Marveltwin
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Original Wikitext content:
[[toc|flat]] =Marveltwin and Marvel= The //marveltwin comma//, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the [[11-limit]] version of marvel, which tempers out 225/224 and 385/384, to get [[13-limit]] 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. =Rank five= Comma: 325/324 13 and 15 limit minimax tuning || [1 0 0 0 0 0> || || [0 1 0 0 0 0> || || [2/3 4/3 1/3 0 0 -1/3> || || [2/3 4/3 -2/3 1 0 -1/3> || || [2/3 4/3 -2/3 0 1 -1/3> || || [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 Map: || <1 0 0 0 0 2] || || <0 1 0 0 0 4] || || <0 0 1 0 0 -2] || || <0 0 0 1 0 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 =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
Original HTML content:
<html><head><title>Marveltwin</title></head><body><!-- ws:start:WikiTextTocRule:8:<img id="wikitext@@toc@@flat" class="WikiMedia WikiMediaTocFlat" title="Table of Contents" src="/site/embedthumbnail/toc/flat?w=100&h=16"/> --><!-- 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:13 --><br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><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 />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Rank five"></a><!-- ws:end:WikiTextHeadingRule:2 -->Rank five</h1>
Comma: 325/324<br />
<br />
13 and 15 limit minimax tuning<br />
<table class="wiki_table">
<tr>
<td>[1 0 0 0 0 0><br />
</td>
</tr>
<tr>
<td>[0 1 0 0 0 0><br />
</td>
</tr>
<tr>
<td>[2/3 4/3 1/3 0 0 -1/3><br />
</td>
</tr>
<tr>
<td>[2/3 4/3 -2/3 1 0 -1/3><br />
</td>
</tr>
<tr>
<td>[2/3 4/3 -2/3 0 1 -1/3><br />
</td>
</tr>
<tr>
<td>[2/3 4/3 -2/3 0 0 2/3><br />
</td>
</tr>
</table>
<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 />
<br />
Map: <br />
<table class="wiki_table">
<tr>
<td><1 0 0 0 0 2]<br />
</td>
</tr>
<tr>
<td><0 1 0 0 0 4]<br />
</td>
</tr>
<tr>
<td><0 0 1 0 0 -2]<br />
</td>
</tr>
<tr>
<td><0 0 0 1 0 0]<br />
</td>
</tr>
<tr>
<td><0 0 0 0 1 0]<br />
</td>
</tr>
</table>
<br />
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:<h1> --><h1 id="toc2"><a name="Rank four"></a><!-- ws:end:WikiTextHeadingRule:4 -->Rank four</h1>
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h2> --><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><1 0 0 -5 0 2]<br />
</td>
</tr>
<tr>
<td><0 1 0 2 0 4]<br />
</td>
</tr>
<tr>
<td><0 0 1 2 0 -2]<br />
</td>
</tr>
<tr>
<td><0 0 0 0 1 0]]<br />
</td>
</tr>
</table>
Edos: 12, 19, 41, 53, 72, 166</body></html>