0 0 2 3}}]
-[|1 0 0 0&gt;, |5/6 1/3 1/2 -1/3&gt;,
+Mapping generators: 2, ~3, ~63/40
-|0 0 1 0&gt;, |5/6 -2/3 1/2 2/3&gt;]
-[[Eigenmonzo]]s (unchanged intervals): 2, 7/6, 5/4
+Minimax tuning:
+* 7-limit minimax:
+: [{{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}}]
+: [[Eigenmonzo]]s (unchanged intervals): 2, 7/6, 5/4
--limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
+* 9-limit minimax:
+: [{{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}}]
+: [[Eigenmonzo]]s (unchanged intervals): 2, 5/4, 9/7
-[|1 0 0 0&gt;, |5/8 1/2 3/8 -1/4&gt;,
+Lattice basis: 63/50 length = 0.8966, 21/20 length = 1.0605
-|0 0 1 0&gt;, |5/4 -1 3/4 1/2&gt;]
-[[Eigenmonzo]]s (unchanged intervals): 2, 5/4, 9/7
-Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605
 Angle(63/50, 21/20) = 97.743 cents
-Map to lattice: [&lt;0 -1 -2 -1|, &lt;0 -1 0 2|]
+Map to lattice: [{{val|0 -1 -2 -1}}, {{val|0 -1 0 2}}]
-[[EDO|EDOs]]: [[12edo|12]], [[26edo|26]], [[27edo|27]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[121edo|121]], [[162edo|162]], [[189edo|189]], [[215edo|215]], [[230edo|230]]
+{{Val list|legend=1| 12, 26, 27, 41, 53, 94, 121, 162, 189, 215, 230 }}
-Badness: 0.000216
+[[Badness]]: 0.000216
-[[Projection pair]]s: 5 3969/800 7 27783/4000 to 2.3.7/5
+[[Projection pair]]s: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5
-==== [[Hobbits|Hobbit bases]]  ====
+==== [[Hobbits|Hobbit bases]] ====
 {2, 3, 7/5} subgroup
@@ Line 39: / Line 38: @@
 : 12800000/12252303, 107163/102400
-=== Nakika  ===
+=== Nakika ===
-Commas: 100/99, 245/242
+[[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 generators: 2, ~3, ~11/7
 Associated linear temperament: [[Tetracot family|octacot]]
-Lattice basis: 11/7 0.798 22/21 0.906
+Lattice basis: 11/7 length = 0.798, 22/21 length = 0.906
 Angle(11/7, 22/21) = 97.747 degrees
-Map to lattice: [&lt;0 1 2 1 2|, &lt;0 -1 0 2 2|]
+Map to lattice: [{{val|0 1 2 1 2}}, {{val|0 -1 0 2 2}}]
-Map: [&lt;1 0 1 4 4|, &lt;0 1 0 -2 -2|, &lt;0 0 2 3 4|]
+Vals: {{Val list| 12, 15, 26, 29, 41 }}
-EDOs: 12, 15, 26, 29, 41
 Badness: 0.000539
-Projection pairs: 5 242/49 7 21296/3087 11 234256/21609 to 2.3.11/7
+Projection pairs: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 to 2.3.11/7
 Scales: [[nakika12]]
+==== 13-limit ====
+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 generators: 2, ~3, ~11/7
+Vals: {{Val list| 12f, 15, 26, 29, 41 }}
 [[Category:Theory]]