Tenney–Euclidean tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 186492863 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 196912296 - 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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-28 15:03:11 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>196912296</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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If the vals defining V are linearly independent, then P = V*(VV*)^(-1)V. I the rows of M are independent, then we likewise have P = I - M*(MM*)^(-1)M. | If the vals defining V are linearly independent, then P = V*(VV*)^(-1)V. I the rows of M are independent, then we likewise have P = I - M*(MM*)^(-1)M. | ||
==Examples== | |||
The val for 5-limit 12et is <12 19 28|. In weighted coordinates, that becomes v12 = <12 19/log2(3) 28/log2(5)| ~ <12.0 11.988 12.059|. If we take this to be a 1x3 matrix and take the pseudoinverse, we get the 3x1 matrix v12` ~ [|0.027706 0.027677 0.027842|]. Then P = v12`v12 is a projection map onto the one-dimensional subspace whose single basis vector is v12. We find that v12.P equals v12 (using a dot to denote matrix multiplication.) | |||
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Spelling this out, if V is a matrix whose rows are vals, then P = V`V is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow">positive-semidefinite</a> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Symmetric_matrix" rel="nofollow">symmetric matrix</a> with rational matrix entries, which exactly specifies the regular temperament defined by the vals of V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M`M. <br /> | Spelling this out, if V is a matrix whose rows are vals, then P = V`V is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Positive-definite_matrix" rel="nofollow">positive-semidefinite</a> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Symmetric_matrix" rel="nofollow">symmetric matrix</a> with rational matrix entries, which exactly specifies the regular temperament defined by the vals of V. If M is a matrix with rows of monzos which spans the subspace of interval space containing the commas, then this same matrix P is given by I - M`M. <br /> | ||
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If the vals defining V are linearly independent, then P = V*(VV*)^(-1)V. I the rows of M are independent, then we likewise have P = I - M*(MM*)^(-1)M.</body></html></pre></div> | If the vals defining V are linearly independent, then P = V*(VV*)^(-1)V. I the rows of M are independent, then we likewise have P = I - M*(MM*)^(-1)M.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="x-Examples"></a><!-- ws:end:WikiTextHeadingRule:12 -->Examples</h2> | |||
The val for 5-limit 12et is &lt;12 19 28|. In weighted coordinates, that becomes v12 = &lt;12 19/log2(3) 28/log2(5)| ~ &lt;12.0 11.988 12.059|. If we take this to be a 1x3 matrix and take the pseudoinverse, we get the 3x1 matrix v12` ~ [|0.027706 0.027677 0.027842|]. Then P = v12`v12 is a projection map onto the one-dimensional subspace whose single basis vector is v12. We find that v12.P equals v12 (using a dot to denote matrix multiplication.)</body></html></pre></div> | |||