Tenney–Euclidean temperament measures: Difference between revisions
Wikispaces>genewardsmith **Imported revision 471686322 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 471802146 - 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>2013-11- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-24 11:32:34 UTC</tt>.<br> | ||
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TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents. From the ratio (||J∧M||/||M||)^2 we obtain C(n, r+1)/(n C(n, r)) = (n-r)/(n (r+1)). If we take the ratio of this for rank one with this for rank r, the "n" cancels, and we get (n-1)/2 * (r+1)/(n-r) = (r+1)(n-1)/(2(n-r)). It follows that multiplying TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank r temperament then | TE simple badness can also be called and considered to be error relative to complexity. By dividing it by TE complexity, we get an error measurement, TE error. Multiplying this by 1200, we get a figure we can consider to be a weighted average with values in cents. From the ratio (||J∧M||/||M||)^2 we obtain C(n, r+1)/(n C(n, r)) = (n-r)/(n (r+1)). If we take the ratio of this for rank one with this for rank r, the "n" cancels, and we get (n-1)/2 * (r+1)/(n-r) = (r+1)(n-1)/(2(n-r)). It follows that multiplying TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank r temperament then | ||
[[math]] | [[math]] | ||
\displaystyle \psi = \sqrt{\frac{(r+1)(n-1 | \displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi | ||
[[math]] | [[math]] | ||
is an adjusted error, which we may call val normalized TE error, which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament.</pre></div> | is an adjusted error, which we may call val normalized TE error, which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament.</pre></div> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\displaystyle \psi = \sqrt{\frac{(r+1)(n-1 | \displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi&lt;br/&gt;[[math]] | ||
--><script type="math/tex">\displaystyle \psi = \sqrt{\frac{(r+1)(n-1 | --><script type="math/tex">\displaystyle \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
is an adjusted error, which we may call val normalized TE error, which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament.</body></html></pre></div> | is an adjusted error, which we may call val normalized TE error, which makes the error of a rank r temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament.</body></html></pre></div> | ||