Archytas family: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
-Archytas temperament [[tempering out|tempers out]] [[64/63]], and thereby identifies the [[otonal]] [[tetrad]] with the [[dominant seventh chord]], which is familiar from [[12edo]] and also a feature of [[22edo]] and [[27edo]].
+{{Technical data page}}
+The '''archytas family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[64/63]]. The head of this family, archytas, tempers out 64/63 alone in the full 7-limit, so it has the same 2.3.7-[[subgroup]] structure as [[archy]] but giving [[prime harmonic|prime]] [[5/1|5]] an independent generator.
-[[Comma]]: 64/63
+See [[Archytas clan]] for the rank-2 temperament without the last generator of archytas, and its various extensions.
--limit minimax
+== Archytas ==
+{{Main| Archytas and ares }}
-[|1 0 0 0&gt;, |2 1/3 0 -1/3&gt;,
+[[Subgroup]]: 2.3.5.7
-|2 -2/3 1 -1/3&gt;, |2 -2/3 0 2/3&gt;]
-[[Eigenmonzo]]s: 2, 6/5, 7/5
+[[Comma list]]: [[64/63]]
--limit minimax
+{{Mapping|legend=1| 1 0 0 6 | 0 1 0 -2 | 0 0 1 0 }}
-[|1 0 0 0&gt;, |3/2 1/2 0 -1/4&gt;,
+: mapping generators: ~2, ~3, ~5
-|3/2 -1/2 1 -1/4&gt;, |3 -1 0 1/2&gt;]
-[[Eigenmonzo]]s: 2, 6/5, 9/7
+[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
-Lattice basis: 3/2 length 1.0508 5/4 length 2.3219
+Lattice basis:
+: 3/2 length = 1.0508, 5/4 length = 2.3219
+: Angle (3/2, 5/4) = 90 degrees
-Angle(3/2, 5/4) = 90 degrees
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}, ~5/4 = 392.3765{{c}}
+: [[error map]]: {{val| -3.045 +2.522 -0.027 +3.952 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.3901{{c}}, ~5/4 = 391.5995{{c}}
+: error map: {{val| 0.000 +7.435 +5.286 +12.394 }}
-Map to lattice: [&lt;0 1 0 -2|, &lt;0 0 1 0|]
+[[Minimax tuning]]:
+* [[7-odd-limit]]
+: {{monzo list| 1 0 0 0 | 2 1/3 0 -1/3 | 2 -2/3 1 -1/3 | 2 -2/3 0 2/3 }}
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7/3
+* [[9-odd-limit]]
+: {{monzo list| 1 0 0 0 | 3/2 1/2 0 -1/4 | 3/2 -1/2 1 -1/4 | 3 -1 0 1/2 }}
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7
-Map: [&lt;1 0 0 6|, &lt;0 1 0 -2|, &lt;0 0 1 0|]
+{{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }}
-[[Generator]]s: 2, 3, 5
+[[Badness]] (Sintel): 0.435
-{{EDOs|legend=1| 5, 7, 10, 12, 22, 27, 49 }}
+Scales: [[archytas12]], [[archytas12synch]]
-Scales:
-* [[archytas12]]
-* [[archytas12synch]]
-Music:
-Chris Vaisvil
-* [http://micro.soonlabel.com/hobbit_scales/daily201031--hobbit-archytas12.mp3 12 archytas hobbits with ale]
-Joel Taylor
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/Archytas12%20sync%20Study%203.mp3 Portrait of a Tragedy (Death of a Son) - Archytas12 sync Study 3]
 == Ares ==
-[[Comma]]s: 64/63, 100/99
+{{Main| Archytas and ares }}
+{{See also| Ptolemismic clan #Ares }}
-[[Minimax tuning]]
-[|1 0 0 0 0&gt;, |3/2 1/2 0 -1/4 0&gt;,
+[[Subgroup]]: 2.3.5.7.11
-|2 0 0 -1/2 1/2&gt;, |3 -1 0 1/2 0&gt;,
-|3 -1 0 -1/2 1&gt;]
-[[Eigenmonzo]]s: 2, 11/9, 9/7
+[[Comma list]]: 64/63, 100/99
-Lattice basis: 3/2 0.9878 6/5 1.4312
+{{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }}
-Angle(3/2, 6/5) = 75.0475 degrees
+[[Mapping to lattice]]: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]
-Map to lattice: [&lt;0 1 1 -2 0|, &lt;0 0 -1 0 -2|]
+Lattice basis:
+: 3/2 length = 0.9878, 6/5 length = 1.4312
+: Angle (3/2, 6/5) = 75.0475 degrees
-Map: [&lt;1 0 0 6 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 0 2|]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1197.0279{{c}}, ~3/2 = 707.9291{{c}}, ~5/4 = 390.4812{{c}}
+: [[error map]]: {{val| -2.972 +3.002 -1.777 +3.427 +1.898 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.6947{{c}}, ~5/4 = 390.0248{{c}}
+: error map: {{val| 0.000 +7.740 +3.711 +11.785 +9.342 }}
-[[Generator]]s: 2, 3, 5
+[[Minimax tuning]]:
+* [[11-odd-limit]]
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 0 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 -1 0 1/2 0 }}, {{monzo| 3 -1 0 -1/2 1 }}]
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11/9.9/7
-{{EDOs|legend=1| 12, 15, 22, 27e, 34d, 49 }}
+{{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}
-Badness: 0.000469
+[[Badness]] (Sintel): 0.563
 Scales: [[ares12]]
 == Vulcan ==
-Commas: 56/55, 64/63
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 56/55, 64/63
-Map: [&lt;1 0 0 6 9|, &lt;0 1 0 -2 -2|, &lt;0 0 1 0 -1|]
+{{Mapping|legend=1| 1 0 0 6 9 | 0 1 0 -2 -2 | 0 0 1 0 -1 }}
-Generators: 2, 3, 5
+[[Optimal tuning]]s:
+* [[CWE]]: ~2 = 1195.4265{{c}}, ~3/2 = 708.2623{{c}}, ~5/4 = 399.7405{{c}}
+: [[error map]]: {{val| -4.573 +1.734 +4.280 -3.644 +9.550 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 711.5280{{c}}, ~5/4 = 400.4798{{c}}
+: error map: {{val| 0.000 +9.573 +14.166 +8.118 +25.146 }}
-{{EDOs|legend=1| 5, 10, 12, 15, 27e }}
+{{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }}
-Badness: 0.000577
+[[Badness]] (Sintel): 0.693
-[[Category:Theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
+[[Category:Pages with mostly numerical content]]
-[[Category:Planar temperament]]
+[[Category:Archytas family| ]] <!-- main article -->
-[[Category:Archytas]]
 [[Category:Rank 3]]