Regular temperament: Difference between revisions
Wikispaces>genewardsmith **Imported revision 255488192 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 278221020 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 14:24:51 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>278221020</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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* **The [[Wedgies and Multivals|wedgie]]** | * **The [[Wedgies and Multivals|wedgie]]** | ||
This uses [[http://en.wikipedia.org/wiki/Exterior_algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[ | This uses [[http://en.wikipedia.org/wiki/Exterior_algebra|multilinear algebra]] to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the [[Interior product|interior product]] of a [[Wedgies and Multivals|wedgie]] for a p-limit temperament with the p-limit monzos. | ||
For example, 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> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> which is also <1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison. | For example, 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> is <0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0> which is also <1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison. | ||
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An <em>abstract regular temperament</em> is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are<br /> | An <em>abstract regular temperament</em> is a <a class="wiki_link" href="/regular%20temperament">regular temperament</a> considered apart from any consideration of tuning, which classifies all of its tunings as tunings of the single abstract temperament. Various methods have been proposed for uniquely characterizing an abstract regular temperament. Among these are<br /> | ||
<br /> | <br /> | ||
<ul><li><strong>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></strong></li></ul>This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class=" | <ul><li><strong>The <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a></strong></li></ul>This uses <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Exterior_algebra" rel="nofollow">multilinear algebra</a> to define a unique reduced wedge product uniquely associated to the abstract regular temperament. The intervals of the temperament, as an abstract group, may be defined by the <a class="wiki_link" href="/Interior%20product">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. <br /> | ||
<br /> | <br /> | ||
For example, 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; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison. <br /> | For example, 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; is &lt;0 -6 7 2|, with 15/14 we get mir v |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir v |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison. <br /> | ||