DR error measures

This is a technical or mathematical page. While the subject may be of some relevance to music, the page treats the subject in technical language.

Least-squares linear error (here linear means "in frequency space, not pitch space") is a proposed error measure for approximations to delta-rational chords. It has the advantage of not fixing a particular interval in the chord when constructing the chord of best fit. However, 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:...:r_n (in increasing order), with n > 1, in the linear domain as an approximation to a fully delta-rational chord with signature +δ₁ +δ₂ ... +δ_n, 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 the closed-form solution

$x=\frac{\sum _{i=1}^n{D}_i}{-n+\sum _{i=1}^n{f}_i},$

which you plug back into

$\sqrt{\sum _{1=1}^n(1+\frac{D_i}{x}-f_i{)}^2}$

to obtain the least-squares linear error.

External links

• Inthar's DR chord explorer (includes least-squares linear error calculation)