Tenney–Euclidean tuning: Difference between revisions

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<math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q))</math>

If ''V'' is the mapping of the [[~~Regular~~ temperament|abstract temperament]] whose rows are (not necessarily independent) vals, then {{nowrap|''VW'' {{=}} ''VW''}} is the mapping in the weighted space. If ''J'' is the row vector of targeted JI intervals (i.e. the [[JIP]]), then {{nowrap|''JW'' {{=}} ''JW''}} is the JI intervals in the weighted space, in the case of Tenney-weighting it is {{val| 1 1 … 1 }}. Let us also denote the row vector of TE generators ''G''. TE tuning then defines a {{w|least squares}} problem of the following overdetermined linear equation system:

If ''V'' is the mapping of the [[regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, then {{nowrap| ''VW'' {{=}} ''VW'' }} is the mapping in the weighted space. If ''J'' is the row vector of targeted JI intervals (i.e. the [[JIP]]), then {{nowrap| ''JW'' {{=}} ''JW'' }} is the JI intervals in the weighted space, in the case of Tenney-weighting it is {{val| 1 1 … 1 }}. Let us also denote the row vector of TE generators ''G''. TE tuning then defines a {{w|least squares}} problem of the following overdetermined linear equation system:

<math>\displaystyle GV_W = J_W</math>

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The system simply says that the sum of (''v''''w'')''kl'' steps of generator ''g''''k'' for all ''k'''s should equal the ''l''-th targeted JI interval (''j''''w'')''l''.

There are a number of methods to solve least squares problems. One common way is to use the ~~{{w|~~Moore–Penrose pseudoinverse}}.

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

== Computation using pseudoinverse ==

The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. ~~To name a few:~~

The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. In this method, the (not necessarily independent) TE generator map ''G'', which correspond to the rows of ''V'' are given by

* If ''A'' is square and invertible, then its pseudoinverse is equal to its inverse; that is, {{nowrap|''A''{{+}} {{=}} ''A''{{inv}}}}

* If ''A'' has rational entries, so does ''A''{{+}}

* {{nowrap|(''A''{{+}}){{+}} {{=}} ''A''}}

* {{nowrap|(''A''{{t}}){{+}} {{=}} (''A''{{+}}){{t}}}}, where ''A''{{t}} is the transpose of ''A''

* ''AA''{{+}} is the orthogonal projection matrix that maps onto the space spanned by the columns of ''A''

* ''A''{{+}}''A'' is the orthogonal projection matrix that maps onto the space spanned by the rows of ''A''

* {{nowrap|''I'' − ''A''{{+}}''A''}}, where ''I'' is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of ''A''

* If the rows of ''A'' are linearly independent, then {{nowrap|''A''{{+}} {{=}} ''A''{{t}}(''AA''{{+}}){{inv}}}}. This means the pseudoinverse can be found in this ~~important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.~~

* ''uA''{{+}} is the nearest point to ''u'' in the subspace spanned by the rows of ''A''; ''A''{{+}}''v'' is the nearest point to ''v'' in the space spanned by the columns of ''A''.

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

<math>\displaystyle G = J_W V_W^+</math>

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

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

<math>\displaystyle T = GV = J_W V_W^+ V</math>

We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''~~~~''W''~~{{~~+}}''V''''W''}} is the orthogonal projection matrix that maps 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 JIP: {{nowrap|''T'' {{=}} ''JP''}} ~~where~~ ''J'' ~~is the JIP~~, which is the nearest point in the subspace corresponding to the temperament to ''J''.

