Delta-rational chord: Difference between revisions

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== Least-squares error ==

Least-squares linear error (here linear means "in frequency space, not pitch space") is the most naive error measure for approximations to DR chords. Like any other numerical measure of concordance, you should take it with a grain of salt.

=== Fully DR ===

The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ1 +δ2 ... +δ''n''

The idea motivating least-squares linear error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order), with {{nowrap|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ1 +δ2 ... +δ''n''

}}, i.e. a chord

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which can be plugged back into the error formula to obtain the error. (We multiply the target DR chord by ''x'' in order to compare it to the approximation on the same isodifferential series.) Note that scaling the delta signature does not affect the error, a critical property for a DR error measure.

The least-squares error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.

The least-squares linear error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.

=== Partially DR ===

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 == Least-squares error ==
+Least-squares linear error (here linear means "in frequency space, not pitch space") is the most naive error measure for approximations to DR chords. Like any other numerical measure of concordance, you should take it with a grain of salt.
 === Fully DR ===
-The idea motivating least-squares error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order), with {{nowrap|''n'' &gt; 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>
+The idea motivating least-squares linear error on a chord as an approximation to a given delta signature is the following: Say we want the error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order), with {{nowrap|''n'' &gt; 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>
 }}, i.e. a chord
@@ Line 93: / Line 94: @@
 which can be plugged back into the error formula to obtain the error. (We multiply the target DR chord by ''x'' in order to compare it to the approximation on the same isodifferential series.) Note that scaling the delta signature does not affect the error, a critical property for a DR error measure.
-The least-squares error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.
+The least-squares linear error measure does not form a metric on the set of delta signatures with a fixed number of terms, since it is not symmetric.
 === Partially DR ===