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Wikispaces>genewardsmith **Imported revision 181289761 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 181292097 - 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>2010-11-19 18: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-19 18:45:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>181292097</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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For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament. | For example, for [[Gamelismic clan|miracle temperament]] [2, 15/14] defines a rank two 7-limit subgroup whose [[Normal lists|normal interval list]] is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament. | ||
</pre></div> | Alternatively, using "v" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0> = <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> = <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> = <1 1 3 3|. Unlike the case for just intonation subgroups, the rank two group for the abstract temperament in terms of interior products is already exactly the same.</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>Generators</title></head><body>A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Generating_set_of_a_group" rel="nofollow">set of generators</a> for a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_%28mathematics%29" rel="nofollow">group</a> is a subset of the elements of the group which is not contained in any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Subgroup" rel="nofollow">proper subgroup</a>, which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian group</a>, it is called a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Finitely_generated_abelian_group" rel="nofollow">finitely generated abelian group</a>.<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>Generators</title></head><body>A <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Generating_set_of_a_group" rel="nofollow">set of generators</a> for a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Group_%28mathematics%29" rel="nofollow">group</a> is a subset of the elements of the group which is not contained in any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Subgroup" rel="nofollow">proper subgroup</a>, which is to say, any subgroup which is not the whole group. If the set is a finite set, the group is called finitely generated. If it is also an <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Abelian_group" rel="nofollow">abelian group</a>, it is called a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Finitely_generated_abelian_group" rel="nofollow">finitely generated abelian group</a>.<br /> | ||
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These two example converge when we seek generators for the abstract temperament rather than any particular tuning of it. One way to obtain these is to use the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos. A less abstract approach is to define a <a class="wiki_link" href="/transversal">transversal</a> for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.<br /> | These two example converge when we seek generators for the abstract temperament rather than any particular tuning of it. One way to obtain these is to use the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/InteriorProduct.html" rel="nofollow">interior product</a> of a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> for a p-limit temperament with the p-limit monzos. A less abstract approach is to define a <a class="wiki_link" href="/transversal">transversal</a> for the temperament by giving generators for a just intonation subgroup which when tempered becomes the notes of the temperament.<br /> | ||
<br /> | <br /> | ||
For example, for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a> [2, 15/14] defines a rank two 7-limit subgroup whose <a class="wiki_link" href="/Normal%20lists">normal interval list</a> is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.</body></html></pre></div> | For example, for <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a> [2, 15/14] defines a rank two 7-limit subgroup whose <a class="wiki_link" href="/Normal%20lists">normal interval list</a> is [2, 15/7]. We might also use [2, 16/15], with a normal interval list [2, 15]. When tempered by miracle, [2, 15/14] and [2, 16/15] lead to the same notes; hence we can use either for a transversal. Either pair may be considered a generating set for the abstract temperament.<br /> | ||
<br /> | |||
Alternatively, using &quot;v&quot; to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir v |1 0 0 0&gt; <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><!-- ws:end:WikiTextHeadingRule:0 --> &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; </h1> | |||
&lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; = &lt;1 1 3 3|. Unlike the case for just intonation subgroups, the rank two group for the abstract temperament in terms of interior products is already exactly the same.</body></html></pre></div> | |||