We may also obtain the TE tuning from a [[projection matrix]]. {{nowrap| ''P''''W'' {{=}} {{subsup|''V''|''W''|+}}''V''''W'' }} is the orthogonal projection matrix that maps onto the space spanned by the rows of ''V''''W''. This space corresponds to the temperament, and so does ''P''''W''. However, ''P''''W'' 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 weighted tuning map ''T''''W'' = ''TW'' giving the weighted tuning of each prime number is found by multiplying by the JIP: {{nowrap| ''T''''W'' {{=}} ''J''''W''''P''''W'' {{=}} ''J''''W''{{subsup|''V''|''W''|+}}''V''''W'' }}, which is the nearest point in the subspace corresponding to the temperament to ''J''''W'', so that {{nowrap| ''T'' {{=}} ''J''''W''''P''''W''''W''{{inv}} {{=}} ''J''''W''{{subsup|''V''|''W''|+}}''V'' }}.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''''W'' is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''''W''{{subsup|''M''|''W''|+}}}} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''''W'' so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.

We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''''W'' is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap| ''P''''W'' {{=}} ''I'' − ''M''''W''{{subsup|''M''|''W''|+}} }} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''''W'' so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.

== Enforcement ==

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We may call the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''JP'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|scalar multiplication}}, ''T''/''T''1, where ''T''1, 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 the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''J''''W''{{subsup|''V''|''W''|+}}''V'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|scalar multiplication}}, ''T''/''t''1, where ''t''1, 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''.

