Radical interval: Difference between revisions
Wikispaces>guest **Imported revision 419039896 - Original comment: ** |
Wikispaces>x31eq **Imported revision 419049108 - 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: | : This revision was by author [[User:x31eq|x31eq]] and made on <tt>2013-03-31 11:42:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>419049108</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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Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it. | Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it. | ||
===Fractional projection maps=== | ===Fractional projection maps=== | ||
A square matrix P is a [[http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29|projection]] if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and Interval Space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v. | A square matrix P is a [[http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29|projection]] if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and Interval Space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v. | ||
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Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.<br /> | Vectors in interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a positive rational number which corresponds to it.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h3&gt; --><h3 id="toc0"><a name="x--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3> | ||
A square matrix P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" rel="nofollow">projection</a> if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvalues</a> of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.<br /> | A square matrix P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" rel="nofollow">projection</a> if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvalues</a> of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v.<br /> | ||