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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Inaccessible}}Todo: Incorporate Hendrix diamond ideas
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:Praimhin|Praimhin]] and made on <tt>2016-08-04 01:12:44 UTC</tt>.<br>
: The original revision id was <tt>588772064</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">The **15-limit tonality diamond** has the following notes:


|| 1/1 || 9/8 || 5/4 || 11/8 || 3/2 || 13/8 || 7/4 || 15/8 ||
The '''15-limit tonality diamond''' has the following notes:
|| 16/9 || 1/1 || 10/9 || 11/9 || 4/3 || 13/9 || 14/9 || 5/3 ||
 
|| 8/5 || 9/5 || 1/1 || 11/10 || 6/5 || 13/10 || 7/5 || 3/2 ||
{| class="wikitable"
|| 16/11 || 18/11 || 20/11 || 1/1 || 12/11 || 13/11 || 14/11 || 15/11 ||
|-
|| 4/3 || 3/2 || 5/3 || 11/6 || 1/1 || 13/12 || 7/6 || 5/4 ||
| | 1/1
|| 16/13 || 18/13 || 20/13 || 22/13 || 24/13 || 1/1 || 14/13 || 15/13 ||
| | 9/8
|| 8/7 || 9/7 || 10/7 || 11/7 || 12/7 || 13/7 || 1/1 || 15/14 ||
| | 5/4
|| 16/15 || 6/5 || 4/3 || 22/15 || 8/5 || 26/15 || 28/15 || 1/1 ||
| | 11/8
==Symmetry group==  
| | 3/2
| | 13/8
| | 7/4
| | 15/8
|-
| | 16/9
| | 1/1
| | 10/9
| | 11/9
| | 4/3
| | 13/9
| | 14/9
| | 5/3
|-
| | 8/5
| | 9/5
| | 1/1
| | 11/10
| | 6/5
| | 13/10
| | 7/5
| | 3/2
|-
| | 16/11
| | 18/11
| | 20/11
| | 1/1
| | 12/11
| | 13/11
| | 14/11
| | 15/11
|-
| | 4/3
| | 3/2
| | 5/3
| | 11/6
| | 1/1
| | 13/12
| | 7/6
| | 5/4
|-
| | 16/13
| | 18/13
| | 20/13
| | 22/13
| | 24/13
| | 1/1
| | 14/13
| | 15/13
|-
| | 8/7
| | 9/7
| | 10/7
| | 11/7
| | 12/7
| | 13/7
| | 1/1
| | 15/14
|-
| | 16/15
| | 6/5
| | 4/3
| | 22/15
| | 8/5
| | 26/15
| | 28/15
| | 1/1
|}
 
==Symmetry group==


The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:
The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:
* 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9
* 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8
* 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8
* 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13


===Orbits and Invariant Subsets===
<ul><li>Transformation ''R'': 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>Transformation ''S'': 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>Transformation ''S''': 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>Transformation ''T'': 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul>
[to be included]</pre></div>
 
<h4>Original HTML content:</h4>
These generators have the relations ''R''² = ''S''² = ''T''² = ''S'''² = I, (''SS''')³ = I, ''RS'' = ''SR'', ''RS''' = ''S'''''R'', and ''T'' commutes with the three other generators. Thus the symmetry group is isomorphic to ''S''₃ × ''C''₂².
<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;15-limit tonality diamond&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;strong&gt;15-limit tonality diamond&lt;/strong&gt; has the following notes:&lt;br /&gt;
&lt;br /&gt;


===Orbits and Invariant Subsets===
The [[Hendrix_diamond|Hendrix diamond]] is invariant under action by ''R,'' ''S''' and ''T'', and the images of the action of ''S'' and ''S''² on the Hendrix diamond are the [[11-Hendrix_diamond|11-Hendrix diamond]] and [[13-Hendrix_diamond|13-Hendrix diamond]] respectively.


&lt;table class="wiki_table"&gt;
Two other interesting invariant subsets are the 5-limit tonality diamond and the tonality diamond constructed from the harmonics 1, 3, 5, 9 and 15.
    &lt;tr&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/8&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;20/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;24/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/13&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;10/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;22/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;8/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;26/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;28/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1/1&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Symmetry group"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Symmetry group&lt;/h2&gt;
[[Category:15-odd-limit]]
&lt;br /&gt;
[[Category:Diamond]]
The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;3:2, 5:4, 7:4, 11:8, 13:8 -&amp;gt; 4:3, 5:3, 14:9, 11:9, 13:9&lt;/li&gt;&lt;li&gt;3:2, 5:4, 7:4, 11:8, 13:8 -&amp;gt; 3:2, 5:4, 11:8, 7:4, 13:8&lt;/li&gt;&lt;li&gt;3:2, 5:4, 7:4, 11:8, 13:8 -&amp;gt; 3:2, 5:4, 7:4, 13:8, 11:8&lt;/li&gt;&lt;li&gt;3:2, 5:4, 7:4, 11:8, 13:8 -&amp;gt; 4:3, 8:5, 8:7, 16:11, 16:13&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Symmetry group-Orbits and Invariant Subsets"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Orbits and Invariant Subsets&lt;/h3&gt;
[to be included]&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/User:AthiTrydhen/15-limit_tonality_diamond"