User:AthiTrydhen/15-limit tonality diamond - Revision history

AthiTrydhen at 05:49, 14 December 2024

2024-12-14T05:49:58Z

← Older revision		Revision as of 05:49, 14 December 2024
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	{{Inaccessible}}		{{Inaccessible}}Todo: Incorporate Hendrix diamond ideas
	{{Todo~~\|inline=1\|merge articles\|comment= → [[15-odd-limit]]}}~~

	The '''15-limit tonality diamond''' has the following notes:		The '''15-limit tonality diamond''' has the following notes:

AthiTrydhen: AthiTrydhen moved page 15-limit tonality diamond to User:AthiTrydhen/15-limit tonality diamond

2024-12-14T05:49:22Z

AthiTrydhen moved page 15-limit tonality diamond to User:AthiTrydhen/15-limit tonality diamond

← Older revision	Revision as of 05:49, 14 December 2024
(No difference)

FloraC: Wtf is this

2024-08-13T10:09:43Z

Wtf is this

← Older revision		Revision as of 10:09, 13 August 2024
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			{{Inaccessible}}
			{{Todo\|inline=1\|merge articles\|comment= → [[15-odd-limit]]}}

	The '''15-limit tonality diamond''' has the following notes:		The '''15-limit tonality diamond''' has the following notes:

Fredg999 category edits: Categories

2023-04-20T20:48:15Z

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>Praimhin: Imported revision 588772130 - Original comment:

2016-08-04T01:23:52Z

**Imported revision 588772130 - Original comment: **

← Older revision		Revision as of 01:23, 4 August 2016
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	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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>		: This revision was by author [[User:Praimhin\|Praimhin]] and made on <tt>2016-08-04 01:23:52 UTC</tt>.<br>
	: The original revision id was <tt>~~588772064~~</tt>.<br>		: The original revision id was <tt>588772130</tt>.<br>
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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:
	* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9		* Transformation //R//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9
	* 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, 11:8, 7:4, 13:8
	* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8		* Transformation //S'//: 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8
	* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13		* 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 //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.

	===Orbits and Invariant Subsets===		===Orbits and Invariant Subsets===
	[~~to be included~~]</pre></div>		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.

			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>
	<h4>Original HTML content:</h4>		<h4>Original HTML 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;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 />		<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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	<br />		<br />
	The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br />		The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br />
	<ul><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul><br />		<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 />
			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 />
			<br />
	<!-- 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>		<!-- 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>
	~~[to be included]~~</body></html></pre></div>		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 />
			<br />
			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>

Wikispaces>Praimhin: Imported revision 588772064 - Original comment:

2016-08-04T01:12:44Z

**Imported revision 588772064 - Original comment: **

Wikispaces>Praimhin: Imported revision 588772052 - Original comment:

2016-08-04T01:11:33Z

**Imported revision 588772052 - Original comment: **

← Older revision		Revision as of 01:11, 4 August 2016
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 01:09:06 UTC</tt>.<br>		: This revision was by author [[User:Praimhin\|Praimhin]] and made on <tt>2016-08-04 01:11:33 UTC</tt>.<br>
	: The original revision id was <tt>~~588772026~~</tt>.<br>		: The original revision id was <tt>588772052</tt>.<br>
	: The revision comment was: <tt></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>		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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	* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8		* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8
	* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8		* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8
	* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13</pre></div>		* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13

			===Orbits===
			[to be included]</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML 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;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 />		<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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	<br />		<br />
	The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br />		The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br />
	<ul><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul></body></html></pre></div>		<ul><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul><br />
			<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Symmetry group-Orbits"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orbits</h3>
			[to be included]</body></html></pre></div>

Wikispaces>Praimhin: Imported revision 588772026 - Original comment:

