Tenney–Euclidean tuning: Difference between revisions

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== 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 [[JIP]] {{val|1 1 … 1}}, and this closest point will define a tuning map which is called TE tuning, 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, 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 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:

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 squares problem]] of the following overdetermined linear equation system:

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''.

where J = J0W, i.e. the JIP, in the case of Tenney-weighted 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]].

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

== Computing TE tuning using pseudoinverse ==

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In the pseudoinverse method, the (not necessarily independent) TE generators which correspond to the rows of V are given by

Applying the weighted val list to the generators, The TE tuning map is given by

We may also obtain the TE tuning from a projection map. P = V+V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether contorsion has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.

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We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = gV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia: Scalar multiplication|scalar multiplication]], T/T1, where T1, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.

We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = GV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia: Scalar multiplication|scalar multiplication]], T/T1, where T1, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.

== Frobenius projection map ==

@@ Line 9: / Line 9: @@
 == 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 [[JIP]] {{val|1 1 … 1}}, and this closest point will define a tuning map which is called TE tuning, 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, 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 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:
+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 squares problem]] of the following overdetermined linear equation system:
-<math>\vec{g}V = J</math>
+<math>GV = J</math>
-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>.
+where J = J<sub>0</sub>W, i.e. the JIP, in the case of Tenney-weighted 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]].
+There are a number of methods to solve least squares problems. One common way is to use the [[Wikipedia: Moore–Penrose pseudoinverse|Moore–Penrose pseudoinverse]].
 == Computing TE tuning using pseudoinverse ==
@@ Line 34: / Line 34: @@
 In the pseudoinverse method, the (not necessarily independent) TE generators which correspond to the rows of V are given by
-<math>\vec{g} = JV^+</math>
+<math>G = JV^+</math>
 Applying the weighted val list to the generators, The TE tuning map is given by
-<math>T = \vec{g}V = JV^+V</math>
+<math>T = GV = JV^+V</math>
 We may also obtain the TE tuning from a projection map. P = V<sup>+</sup>V is the orthogonal projection map onto the space spanned by the rows of V. This space corresponds to the temperament, and so does P. However, P is independent of how the temperament is defined; it does not depend on whether the vals are linearly independent, how many of them there are, or whether contorsion has been removed. The tuning map giving the tuning of each prime number is found by multiplying by the JI map: JP where J is the JI map, which is the nearest point in the subspace corresponding to the temperament to J.
@@ Line 47: / Line 47: @@
 {{see also| POTE tuning }}
-We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = gV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia: Scalar multiplication|scalar multiplication]], T/T<sub>1</sub>, where T<sub>1</sub>, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.
+We may call pure-octaves Tenney-Euclidean tuning the '''POTE tuning'''. If T = JP = GV is the TE tuning map, then a corresponding pure-octaves map can be found by [[wikipedia: Scalar multiplication|scalar multiplication]], T/T<sub>1</sub>, where T<sub>1</sub>, the first entry, is the tuning of 2. The justification for this is that T does not only define a point, but a line through the origin lying in the subspace defining the temperament, or in other words, a point in the linear subspace of projective space corresponding to the temperament, and hence is a projective object. Another way to say this is that T defines not only the closest point to J, but the closest direction in terms of angular measure between the line through T and the line through J.
 == Frobenius projection map ==