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 TE-weighted basis.

'''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 ''k'' 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 ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' 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 ([[Wikipedia: 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.

== ~~Introduction ==~~

== Definition ==

If we have ''k'' 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 ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' 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 ([[Wikipedia: Root mean square|root mean squared]]) tuning.

~~== 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.

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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 TE-weighted basis.
+'''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 ''k'' 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 ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' 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 ([[Wikipedia: 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.
-== Introduction ==
+== Definition ==
-If we have ''k'' 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 ''k'' in the prime limit ''p'', where ''p'' is the ''n''-th prime. Similarly, starting from ''n'' - ''k'' independent commas for the same regular temperament, the corresponding monzos span an ''n'' - ''k'' 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 ([[Wikipedia: Root mean square|root mean squared]]) tuning.
-== 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.