Tenney–Euclidean tuning: Difference between revisions

Line 8:

== TE tuning ==

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 ~~JI point~~ {{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.

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 p. TE tuning then defines a [[~~wikipedia~~:~~Least_squares~~|least square]] problem of the following overdetermined linear equation system:

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:

where J = pW, i.e. the ~~JI point~~, in the case of TE ~~weighting~~ it is {{val|1 1 … 1}}. The system simply says that the sum of V''ij'' steps of generator g''i'' for all ''i'' should equal the ''j''-th targeted JI interval J''j''.

where J = J0W, i.e. the JIP, in the case of TE tuning space it is {{val|1 1 … 1}}. The system simply says that the sum of V''ij'' steps of generator g''i'' for all ''i'' should equal the ''j''-th targeted JI interval J''j''.

There are a number of methods to solve least square problems. One common way is to use the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]].

@@ Line 8: / Line 8: @@
 == TE tuning ==
-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 JI point {{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.
+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 p. TE tuning then defines a [[wikipedia:Least_squares|least square]] problem of the following overdetermined linear equation system:
+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:
 <math>\vec{g}V = J</math>
-where J = pW, i.e. the JI point, in the case of TE weighting it is {{val|1 1 … 1}}. The system simply says that the sum of V<sub>''ij''</sub> steps of generator g<sub>''i''</sub> for all ''i'' should equal the ''j''-th targeted JI interval J<sub>''j''</sub>.
+where J = J<sub>0</sub>W, i.e. the JIP, in the case of TE tuning space it is {{val|1 1 … 1}}. The system simply says that the sum of V<sub>''ij''</sub> steps of generator g<sub>''i''</sub> for all ''i'' should equal the ''j''-th targeted JI interval J<sub>''j''</sub>.
 There are a number of methods to solve least square problems. One common way is to use the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]].