Archytas family: Difference between revisions

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- CTE & POTE tunings
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-10 19:35:43 UTC</tt>.<br>
: The original revision id was <tt>192308444</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">Archytas temperament 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]].


[[Comma]]: 64/63
See [[Archytas clan]] for the rank-2 temperament without the last generator of archytas, and its various extensions.


7-limit minimax
== Archytas ==
[|1 0 0 0&gt;, |2 1/3 0 -1/3&gt;,
{{Main| Archytas and ares }}
|2 -2/3 1 -1/3&gt;, |2 -2/3 0 2/3&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 6/5, 7/5


9-limit minimax
[[Subgroup]]: 2.3.5.7
[|1 0 0 0&gt;, |3/2 1/2 0 -1/4&gt;,
|3/2 -1/2 1 -1/4&gt;, |3 -1 0 1/2&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 6/5, 9/7


Lattice basis: 3/2 length 1.0508 5/4 length 2.3219
[[Comma list]]: [[64/63]]
Angle(3/2, 5/4) = 90 degrees
Map to lattice: [&lt;0 1 0 -2|, &lt;0 0 1 0|]


Map: [&lt;1 0 0 6|, &lt;0 1 0 -2|, &lt;0 0 1 0|]
{{Mapping|legend=1| 1 0 0 6 | 0 1 0 -2 | 0 0 1 0 }}
[[Generator|Generators]]: 2, 3, 5
[[edo|Edos]]: [[10edo|10]], [[12edo|12]], [[15edo|15]], [[22edo|22]], [[27edo|27]], [[37edo|37]], [[49edo|49]], [[59edo|59]]
Scales: [[archytas12]]


Music: Chris Vaisvil
: mapping generators: ~2, ~3, ~5
[[http://micro.soonlabel.com/hobbit_scales/daily201031--hobbit-archytas12.mp3|12 archytas hobbits with ale]]
==11 limit children==


==Ares==
[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
[[Comma|Commas]]: 64/63, 100/99


[[Minimax tuning]]
Lattice basis:
[|1 0 0 0 0&gt;, |3/2 1/2 0 -1/4 0&gt;,
: 3/2 length = 1.0508, 5/4 length = 2.3219
|2 0 0 -1/2 1/2&gt;, |3 -1 0 1/2 0&gt;,
: Angle (3/2, 5/4) = 90 degrees
|3 -1 0 -1/2 1&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 11/9, 9/7


Lattice basis: 3/2 0.9878 6/5 1.4312
[[Optimal tuning]]s:
Angle(3/2, 6/5) = 75.0475 degrees
* [[WE]]: ~2 = 1196.9552{{c}}, ~3/2 = 707.5215{{c}}, ~5/4 = 392.3765{{c}}
Map to lattice: [&lt;0 1 1 -2 0|, &lt;0 0 -1 0 -2|]
: [[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: [&lt;1 0 0 6 2|, &lt;0 1 0 -2 -2|, &lt;0 0 1 0 2|]
[[Minimax tuning]]:  
[[Generator|Generators]]: 2, 3, 5
* [[7-odd-limit]]
[[edo|Edos]]: [[12edo|12]], [[15edo|15]], [[22edo|22]], [[37edo|37]], [[49edo|49]]
: {{monzo list| 1 0 0 0 | 2 1/3 0 -1/3 | 2 -2/3 1 -1/3 | 2 -2/3 0 2/3 }}
Badness: 0.000469</pre></div>
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.7/3
<h4>Original HTML content:</h4>
* [[9-odd-limit]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Archytas family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Archytas temperament tempers out 64/63, and thereby identifies the otonal tetrad with the dominant seventh chord, which is familiar from &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; and also a feature of &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; and &lt;a class="wiki_link" href="/27edo"&gt;27edo&lt;/a&gt;.&lt;br /&gt;
: {{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 }}
&lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3.9/7
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt;: 64/63&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd }}
7-limit minimax&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |2 1/3 0 -1/3&amp;gt;, &lt;br /&gt;
[[Badness]] (Sintel): 0.435
|2 -2/3 1 -1/3&amp;gt;, |2 -2/3 0 2/3&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 6/5, 7/5&lt;br /&gt;
Scales: [[archytas12]], [[archytas12synch]]
&lt;br /&gt;
 
