Tenney–Euclidean tuning: Difference between revisions
Wikispaces>hstraub **Imported revision 245007381 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 248588449 - Original comment: ** |
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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> | ||
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=Pure octaves TE tuning= | =Pure octaves TE tuning= | ||
(alias **[[POTE tuning]]**) | |||
If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by [[http://en.wikipedia.org/wiki/Scalar_multiplication|scalar multiplication]], T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J. We may call pure-octaves Tenney-Euclidean tuning the [[POTE tuning]]. | If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by [[http://en.wikipedia.org/wiki/Scalar_multiplication|scalar multiplication]], T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J. We may call pure-octaves Tenney-Euclidean tuning the [[POTE tuning]]. | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Pure octaves TE tuning"></a><!-- ws:end:WikiTextHeadingRule:8 -->Pure octaves TE tuning</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Pure octaves TE tuning"></a><!-- ws:end:WikiTextHeadingRule:8 -->Pure octaves TE tuning</h1> | ||
If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow">scalar multiplication</a>, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J. We may call pure-octaves Tenney-Euclidean tuning the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a>.<br /> | (alias <strong><a class="wiki_link" href="/POTE%20tuning">POTE tuning</a></strong>)<br /> | ||
If T = JP is the TE tuning map, then a corresponding pure-octaves map can be found by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Scalar_multiplication" rel="nofollow">scalar multiplication</a>, T/T[1], where T[1], the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J. We may call pure-octaves Tenney-Euclidean tuning the <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a>.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="The Frobenius projection map"></a><!-- ws:end:WikiTextHeadingRule:10 -->The Frobenius projection map</h1> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="The Frobenius projection map"></a><!-- ws:end:WikiTextHeadingRule:10 -->The Frobenius projection map</h1> | ||