Tenney–Euclidean tuning: Difference between revisions

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'''Tenney–Euclidean tuning''' ('''TE tuning'''), also known as '''TOP–RMS tuning''', is a tuning technique for regular temperaments which leads to the least sum of squared errors of the Tenney-weighted basis.

If we have ''r'' linearly independent [[Vals and tuning space|vals]] of dimension ''n'', they will span a subspace of [[Vals and tuning space|tuning space]]. This subspace defines a regular temperament of rank ''r'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from {{nowrap|''n'' − ''r''}} independent commas for the same regular temperament, the corresponding monzos span an {{nowrap|''n'' − ''r''}} dimensional subspace of [[~~Monzos~~ and interval space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is the weighted RMS ({{w|root-mean-squared}}) tuning discussed right here.

If we have ''r'' linearly independent [[Vals and tuning space|vals]] of dimension ''n'', they will span a subspace of [[Vals and tuning space|tuning space]]. This subspace defines a regular temperament of rank ''r'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from {{nowrap|''n'' − ''r''}} independent commas for the same regular temperament, the corresponding monzos span an {{nowrap|''n'' − ''r''}} dimensional subspace of [[monzos and interval space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is the weighted RMS ({{w|root mean square|root-mean-squared}}) tuning discussed right here.

TE tuning can be viewed as a variant of [[TOP tuning]] since it employs the [[Tenney–Euclidean metrics #TE norm|TE norm]] in place of the [[Tenney height]] as in TOP tuning. Just as TOP tuning minimizes the maximum Tenney-weighted ''L''1 error of any interval, TE tuning minimizes the maximum Tenney-weighted ''L''2 error of any interval.

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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, 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 ''W'' is the weighting matrix. For the prime basis Q = {{val| 2 3 5 … ''p'' }},

In more pragmatic terms, suppose ''W'' is the weighting matrix. For the prime basis ''Q'' = {{val| 2 3 5 … ''p'' }},

<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 ~~[[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 {{w|Moore–Penrose pseudoinverse}}.

== Computation using pseudoinverse ==

@@ Line 2: / Line 2: @@
 '''Tenney–Euclidean tuning''' ('''TE tuning'''), also known as '''TOP–RMS tuning''', is a tuning technique for regular temperaments which leads to the least sum of squared errors of the Tenney-weighted basis.
-If we have ''r'' linearly independent [[Vals and tuning space|vals]] of dimension ''n'', they will span a subspace of [[Vals and tuning space|tuning space]]. This subspace defines a regular temperament of rank ''r'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from {{nowrap|''n'' − ''r''}} independent commas for the same regular temperament, the corresponding monzos span an {{nowrap|''n'' − ''r''}} dimensional subspace of [[Monzos and interval space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is the weighted RMS ({{w|root-mean-squared}}) tuning discussed right here.
+If we have ''r'' linearly independent [[Vals and tuning space|vals]] of dimension ''n'', they will span a subspace of [[Vals and tuning space|tuning space]]. This subspace defines a regular temperament of rank ''r'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from {{nowrap|''n'' − ''r''}} independent commas for the same regular temperament, the corresponding monzos span an {{nowrap|''n'' − ''r''}} dimensional subspace of [[monzos and interval space|interval space]]. Both the subspace of tuning space and the subspace of interval space characterize the temperament completely. A question then arises as to how to choose a specific tuning for this temperament, which is the same as asking how to choose a point (vector) in this subspace of tuning space which provides a good tuning. One answer to this is the weighted RMS ({{w|root mean square|root-mean-squared}}) tuning discussed right here.
 TE tuning can be viewed as a variant of [[TOP tuning]] since it employs the [[Tenney–Euclidean metrics #TE norm|TE norm]] in place of the [[Tenney height]] as in TOP tuning. Just as TOP tuning minimizes the maximum Tenney-weighted ''L''<sub>1</sub> error of any interval, TE tuning minimizes the maximum Tenney-weighted ''L''<sub>2</sub> error of any interval.
@@ Line 11: / Line 11: @@
 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 ''W'' is the weighting matrix. For the prime basis Q = {{val| 2 3 5 … ''p'' }},
+In more pragmatic terms, suppose ''W'' is the weighting matrix. For the prime basis ''Q'' = {{val| 2 3 5 … ''p'' }},
 <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 21: / Line 21: @@
 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 [[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 {{w|Moore–Penrose pseudoinverse}}.
 == Computation using pseudoinverse ==