9-limit minimax&lt;br /&gt;
== Ares ==
[|1 0 0 0&amp;gt;, |3/2 1/2 0 -1/4&amp;gt;, &lt;br /&gt;
{{Main| Archytas and ares }}
|3/2 -1/2 1 -1/4&amp;gt;, |3 -1 0 1/2&amp;gt;]&lt;br /&gt;
{{See also| Ptolemismic clan #Ares }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 6/5, 9/7&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Lattice basis: 3/2 length 1.0508 5/4 length 2.3219&lt;br /&gt;
 
Angle(3/2, 5/4) = 90 degrees&lt;br /&gt;
[[Comma list]]: 64/63, 100/99
Map to lattice: [&amp;lt;0 1 0 -2|, &amp;lt;0 0 1 0|]&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 0 0 6 2 | 0 1 0 -2 -2 | 0 0 1 0 2 }}
Map: [&amp;lt;1 0 0 6|, &amp;lt;0 1 0 -2|, &amp;lt;0 0 1 0|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
[[Mapping to lattice]]: [{{val| 0 1 1 -2 0 }}, {{val| 0 0 -1 0 -2 }}]
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/10edo"&gt;10&lt;/a&gt;, &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49&lt;/a&gt;, &lt;a class="wiki_link" href="/59edo"&gt;59&lt;/a&gt;&lt;br /&gt;
 
Scales: &lt;a class="wiki_link" href="/archytas12"&gt;archytas12&lt;/a&gt;&lt;br /&gt;
Lattice basis:
&lt;br /&gt;
: 3/2 length = 0.9878, 6/5 length = 1.4312
Music: Chris Vaisvil &lt;br /&gt;
: Angle (3/2, 6/5) = 75.0475 degrees
&lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/hobbit_scales/daily201031--hobbit-archytas12.mp3" rel="nofollow"&gt;12 archytas hobbits with ale&lt;/a&gt;&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-11 limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;11 limit children&lt;/h2&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[WE]]: ~2 = 1197.0279{{c}}, ~3/2 = 707.9291{{c}}, ~5/4 = 390.4812{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Ares"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Ares&lt;/h2&gt;
: [[error map]]: {{val| -2.972 +3.002 -1.777 +3.427 +1.898 }}
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 64/63, 100/99&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.6947{{c}}, ~5/4 = 390.0248{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +7.740 +3.711 +11.785 +9.342 }}
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |3/2 1/2 0 -1/4 0&amp;gt;, &lt;br /&gt;
[[Minimax tuning]]:  
|2 0 0 -1/2 1/2&amp;gt;, |3 -1 0 1/2 0&amp;gt;, &lt;br /&gt;
* [[11-odd-limit]]
|3 -1 0 -1/2 1&amp;gt;]&lt;br /&gt;
: [{{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 }}]
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 11/9, 9/7&lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.11/9.9/7
&lt;br /&gt;
 
Lattice basis: 3/2 0.9878 6/5 1.4312&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 15, 22, 27e, 34d, 49, 181bbcddeeee }}
Angle(3/2, 6/5) = 75.0475 degrees&lt;br /&gt;
 
Map to lattice: [&amp;lt;0 1 1 -2 0|, &amp;lt;0 0 -1 0 -2|]&lt;br /&gt;
[[Badness]] (Sintel): 0.563
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 6 2|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 1 0 2|]&lt;br /&gt;
Scales: [[ares12]]
&lt;a class="wiki_link" href="/Generator"&gt;Generators&lt;/a&gt;: 2, 3, 5&lt;br /&gt;
 
