Delta-rational chord: Difference between revisions

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=== Measuring the error of an approximate fully DR chord ===

==== ~~Based on the least~~-squares ~~problem on delta~~ error ====

==== Least-squares error measures ====

Say we want the error of a chord 1:''r''1:''r''2:...:''r''''n'' (in increasing order) in the linear domain as an approximation to a fully delta-rational chord with signature +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 1:''r''1:''r''2:...:''r''''n'' chord by α in order to compare it to the target DR chord on the same isodifferential series.)

This error measure is called '''least-squares delta error'''~~{{idiosyncratic}} (LSD error)~~. ~~LSD~~ error does not depend on whether the chord whose error is being measured is 1:''r''1:''r''2:...:''r''''n'' or the same chord linearly shifted to have root α. Unfortunately, ~~LSD~~ error does not form a metric on the set of delta signatures with a fixed number of terms.

This error measure is called '''least-squares delta error'''. Least-squares delta error does not depend on whether the chord whose error is being measured is 1:''r''1:''r''2:...:''r''''n'' or the same chord linearly shifted to have root α. Unfortunately, this error measure does not form a metric on the set of delta signatures with a fixed number of terms.

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==== Partially DR ====

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</math>

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==== Metric error measures ====

Another possible approach is to view delta signatures as vectors of positive reals by fixing the first element as 1) and measure the induced <math>L^p</math> distance.

== DR and RTT ==

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 === Measuring the error of an approximate fully DR chord ===
-==== Based on the least-squares problem on delta error ====
+==== Least-squares error measures ====
 Say we want the error of a chord 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> (in increasing order) in the linear domain as an approximation to a fully delta-rational chord with signature +δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>, i.e. a chord
@@ Line 73: / Line 73: @@
 which can be plugged back into the error formula to obtain the error. (We multiply the 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> chord by α in order to compare it to the target DR chord on the same isodifferential series.)
-This error measure is called '''least-squares delta error'''{{idiosyncratic}} (LSD error). LSD error does not depend on whether the chord whose error is being measured is 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> or the same chord linearly shifted to have root α. Unfortunately, LSD error does not form a metric on the set of delta signatures with a fixed number of terms.
+This error measure is called '''least-squares delta error'''. Least-squares delta error does not depend on whether the chord whose error is being measured is 1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub> or the same chord linearly shifted to have root α. Unfortunately, this error measure does not form a metric on the set of delta signatures with a fixed number of terms.
 <!--
 ==== Partially DR ====
@@ Line 91: / Line 91: @@
 </math>
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+==== Metric error measures ====
+Another possible approach is to view delta signatures as vectors of positive reals by fixing the first element as 1) and measure the induced <math>L^p</math> distance.
 == DR and RTT ==