Archytas family: Difference between revisions

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-The '''archytas family''' of temperaments [[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.
-= Archytas =
+See [[Archytas clan]] for the rank-2 temperament without the last generator of archytas, and its various extensions.
-Period: 1\1
+== Archytas ==
+{{Main| Archytas and ares }}
-Optimal ([[POTE]]) generators: ~3/2 = 709.3213, ~5/4 = 393.3747
+[[Subgroup]]: 2.3.5.7
-EDO generators: [[15edo|(9, 5)\15]], [[22edo|(13, 7)\22]], [[27edo|(16, 9)\27]]
+[[Comma list]]: [[64/63]]
-Scales: [[archytas12]], [[archytas12synch]]
+{{Mapping|legend=1| 1 0 0 6 | 0 1 0 -2 | 0 0 1 0 }}
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-[[Comma list]]: 64/63
-Mapping: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
-Mapping [[generator]]s: ~2, ~3, ~5
+: mapping generators: ~2, ~3, ~5
-Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
+[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
 Lattice basis:
@@ Line 27: / Line 21: @@
 : Angle (3/2, 5/4) = 90 degrees
-Minimax tuning:
+[[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 }}
+[[Minimax tuning]]:
 * [[7-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}]
+: {{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]]s: 2, 6/5, 7/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7/3
 * [[9-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}]
+: {{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]]s: 2, 6/5, 9/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7
-{{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }}
+{{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }}
-</div></div>
+[[Badness]] (Sintel): 0.435
-; Music
+Scales: [[archytas12]], [[archytas12synch]]
-* 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 =
+== Ares ==
+{{Main| Archytas and ares }}
+{{See also| Ptolemismic clan #Ares }}
-Period: 1\1
+[[Subgroup]]: 2.3.5.7.11
-Optimal ([[POTE]]) generators: ~3/2 = 709.6869, ~5/4 = 391.4507
-EDO generators: [[15edo|(9, 5)\15]], [[22edo|(13, 7)\22]], [[27edo|(16, 9)\27]]
-Scales: [[ares12]]
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
 [[Comma list]]: 64/63, 100/99
-Mapping: [{{val| 1 0 0 6 2 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 0 2 }}]
+{{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }}
-Mapping [[generator]]s: ~2, ~3, ~5
+[[Mapping to lattice]]: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]
-Map to lattice: [{{val| 0 1 1 -2 0 }}, {{val| 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
+[[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 }}
 [[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]]s: 2, 11/9, 9/7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11/9.9/7
-{{Val list|legend=1| 12, 15, 22, 27e, 34d, 49 }}
+{{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}
-Badness: 0.469 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.563
-</div></div>
+Scales: [[ares12]]
-= Vulcan =
-Period: 1\1
-Optimal ([[POTE]]) generators: ~3/2 = 710.9720, ~5/4 = 401.2699
-EDO generators: [[10edo|(6, 3)\10]], [[15edo|(9, 5)\15]], [[17edo|(10, 6)\17]]
-Scales:
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
-<div style="line-height:1.6;">Technical data</div>
-<div class="mw-collapsible-content">
-Comma list: 56/55, 64/63
+== Vulcan ==
+[[Subgroup]]: 2.3.5.7.11
-Mapping: [{{val| 1 0 0 6 9 }}, {{val| 0 1 0 -2 -2 }}, {{val| 0 0 1 0 -1 }}]
+[[Comma list]]: 56/55, 64/63
-Mapping generators: ~2, ~3, ~5
+{{Mapping|legend=1| 1 0 0 6 9 | 0 1 0 -2 -2 | 0 0 1 0 -1 }}
-{{Val list|legend=1| 5, 10, 12, 15, 27e }}
+[[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 }}
-Badness: 0.577 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }}
-</div></div>
+[[Badness]] (Sintel): 0.693
-[[Category:Theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
+[[Category:Pages with mostly numerical content]]
-[[Category:Archytas]]
+[[Category:Archytas family| ]] <!-- main article -->
 [[Category:Rank 3]]