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/15edo"&gt;15&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37&lt;/a&gt;, &lt;a class="wiki_link" href="/49edo"&gt;49&lt;/a&gt;&lt;br /&gt;
== Vulcan ==
Badness: 0.000469&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 56/55, 64/63
 
{{Mapping|legend=1| 1 0 0 6 9 | 0 1 0 -2 -2 | 0 0 1 0 -1 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 5, 10, 12, 15, 27e }}
 
[[Badness]] (Sintel): 0.693
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Archytas family| ]] <!-- main article -->
[[Category:Rank 3]]

Latest revision as of 10:00, 14 July 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The archytas family of rank-3 temperaments 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 5 an independent generator.

See Archytas clan for the rank-2 temperament without the last generator of archytas, and its various extensions.

Archytas

Main article: Archytas and ares

Subgroup: 2.3.5.7

Comma list: 64/63

Mapping: [⟨1 0 0 6], ⟨0 1 0 -2], ⟨0 0 1 0]]

mapping generators: ~2, ~3, ~5

Mapping to lattice: [⟨0 1 0 -2], ⟨0 0 1 0]]

Lattice basis:

3/2 length = 1.0508, 5/4 length = 2.3219
Angle (3/2, 5/4) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1196.9552 ¢, ~3/2 = 707.5215 ¢, ~5/4 = 392.3765 ¢
error map: ⟨-3.045 +2.522 -0.027 +3.952]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.3901 ¢, ~5/4 = 391.5995 ¢
error map: ⟨0.000 +7.435 +5.286 +12.394]

Minimax tuning:

[[1 0 0 0⟩, [2 1/3 0 -1/3⟩, [2 -2/3 1 -1/3⟩, [2 -2/3 0 2/3⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.7/3
[[1 0 0 0⟩, [3/2 1/2 0 -1/4⟩, [3/2 -1/2 1 -1/4⟩, [3 -1 0 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.5/3.9/7

Optimal ET sequence: 5, 7, 10, 12, 15, 22, 27, 39d, 49, 88bcd, 110bcdd

Badness (Sintel): 0.435

Scales: archytas12, archytas12synch

Ares

Main article: Archytas and ares
See also: Ptolemismic clan § Ares

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99

Mapping: [⟨1 0 0 6 2], ⟨0 1 0 -2 -2], ⟨0 0 1 0 2]]

Mapping to lattice: [⟨0 1 1 -2 0], ⟨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 tunings:

  • WE: ~2 = 1197.0279 ¢, ~3/2 = 707.9291 ¢, ~5/4 = 390.4812 ¢
error map: ⟨-2.972 +3.002 -1.777 +3.427 +1.898]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.6947 ¢, ~5/4 = 390.0248 ¢
error map: ⟨0.000 +7.740 +3.711 +11.785 +9.342]

Minimax tuning:

[[1 0 0 0 0⟩, [3/2 1/2 0 -1/4 0⟩, [2 0 0 -1/2 1/2⟩, [3 -1 0 1/2 0⟩, [3 -1 0 -1/2 1⟩]
unchanged-interval (eigenmonzo) basis: 2.11/9.9/7

Optimal ET sequence: 12, 15, 22, 27e, 34d, 49, 181bbcddeeee

Badness (Sintel): 0.563

Scales: ares12

Vulcan

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63

Mapping: [⟨1 0 0 6 9], ⟨0 1 0 -2 -2], ⟨0 0 1 0 -1]]

Optimal tunings:

  • CWE: ~2 = 1195.4265 ¢, ~3/2 = 708.2623 ¢, ~5/4 = 399.7405 ¢
error map: ⟨-4.573 +1.734 +4.280 -3.644 +9.550]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 711.5280 ¢, ~5/4 = 400.4798 ¢
error map: ⟨0.000 +9.573 +14.166 +8.118 +25.146]

Optimal ET sequence: 5, 10, 12, 15, 27e

Badness (Sintel): 0.693

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