Error measures for DR chords: Difference between revisions

The idea motivating least-squares error measures 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):

We want the error of a chord 1:''r''₁:''r''₂:...:''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

with root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature +δ₁ +δ₂ ... +δ_''n'' written cumulatively.

We want to measure the error ''without having to fix any dyad'' (as one might naively fix a dyad and measure errors in the other deltas in relation to the fixed dyad). To do this we solve a least-squares error problem: use a root-sum-square error function and optimize ''x'' (and any free deltas) to minimize that function.

We arrive at the following general formula: Let <math>[n] = \{0, 1, 2, ..., n\},</math> let <math>I \subseteq {[n] \choose 2}</math> be the error model, and let <math>f_D</math> represent the domain function (identity for linear, or <math>\log</math>). Then the error function to be minimized by optimizing <math>x</math> and any free deltas is:

|+ Error function for various modes and error models

!|Domain

!|Error model

!|Error function

!rowspan="3"|Linear

!|Rooted

||<math>\sqrt{\sum_{i=1}^n \Bigg( \frac{x + D_i}{x} - r_i \Bigg)^2 } = \sqrt{\sum_{i=1}^n \Bigg( 1 + \frac{D_i}{x} - r_i \Bigg)^2 }</math>

!|Pairwise

||<math>\sqrt{\sum_{0\leq i<j\leq n} \Bigg( \frac{x + D_j}{x + D_i} - \frac{r_j}{r_i} \Bigg)^2 }</math>

!|All-steps

||<math>\sqrt{\sum_{0\leq i<n} \Bigg( \frac{x + D_{i+1}}{x + D_i} - \frac{r_{i+1}}{r_i} \Bigg)^2 }</math>

!rowspan="3"|Logarithmic<br/>(nepers)

!|Rooted

||<math>\sqrt{\sum_{i=1}^n \Bigg(\log \frac{x + D_i}{r_i x} \Bigg)^2}</math>

!|Pairwise

||<math>\sqrt{\sum_{0\leq i<j\leq n} \Bigg(\log \frac{x + D_j}{x + D_i} - \log \frac{r_j}{r_i} \Bigg)^2}</math>

!|All-steps

||<math>\sqrt{\sum_{0\leq i<n} \Bigg(\log \frac{x + D_{i+1}}{x + D_i} - \log \frac{r_{i+1}}{r_i} \Bigg)^2}</math>

This section gets into the depths of mathematical optimization methods used to minimize DR error. (Optimization is a whole field and there are many different methods; the reason these methods exist is to find solutions when finding them symbolically is infeasible.)

<math>x = \frac{\sum_{i=1}^n D_i }{-n + \sum_{i=1}^n r_i},</math>

<math>\sqrt{\sum_{1=1}^n \Bigg( 1 + \frac{D_i}{x} - r_i \Bigg)^2 }</math>

We can set the partial derivatives with respect to ''x'' and ''y'' of the inner expression equal to zero (since the derivative of sqrt() is never 0) and use SymPy to solve the system symbolically:

The unique solution with ''x'' > 0 is

<syntaxhighlight>

We let ''x''₁ = ''x'' and include additional free variables ''x''₂, ..., ''x''_n, one for every additional +?, after coalescing strings of consecutive +?'s into one +? and after trimming leading and trailing free delta segments.

* The error function is smooth, allowing use of gradients

* Naturally deals with the ''x'' > 0 constraint using a log barrier and minimizing the transformed function using the unconstrained BFGS method

It is a ''quasi-Newton method'' because it uses an approximation of the Hessian (matrix of mixed second partial derivatives) of the error function at each step.

In the Python implementation below, <code>x</code> represents the vector <math>(x_1, x_2, ..., x_n),</math> <code>x0</code> is the initial guess for the solution, and <code>f</code> is the error function.

<syntaxhighlight>

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* [https://inthar-raven.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]