Marveltwin: Difference between revisions
Wikispaces>genewardsmith **Imported revision 238014607 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 238017831 - Original comment: ** |
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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:genewardsmith|genewardsmith]] and made on <tt>2011-06-21 15: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-21 15:14:24 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>238017831</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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|| [2/3 4/3 -2/3 0 0 2/3> || | || [2/3 4/3 -2/3 0 0 2/3> || | ||
Fifths are pure; 5, 7, 11 and 13 are all flat by (325/324)^(1/3), which is 1.778 cents. | |||
Eigenmonzo subgroup: 2.3.7/5.11/5.13/5 | Eigenmonzo subgroup: 2.3.7/5.11/5.13/5 | ||
| Line 27: | Line 28: | ||
|| <1 0 0 0 0 2] || | || <1 0 0 0 0 2] || | ||
|| <0 1 0 0 0 4] || | || <0 1 0 0 0 4] || | ||
|| <0 0 1 0 0 -2] || | || <0 0 1 0 0 -2] || | ||
|| <0 0 0 1 0 0] || | || <0 0 0 1 0 0] || | ||
|| <0 0 0 0 1 0] || | || <0 0 0 0 1 0] || | ||
Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 | Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 | ||
</pre></div> | |||
=Rank four= | |||
==225/224== | |||
13-limit eigenmonzo subgroup: 2.7.11/5.13/5 | |||
15-limit eigenmonzo subgroup: 2.7.15/11.15/13 | |||
Map: | |||
|| <1 0 0 -5 0 2] || | |||
|| <0 1 0 2 0 4] || | |||
|| <0 0 1 2 0 -2] || | |||
|| <0 0 0 0 1 0]] || | |||
Edos: 12, 19, 41, 53, 72, 166</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>Marveltwin</title></head><body><!-- ws:start:WikiTextTocRule: | <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>Marveltwin</title></head><body><!-- ws:start:WikiTextTocRule:8:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><a href="#Marveltwin and Marvel">Marveltwin and Marvel</a><!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --> | <a href="#Rank five">Rank five</a><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --> | <a href="#Rank four">Rank four</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | ||
<!-- ws:end:WikiTextTocRule: | <!-- ws:end:WikiTextTocRule:13 --><br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Marveltwin and Marvel"></a><!-- ws:end:WikiTextHeadingRule:0 -->Marveltwin and Marvel</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Marveltwin and Marvel"></a><!-- ws:end:WikiTextHeadingRule:0 -->Marveltwin and Marvel</h1> | ||
The <em>marveltwin comma</em>, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the <a class="wiki_link" href="/11-limit">11-limit</a> version of marvel, which tempers out 225/224 and 385/384, to get <a class="wiki_link" href="/13-limit">13-limit</a> marvel. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.<br /> | The <em>marveltwin comma</em>, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the <a class="wiki_link" href="/11-limit">11-limit</a> version of marvel, which tempers out 225/224 and 385/384, to get <a class="wiki_link" href="/13-limit">13-limit</a> marvel. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.<br /> | ||
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<br /> | <br /> | ||
Fifths are pure; 5, 7, 11 and 13 are all flat by (325/324)^(1/3), which is 1.778 cents. <br /> | |||
Eigenmonzo subgroup: 2.3.7/5.11/5.13/5<br /> | Eigenmonzo subgroup: 2.3.7/5.11/5.13/5<br /> | ||
<br /> | <br /> | ||
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<td>&lt;0 0 1 0 0 -2]<br /> | <td>&lt;0 0 1 0 0 -2]<br /> | ||
</td> | </td> | ||
</tr> | |||
< | <tr> | ||
<td>&lt;0 0 0 1 0 0]<br /> | <td>&lt;0 0 0 1 0 0]<br /> | ||
</td> | </td> | ||
| Line 102: | Line 117: | ||
<br /> | <br /> | ||
Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333</body></html></pre></div> | Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Rank four"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank four</h1> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Rank four-225/224"></a><!-- ws:end:WikiTextHeadingRule:6 -->225/224</h2> | |||
<br /> | |||
13-limit eigenmonzo subgroup: 2.7.11/5.13/5<br /> | |||
15-limit eigenmonzo subgroup: 2.7.15/11.15/13<br /> | |||
<br /> | |||
Map:<br /> | |||
<table class="wiki_table"> | |||
<tr> | |||
<td>&lt;1 0 0 -5 0 2]<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>&lt;0 1 0 2 0 4]<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>&lt;0 0 1 2 0 -2]<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td>&lt;0 0 0 0 1 0]]<br /> | |||
</td> | |||
</tr> | |||
</table> | |||
Edos: 12, 19, 41, 53, 72, 166</body></html></pre></div> | |||