Target tuning
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- This revision was by author genewardsmith and made on 2010-11-17 18:06:02 UTC.
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Original Wikitext content:
By a //target tuning// is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.
==Least squares tunings==
By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0<r<1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, sum_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.
We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are vals spanning the temperament. From U we form the matrix V by taking the [[normal lists|normal val list]] for U and removing the first ("period") row. A list of [[fractional monzos|eigenmonzos]] E = VR*R, where the asterisk denotes the matrix transpose, can now be obtained by matrix multiplication. The [[fractional monzos|projection matrix]] for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E as left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.
Original HTML content:
<html><head><title>Target tunings</title></head><body>By a <em>target tuning</em> is meant a tuning for a regular temperament which has been optimized with respect to a set of target intervals. Usually this set of intervals is derived from a finite set of octave equivalence classes, and octaves are taken to be pure 2s. Moreover, normally the intervals in question are rational numbers. Below we will make all of these assumptions, and will discuss the two most important target tunings, minimax and least squares.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Least squares tunings"></a><!-- ws:end:WikiTextHeadingRule:0 -->Least squares tunings</h2> <br /> By assumption we have a finite set of rational intervals defining interval classes, which we may take to be octave reduced to 0<r<1, where the intervals are expressed logarithmically in terms of log base two. The least squares tuning T is the tuning for a particular regular temperament which minimizes the sum of the squares of the errors, sum_i (T(q_i) - log2(q_i))^2, where q_i are the rational intervals of the target set. Note that most commonly, the target set is a tonality diamond, since these are the intervals in an overtone series up to some odd integer d, and hence may be considered the consonances of the d odd limit.<br /> <br /> We may find the least squares tuning in various ways, one of which is to start from a matrix R whose rows are the monzos of the target set, and a matrix U whose rows are vals spanning the temperament. From U we form the matrix V by taking the <a class="wiki_link" href="/normal%20lists">normal val list</a> for U and removing the first ("period") row. A list of <a class="wiki_link" href="/fractional%20monzos">eigenmonzos</a> E = VR*R, where the asterisk denotes the matrix transpose, can now be obtained by matrix multiplication. The <a class="wiki_link" href="/fractional%20monzos">projection matrix</a> for the least squares tuning is then the square matrix with fractional monzo rows, the rows of E as left eigenvectors with eigenvalue one, and any basis for the commas of the temperament as left eigenvectors with eigenvalues zero.</body></html>