=== Constrained TE tuning ===

@@ Line 14: / Line 14: @@
 <math>\displaystyle W = \operatorname {diag} (1/\log_2 (Q))</math>
-If ''V'' is the mapping of the [[Regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, then {{nowrap|''V<sub>W</sub>'' {{=}} ''VW''}} is the mapping in the weighted space. If ''J'' is the row vector of targeted JI intervals (i.e. the [[JIP]]), then {{nowrap|''J<sub>W</sub>'' {{=}} ''JW''}} is the JI intervals in the weighted space, in the case of Tenney-weighting it is {{val| 1 1 … 1 }}. Let us also denote the row vector of TE generators ''G''. TE tuning then defines a {{w|least squares}} problem of the following overdetermined linear equation system:
+If ''V'' is the mapping of the [[regular temperament|abstract temperament]] whose rows are (not necessarily independent) vals, then {{nowrap| ''V<sub>W</sub>'' {{=}} ''VW'' }} is the mapping in the weighted space. If ''J'' is the row vector of targeted JI intervals (i.e. the [[JIP]]), then {{nowrap| ''J<sub>W</sub>'' {{=}} ''JW'' }} is the JI intervals in the weighted space, in the case of Tenney-weighting it is {{val| 1 1 … 1 }}. Let us also denote the row vector of TE generators ''G''. TE tuning then defines a {{w|least squares}} problem of the following overdetermined linear equation system:
 <math>\displaystyle GV_W = J_W</math>
@@ Line 20: / Line 20: @@
 The system simply says that the sum of (''v''<sub>''w''</sub>)<sub>''kl''</sub> steps of generator ''g''<sub>''k''</sub> for all ''k'''s should equal the ''l''-th targeted JI interval (''j''<sub>''w''</sub>)<sub>''l''</sub>.
-There are a number of methods to solve least squares problems. One common way is to use the {{w|Moore–Penrose pseudoinverse}}.
+There are a number of methods to solve least squares problems. One common way is to use the [[Moore–Penrose pseudoinverse]].
 == Computation using pseudoinverse ==
-The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. To name a few:
+The Moore–Penrose pseudoinverse, denoted ''A''{{+}}, is a generalization of the inverse matrix with which it shares a lot of properties. In this method, the (not necessarily independent) TE generator map ''G'', which correspond to the rows of ''V'' are given by
-* If ''A'' is square and invertible, then its pseudoinverse is equal to its inverse; that is, {{nowrap|''A''{{+}} {{=}} ''A''{{inv}}}}
-* If ''A'' has rational entries, so does ''A''{{+}}
-* {{nowrap|(''A''{{+}}){{+}} {{=}} ''A''}}
-* {{nowrap|(''A''{{t}}){{+}} {{=}} (''A''{{+}}){{t}}}}, where ''A''{{t}} is the transpose of ''A''
-* ''AA''{{+}} is the orthogonal projection matrix that maps onto the space spanned by the columns of ''A''
-* ''A''{{+}}''A'' is the orthogonal projection matrix that maps onto the space spanned by the rows of ''A''
-* {{nowrap|''I'' − ''A''{{+}}''A''}}, where ''I'' is the identity matrix, is the orthogonal projection matrix that maps onto the kernel, or null space, of ''A''
-* If the rows of ''A'' are linearly independent, then {{nowrap|''A''{{+}} {{=}} ''A''{{t}}(''AA''{{+}}){{inv}}}}. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.
-* ''uA''{{+}} is the nearest point to ''u'' in the subspace spanned by the rows of ''A''; ''A''{{+}}''v'' is the nearest point to ''v'' in the space spanned by the columns of ''A''.
-In the pseudoinverse method, the (not necessarily independent) TE generators which correspond to the rows of ''V'' are given by
 <math>\displaystyle G = J_W V_W^+</math>
-Applying the weighted val list to the generators, The TE tuning map is given by
+Applying the val list to the generators, The TE tuning map is given by
 <math>\displaystyle T = GV = J_W V_W^+ V</math>
-We may also obtain the TE tuning from a projection matrix. {{nowrap|''P'' {{=}} ''V''<sub>''W''</sub>{{+}}''V''<sub>''W''</sub>}} is the orthogonal projection matrix that maps 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 JIP: {{nowrap|''T'' {{=}} ''JP''}} where ''J'' is the JIP, which is the nearest point in the subspace corresponding to the temperament to ''J''.
+We may also obtain the TE tuning from a [[projection matrix]]. {{nowrap| ''P''<sub>''W''</sub> {{=}} {{subsup|''V''|''W''|+}}''V''<sub>''W''</sub> }} is the orthogonal projection matrix that maps onto the space spanned by the rows of ''V''<sub>''W''</sub>. This space corresponds to the temperament, and so does ''P''<sub>''W''</sub>. However, ''P''<sub>''W''</sub> 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 weighted tuning map ''T''<sub>''W''</sub> = ''TW'' giving the weighted tuning of each prime number is found by multiplying by the JIP: {{nowrap| ''T''<sub>''W''</sub> {{=}} ''J''<sub>''W''</sub>''P''<sub>''W''</sub> {{=}} ''J''<sub>''W''</sub>{{subsup|''V''|''W''|+}}''V''<sub>''W''</sub> }}, which is the nearest point in the subspace corresponding to the temperament to ''J''<sub>''W''</sub>, so that {{nowrap| ''T'' {{=}} ''J''<sub>''W''</sub>''P''<sub>''W''</sub>''W''{{inv}} {{=}} ''J''<sub>''W''</sub>{{subsup|''V''|''W''|+}}''V'' }}.
-We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''<sub>''W''</sub> is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap|''P'' {{=}} ''I'' − ''M''<sub>''W''</sub>{{subsup|''M''|''W''|+}}}} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''<sub>''W''</sub> so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
+We may find the same projection matrix starting from a list of weighted monzos rather than vals. If ''M''<sub>''W''</sub> is a rank-''n'' matrix whose columns are weighted monzos, and ''I'' is the ''n''×''n'' identity matrix, then {{nowrap| ''P''<sub>''W''</sub> {{=}} ''I'' − ''M''<sub>''W''</sub>{{subsup|''M''|''W''|+}} }} is the same projection matrix as {{subsup|''V''|''W''|+}}''V''<sub>''W''</sub> so long as the temperament defined by the vals is the same as the temperament defined by the monzos. Again, it is irrelevant if the monzos are independent or how many of them there are.
 == Enforcement ==
@@ Line 51: / Line 39: @@
 {{Main| POTE tuning }}
-We may call the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''JP'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|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 the pure-octave Tenney−Euclidean tuning the ''POTE tuning''. If {{nowrap|''T'' {{=}} ''J''<sub>''W''</sub>{{subsup|''V''|''W''|+}}''V'' {{=}} ''GV''}} is the TE tuning map, then a corresponding pure-octaves map can be found by {{w|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''.
 === Constrained TE tuning ===