Tenney–Euclidean tuning: Difference between revisions

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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.

In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''.

== Definition ==

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If the vals defining V are linearly independent, then P = VT(VVT)-1V. If the columns of M are independent, then we likewise have P = I - M(MTM)-1MT.

=== Benedetti-Euclidean tuning ===

In D&D's guide, the systematic name for Frobenius tuning is ''minimax-E-copfr-S''.

'''Benedetti-Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson height]]. For Q = {{val| 2 3 5 … }}, the weighting matrix has the form

=== Benedetti/Wilson-Euclidean tuning ===

'''Benedetti/Wilson-Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson height]]. For Q = {{val| 2 3 5 … }}, the weighting matrix has the form

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

In D&D's guide, the systematic name for BE tuning is ''minimax-E-sopfr-S''.

== Examples ==

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 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.
+In [[D&D's guide]], the systematic name for TE tuning is ''minimax-ES''.
 == Definition ==
@@ Line 80: / Line 82: @@
 If the vals defining V are linearly independent, then P = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>V. If the columns of M are independent, then we likewise have P = I - M(M<sup>T</sup>M)<sup>-1</sup>M<sup>T</sup>.
-=== Benedetti-Euclidean tuning ===
+In D&D's guide, the systematic name for Frobenius tuning is ''minimax-E-copfr-S''.
-'''Benedetti-Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson height]]. For Q = {{val| 2 3 5 … }}, the weighting matrix has the form
+=== Benedetti/Wilson-Euclidean tuning ===
+'''Benedetti/Wilson-Euclidean tuning''' ('''BE tuning''') adopts the Benedetti or Wilson weight in place of Tenney weight, based on the dual norm of [[Wilson height]]. For Q = {{val| 2 3 5 … }}, the weighting matrix has the form
 <math>\displaystyle W = \operatorname {diag} (1/Q)</math>
+In D&D's guide, the systematic name for BE tuning is ''minimax-E-sopfr-S''.
 == Examples ==