DR error measures: Difference between revisions
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<adv>{{adv|'''Least-squares linear error''' (here ''linear'' means "in frequency space, not pitch space") is the most naive error measure for approximations to [[delta-rational chord]]s. Like any other numerical measure of concordance or error, you should take it with a grain of salt. | <adv>{{adv|'''Least-squares linear error''' (here ''linear'' means "in frequency space, not pitch space") is the most naive error measure for approximations to [[delta-rational chord]]s. Like any other numerical measure of concordance or error, you should take it with a grain of salt. | ||
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 ''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 | The idea motivating least-squares linear error on a chord as an approximation to a given delta signature is the following (for simplicity, let’s talk about the fully DR case first): | ||
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 | |||
x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l. | x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l. | ||
Revision as of 23:14, 7 December 2025
<adv>Least-squares linear error (here linear means "in frequency space, not pitch space") is the most naive error measure for approximations to delta-rational chords. Like any other numerical measure of concordance or error, you should take it with a grain of salt.
The idea motivating least-squares linear error on a chord as an approximation to a given delta signature is the following (for simplicity, let’s talk about the fully DR case first):
Say we want the error of a chord 1:r1:r2:...:rn (in increasing order), with n > 1, in the linear domain as an approximation to a fully delta-rational chord with signature +δ1 +δ2 ... +δn, i.e. a chord
x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l.
Minimizing the least-squares frequency-domain error by varying x gives you
x = \frac{\sum_{i=1}^n D_i }{-n + \sum_{i=1}^n f_i},
which you plug back into
\sqrt{\sum_{i=1}^n \Bigg( 1 + \frac{D_i}{x} - f_i \Bigg)^2 }
to obtain the least-squares linear error.</adv>
