Delta-rational chord: Difference between revisions
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== Measuring the error of an approximate fully DR chord == | == Measuring the error of an approximate fully DR chord == | ||
=== Least-squares error measures === | === Least-squares error measures === | ||
==== Naive least-squares error ==== | |||
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 ''n'' > 1, 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 | 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 ''n'' > 1, 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 | ||
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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 x in order to compare it to the target DR chord on the same isodifferential series.) | 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 x in order to compare it to the target DR chord on the same isodifferential series.) | ||
This error measure is called '''least-squares | This error measure is called '''naive least-squares 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 x. Unfortunately, this error measure does not form a metric on the set of delta signatures with a fixed number of terms. | ||
This error measure was found by Inthar and groundfault. | This error measure was found by Inthar and groundfault. | ||
==== Symmetric least-squares error ==== | |||
This error is found by solving | |||
<math> | |||
\displaystyle{ \min_x \sqrt{ \sum_{i=1}^n \Bigg( E_ix - D_i \Bigg)^2 + \sum_{i=1}^n \Bigg( D_ix - E_i \Bigg)^2 } } | |||
</math> | |||
in order to make error({D_i}, {E_i}) = error({E_i}, {D_i}). | |||
== DR and RTT == | == DR and RTT == | ||