Tenney–Euclidean temperament measures: Difference between revisions
Wikispaces>genewardsmith **Imported revision 154467605 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 154974821 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-08-03 01:12:08 UTC</tt>.<br> | ||
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===Wedgie Complexity=== | ===Wedgie Complexity=== | ||
Given a [[Wedgies and Multivals|wedgie]] W, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||W|| is a measure of the //complexity// of W; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the numbner of scale steps it takes to reach an octave. | Given a [[Wedgies and Multivals|wedgie]] W, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||W|| is a measure of the //complexity// of W; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the numbner of scale steps it takes to reach an octave. This complexity and related measures have been [[http://x31eq.com/temper/primerr.pdf|extensively studied]] by [[Graham Breed]]. | ||
Wedgie complexity is easily computed if a routine for computing multivectors is available. However such a routine is not required, as it can also be computed using the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gramian]]. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the [[http://en.wikipedia.org/wiki/Dot_product|dot product]] of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the RMS norm we have | Wedgie complexity is easily computed if a routine for computing multivectors is available. However such a routine is not required, as it can also be computed using the [[http://en.wikipedia.org/wiki/Gramian_matrix|Gramian]]. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the [[http://en.wikipedia.org/wiki/Dot_product|dot product]] of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the RMS norm we have | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc0"><a name="x--Wedgie Complexity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Wedgie Complexity</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc0"><a name="x--Wedgie Complexity"></a><!-- ws:end:WikiTextHeadingRule:2 -->Wedgie Complexity</h3> | ||
Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> W, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||W|| is a measure of the <em>complexity</em> of W; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the numbner of scale steps it takes to reach an octave.<br /> | Given a <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a> W, that is a canonically reduced r-val correspondng to a temperament of rank r, the norm ||W|| is a measure of the <em>complexity</em> of W; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the numbner of scale steps it takes to reach an octave. This complexity and related measures have been <a class="wiki_link_ext" href="http://x31eq.com/temper/primerr.pdf" rel="nofollow">extensively studied</a> by <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a>.<br /> | ||
<br /> | <br /> | ||
Wedgie complexity is easily computed if a routine for computing multivectors is available. However such a routine is not required, as it can also be computed using the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow">Gramian</a>. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dot_product" rel="nofollow">dot product</a> of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the RMS norm we have <br /> | Wedgie complexity is easily computed if a routine for computing multivectors is available. However such a routine is not required, as it can also be computed using the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gramian_matrix" rel="nofollow">Gramian</a>. This is the determinant of the square matrix, called the Gram matrix, defined from a list of r vectors, whose (i,j)-th entry is vi.vj, the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dot_product" rel="nofollow">dot product</a> of the ith vector with the jth vector. The square of the ordinary Euclidean norm of a multivector is the Gramian of the vectors wedged together to define it, and hence in terms of the RMS norm we have <br /> | ||