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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>
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| : This revision was by author [[User:Praimhin|Praimhin]] and made on <tt>2016-08-04 01:23:52 UTC</tt>.<br>
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| : The original revision id was <tt>588772130</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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:
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| || 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 |
| | |} |
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| | ==Symmetry group== |
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| 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: |
| * Transformation //R//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9
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| * Transformation //S//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8
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| * Transformation //S'//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8
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| * Transformation //T//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13
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| 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.
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| ===Orbits and Invariant Subsets===
| | <ul><li>Transformation ''R'': 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9</li><li>Transformation ''S'': 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8</li><li>Transformation ''S''': 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8</li><li>Transformation ''T'': 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13</li></ul> |
| The [[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]] and [[13-Hendrix diamond]] respectively.
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| 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.</pre></div>
| | 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. |
| <h4>Original HTML content:</h4>
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| <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"><html><head><title>15-limit tonality diamond</title></head><body>The <strong>15-limit tonality diamond</strong> has the following notes:<br />
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| | ===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. |
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| <table class="wiki_table">
| | 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. |
| <tr>
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| <td>1/1<br />
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| </td>
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| <td>9/8<br />
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| </td>
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| <td>5/4<br />
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| </td>
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| <td>11/8<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| <td>13/8<br />
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| </td>
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| <td>7/4<br />
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| </td>
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| <td>15/8<br />
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| </td>
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| </tr>
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| <tr>
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| <td>16/9<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>10/9<br />
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| </td>
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| <td>11/9<br />
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| </td>
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| <td>4/3<br />
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| </td>
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| <td>13/9<br />
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| </td>
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| <td>14/9<br />
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| </td>
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| <td>5/3<br />
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| </td>
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| </tr>
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| <tr>
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| <td>8/5<br />
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| </td>
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| <td>9/5<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>11/10<br />
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| </td>
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| <td>6/5<br />
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| </td>
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| <td>13/10<br />
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| </td>
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| <td>7/5<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| </tr>
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| <tr>
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| <td>16/11<br />
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| </td>
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| <td>18/11<br />
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| </td>
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| <td>20/11<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>12/11<br />
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| </td>
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| <td>13/11<br />
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| </td>
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| <td>14/11<br />
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| </td>
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| <td>15/11<br />
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| </td>
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| </tr>
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| <tr>
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| <td>4/3<br />
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| </td>
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| <td>3/2<br />
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| </td>
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| <td>5/3<br />
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| </td>
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| <td>11/6<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>13/12<br />
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| </td>
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| <td>7/6<br />
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| </td>
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| <td>5/4<br />
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| </td>
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| </tr>
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| <tr>
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| <td>16/13<br />
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| </td>
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| <td>18/13<br />
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| </td>
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| <td>20/13<br />
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| </td>
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| <td>22/13<br />
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| </td>
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| <td>24/13<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>14/13<br />
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| </td>
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| <td>15/13<br />
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| </td>
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| </tr>
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| <tr>
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| <td>8/7<br />
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| </td>
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| <td>9/7<br />
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| </td>
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| <td>10/7<br />
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| </td>
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| <td>11/7<br />
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| </td>
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| <td>12/7<br />
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| </td>
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| <td>13/7<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| <td>15/14<br />
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| </td>
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| </tr>
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| <tr>
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| <td>16/15<br />
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| </td>
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| <td>6/5<br />
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| </td>
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| <td>4/3<br />
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| </td>
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| <td>22/15<br />
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| </td>
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| <td>8/5<br />
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| </td>
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| <td>26/15<br />
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| </td>
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| <td>28/15<br />
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| </td>
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| <td>1/1<br />
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| </td>
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| </tr>
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| </table>
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| <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Symmetry group"></a><!-- ws:end:WikiTextHeadingRule:0 -->Symmetry group</h2>
| | [[Category:15-odd-limit]] |
| <br />
| | [[Category:Diamond]] |
| The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br />
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| <ul><li>Transformation <em>R</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>Transformation <em>S</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>Transformation <em>S'</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>Transformation <em>T</em>: 3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul><br />
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| These generators have the relations <em>R</em>² = <em>S</em>² = <em>T</em>² = <em>S</em>'² = I, (<em>SS</em>')³ = I, <em>RS</em> = <em>SR</em>, <em>RS</em>' = <em>S</em>'<em>R</em>, and <em>T</em> commutes with the three other generators. Thus the symmetry group is isomorphic to <em>S</em>₃ × <em>C</em>₂².<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Symmetry group-Orbits and Invariant Subsets"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orbits and Invariant Subsets</h3>
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| The <a class="wiki_link" href="/Hendrix%20diamond">Hendrix diamond</a> is invariant under action by <em>R,</em> <em>S</em>' and <em>T</em>, and the images of the action of <em>S</em> and <em>S</em>² on the Hendrix diamond are the <a class="wiki_link" href="/11-Hendrix%20diamond">11-Hendrix diamond</a> and <a class="wiki_link" href="/13-Hendrix%20diamond">13-Hendrix diamond</a> respectively.<br />
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| <br />
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| 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.</body></html></pre></div>
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This page or section may be difficult to understand to those unfamiliar with the mathematical concepts involved. A more accessible version will be worked on; in the meantime, feel free to ask questions in the Xenharmonic Alliance Discord server or Facebook group.
|
Todo: Incorporate Hendrix diamond ideas
The 15-limit tonality diamond has the following notes:
| 1/1
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9/8
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5/4
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11/8
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3/2
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13/8
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7/4
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15/8
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| 16/9
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1/1
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10/9
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11/9
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4/3
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13/9
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14/9
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5/3
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| 8/5
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9/5
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1/1
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11/10
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6/5
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13/10
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7/5
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3/2
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| 16/11
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18/11
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20/11
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1/1
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12/11
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13/11
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14/11
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15/11
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| 4/3
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3/2
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5/3
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11/6
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1/1
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13/12
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7/6
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5/4
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| 16/13
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18/13
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20/13
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22/13
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24/13
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1/1
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14/13
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15/13
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| 8/7
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9/7
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10/7
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11/7
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12/7
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13/7
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1/1
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15/14
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| 16/15
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6/5
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4/3
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22/15
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8/5
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26/15
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28/15
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1/1
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Symmetry group
The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:
- Transformation R: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9
- Transformation S: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8
- Transformation S': 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8
- Transformation T: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13
These generators have the relations R² = S² = T² = S² = I, (SS)³ = I, RS = SR, RS = SR, and T commutes with the three other generators. Thus the symmetry group is isomorphic to S₃ × C₂².
Orbits and Invariant Subsets
The Hendrix diamond is invariant under action by R, S' and T, and the images of the action of S and S² on the Hendrix diamond are the 11-Hendrix diamond and 13-Hendrix diamond respectively.
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.
