Tenney–Euclidean tuning: Difference between revisions

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If we put the weighted Euclidean metric on tuning space, leading to TE tuning space in weighted coordinates, it is easy to find the nearest point in the subspace to the [[JIP]] {{val|1 1 … 1}}, and this closest point will define a tuning map which is called TE tuning (or TOP-RMS), a tuning which has been extensively studied by [[Graham Breed]]. We may also keep unweighted coordinates and use the TE norm on tuning space; in these coordinates the JI point is {{val|1 log23 … log2''p''}}. The two approaches are equivalent.

In more pragmatic terms, suppose M is the known mapping of the [[Regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, and W the weighting matrix. In this case, W''ij'' = 1/log2''p'' if ''i'' = ''j'', and 0 otherwise. Then V = MW is the mapping in the weighted basis. Let's also denote the row vector of TE generators g, the row vector of targeted JI intervals J0. TE tuning then defines a [[Wikipedia: Least squares|least square]] problem of the following overdetermined linear equation system:

In more pragmatic terms, suppose A is the known mapping of the [[Regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, and W the weighting matrix. In this case, W''ij'' = 1/log2''p'' if ''i'' = ''j'', and 0 otherwise. Then V = AW is the mapping in the weighted basis. Let's also denote the row vector of TE generators g, the row vector of targeted JI intervals J0. TE tuning then defines a [[Wikipedia: Least squares|least square]] problem of the following overdetermined linear equation system:

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 If we put the weighted Euclidean metric on tuning space, leading to TE tuning space in weighted coordinates, it is easy to find the nearest point in the subspace to the [[JIP]] {{val|1 1 … 1}}, and this closest point will define a tuning map which is called TE tuning (or TOP-RMS), a tuning which has been extensively studied by [[Graham Breed]]. We may also keep unweighted coordinates and use the TE norm on tuning space; in these coordinates the JI point is {{val|1 log<sub>2</sub>3 … log<sub>2</sub>''p''}}. The two approaches are equivalent.
-In more pragmatic terms, suppose M is the known mapping of the [[Regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, and W the weighting matrix. In this case, W<sub>''ij''</sub> = 1/log<sub>2</sub>''p'' if ''i'' = ''j'', and 0 otherwise. Then V = MW is the mapping in the weighted basis. Let's also denote the row vector of TE generators g, the row vector of targeted JI intervals J<sub>0</sub>. TE tuning then defines a [[Wikipedia: Least squares|least square]] problem of the following overdetermined linear equation system:
+In more pragmatic terms, suppose A is the known mapping of the [[Regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, and W the weighting matrix. In this case, W<sub>''ij''</sub> = 1/log<sub>2</sub>''p'' if ''i'' = ''j'', and 0 otherwise. Then V = AW is the mapping in the weighted basis. Let's also denote the row vector of TE generators g, the row vector of targeted JI intervals J<sub>0</sub>. TE tuning then defines a [[Wikipedia: Least squares|least square]] problem of the following overdetermined linear equation system:
 <math>\vec{g}V = J</math>