2016-08-04T01:09:06Z

**Imported revision 588772026 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference: 
: This revision was by author [[User:Praimhin|Praimhin]] and made on <tt>2016-08-04 01:09:06 UTC</tt>. 
: The original revision id was <tt>588772026</tt>. 
: The revision comment was: <tt></tt> 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it. 
<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 ||
|| 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:
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 5:3, 14:9, 11:9, 13:9
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 11:8, 7:4, 13:8
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 3:2, 5:4, 7:4, 13:8, 11:8
* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13</pre></div>
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>15-limit tonality diamond</title></head><body>The 15-limit tonality diamond has the following notes: 
 

<table class="wiki_table">
<tr>
<td>1/1 
</td>
<td>9/8 
</td>
<td>5/4 
</td>
<td>11/8 
</td>
<td>3/2 
</td>
<td>13/8 
</td>
<td>7/4 
</td>
<td>15/8 
</td>
</tr>
<tr>
<td>16/9 
</td>
<td>1/1 
</td>
<td>10/9 
</td>
<td>11/9 
</td>
<td>4/3 
</td>
<td>13/9 
</td>
<td>14/9 
</td>
<td>5/3 
</td>
</tr>
<tr>
<td>8/5 
</td>
<td>9/5 
</td>
<td>1/1 
</td>
<td>11/10 
</td>
<td>6/5 
</td>
<td>13/10 
</td>
<td>7/5 
</td>
<td>3/2 
</td>
</tr>
<tr>
<td>16/11 
</td>
<td>18/11 
</td>
<td>20/11 
</td>
<td>1/1 
</td>
<td>12/11 
</td>
<td>13/11 
</td>
<td>14/11 
</td>
<td>15/11 
</td>
</tr>
<tr>
<td>4/3 
</td>
<td>3/2 
</td>
<td>5/3 
</td>
<td>11/6 
</td>
<td>1/1 
</td>
<td>13/12 
</td>
<td>7/6 
</td>
<td>5/4 
</td>
</tr>
<tr>
<td>16/13 
</td>
<td>18/13 
</td>
<td>20/13 
</td>
<td>22/13 
</td>
<td>24/13 
</td>
<td>1/1 
</td>
<td>14/13 
</td>
<td>15/13 
</td>
</tr>
<tr>
<td>8/7 
</td>
<td>9/7 
</td>
<td>10/7 
</td>
<td>11/7 
</td>
<td>12/7 
</td>
<td>13/7 
</td>
<td>1/1 
</td>
<td>15/14 
</td>
</tr>
<tr>
<td>16/15 
</td>
<td>6/5 
</td>
<td>4/3 
</td>
<td>22/15 
</td>
<td>8/5 
</td>
<td>26/15 
</td>
<td>28/15 
</td>
<td>1/1 
</td>
</tr>
</table>

<h2 id="toc0"><a name="x-Symmetry group"></a>Symmetry group</h2>
 
The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators: 
<ul><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul></body></html></pre></div>

← Older revision		Revision as of 20:48, 20 April 2023
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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.		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.

			[[Category:15-odd-limit]]
			[[Category:Diamond]]

← Older revision		Revision as of 01:12, 4 August 2016
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		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 01:11:33 UTC</tt>.<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>~~588772052~~</tt>.<br>		: The original revision id was <tt>588772064</tt>.<br>
	: The revision comment was: <tt></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>		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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	* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13		* 3:2, 5:4, 7:4, 11:8, 13:8 -> 4:3, 8:5, 8:7, 16:11, 16:13

	===Orbits===		===Orbits and Invariant Subsets===
	[to be included]</pre></div>		[to be included]</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
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	The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br />		The symmetry group of the 15-limit tonality diamond has 24 elements. The following are its generators:<br />
	<ul><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul><br />		<ul><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 5:3, 14:9, 11:9, 13:9</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 11:8, 7:4, 13:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 3:2, 5:4, 7:4, 13:8, 11:8</li><li>3:2, 5:4, 7:4, 11:8, 13:8 -&gt; 4:3, 8:5, 8:7, 16:11, 16:13</li></ul><br />
	<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Symmetry group-Orbits"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orbits</h3>		<!-- 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>
	[to be included]</body></html></pre></div>		[to be included]</body></html></pre></div>