Lesfip scales
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- This revision was by author genewardsmith and made on 2011-01-10 17:15:13 UTC.
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Original Wikitext content:
A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again. A lesfip scale may be found by choosing a [[periodic scale|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[circulating temperaments]] Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale. Examples of lesfip scales can be found at [[Scalesmith]].
Original HTML content:
<html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br /> <br /> A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous <a class="wiki_link" href="/circulating%20temperaments">circulating temperaments</a> Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, <a class="wiki_link" href="/val-werck">val-werck</a>. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.<br /> <br /> Examples of lesfip scales can be found at <a class="wiki_link" href="/Scalesmith">Scalesmith</a>.</body></html>