DR error measures - Revision history

Inthar: /* Solution methods */

2026-02-24T04:22:14Z

Solution methods

← Older revision		Revision as of 04:22, 24 February 2026
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	== Solution methods ==		== Solution methods ==
	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 most of those methods exist is to find solutions when finding ~~them symbolically~~ is infeasible.)		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 most of those methods exist is to find solutions when finding formulas for the solutions is infeasible.)
	=== Grid method (FDR case) ===		=== Grid method (FDR case) ===
	<syntaxhighlight lang="py" line="1">		<syntaxhighlight lang="py" line="1">

Inthar at 15:36, 9 February 2026

2026-02-09T15:36:28Z

← Older revision		Revision as of 15:36, 9 February 2026
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	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 most of those methods exist is to find solutions when finding them symbolically is infeasible.)		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 most of those methods exist is to find solutions when finding them symbolically is infeasible.)
	=== Grid method (FDR case) ===		=== Grid method (FDR case) ===
	<syntaxhighlight lang="py">		<syntaxhighlight lang="py" line="1">
	class GridSolution:		class GridSolution:
	def __init__(self, x, fx):		def __init__(self, x, fx):
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	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 objective function.		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 objective function.

	<syntaxhighlight lang="py">		<syntaxhighlight lang="py" line="1">
	import numpy as np		import numpy as np
	import math		import math
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	L-BFGS-B is an approximation to BFGS-B limiting memory usage, particularly suited for high-dimensional problems but nevertheless agreeing very well with BFGS-B for typical DR test cases. Python code for the L-BFGS method is provided below.		L-BFGS-B is an approximation to BFGS-B limiting memory usage, particularly suited for high-dimensional problems but nevertheless agreeing very well with BFGS-B for typical DR test cases. Python code for the L-BFGS method is provided below.

	<syntaxhighlight lang="py">		<syntaxhighlight lang="py" line="1">
	import numpy as np		import numpy as np
	import math		import math

Inthar at 02:19, 5 January 2026

2026-01-05T02:19:29Z

← Older revision		Revision as of 02:19, 5 January 2026
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	\|\|<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>		\|\|<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~~)		!rowspan="3"\|Logarithmic<br/>(nataves)
	!\|Rooted		!\|Rooted
	\|\|<math>\sqrt{\sum_{i=1}^n \Bigg(\log \frac{x + D_i}{r_i x} \Bigg)^2}</math>		\|\|<math>\sqrt{\sum_{i=1}^n \Bigg(\log \frac{x + D_i}{r_i x} \Bigg)^2}</math>
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	\|\|<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>		\|\|<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>
	\|}		\|}
	To convert ~~nepers~~ to cents, multiply by <math>\frac{1200}{\log 2}.</math>		To convert nataves to cents, multiply by <math>\frac{1200}{\log 2}.</math>

	== Solution methods ==		== Solution methods ==

Inthar: /* External links */

2026-01-03T16:36:26Z

External links

← Older revision		Revision as of 16:36, 3 January 2026
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	== External links ==		== External links ==
	* [https://~~inthar-raven~~.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]		* [https://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]
	{{cat\|Atypical ratios}}		{{cat\|Atypical ratios}}

Inthar: /* Domain and error model */

2025-12-27T21:31:34Z

Domain and error model

← Older revision		Revision as of 21:31, 27 December 2025
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	** ''all-steps'': Only intervals between adjacent notes are chosen.		** ''all-steps'': Only intervals between adjacent notes are chosen.

	We arrive at the following general formula: Let <math>I \subseteq {\{0, 1, 2, ..., 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 objective function (measuring the error) to be minimized by optimizing <math>x</math> and any free deltas is:		We arrive at the following general formula: Let <math>I \subseteq {\{0, 1, 2, ..., 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 objective function (measuring the error) to be minimized by optimizing <math>x</math> and any free deltas is:

	<math>		<math>

Inthar: /* Domain and error model */

2025-12-27T21:31:23Z

Domain and error model

← Older revision		Revision as of 21:31, 27 December 2025
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	** ''all-steps'': Only intervals between adjacent notes are chosen.		** ''all-steps'': Only intervals between adjacent notes are chosen.

	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 objective function (measuring the error) to be minimized by optimizing <math>x</math> and any free deltas is:		We arrive at the following general formula: Let <math>I \subseteq {\{0, 1, 2, ..., 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 objective function (measuring the error) to be minimized by optimizing <math>x</math> and any free deltas is:

	<math>		<math>

Inthar: /* Domain and error model */

2025-12-27T21:30:34Z

Domain and error model

← Older revision		Revision as of 21:30, 27 December 2025
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	* the collection of intervals to sum over (which we call the ''error model''):		* the collection of intervals to sum over (which we call the ''error model''):
	** ''rooted'': Only intervals from the root real-valued harmonic ''x'' are chosen.		** ''rooted'': Only intervals from the root real-valued harmonic ''x'' are chosen.
	** ''pairwise'': All intervals in the chord are ~~compared~~.		** ''pairwise'': All intervals in the chord are chosen.
	** ''all-steps'': Only intervals between adjacent notes are ~~compared~~.		** ''all-steps'': Only intervals between adjacent notes are chosen.

	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 objective function (measuring the error) to be minimized by optimizing <math>x</math> and any free deltas is:		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 objective function (measuring the error) to be minimized by optimizing <math>x</math> and any free deltas is:

Inthar at 17:09, 27 December 2025

2025-12-27T17:09:21Z

← Older revision		Revision as of 17:09, 27 December 2025
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	with some root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature +δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub> written cumulatively.		with some root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature +δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub> written cumulatively.

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

	== Domain and error model ==		== Domain and error model ==

Inthar: /* Analytic */

2025-12-27T01:05:59Z

Analytic

@@ Line 58: / Line 58: @@
 == Solution methods ==
 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 most of those methods exist is to find solutions when finding them symbolically is infeasible.)
 === Grid method (FDR case) ===
 <syntaxhighlight lang="py">

Inthar: /* Fully DR, rooted linear */

2025-12-26T04:54:40Z

Fully DR, rooted linear

← Older revision		Revision as of 04:54, 26 December 2025
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	which can be plugged back into		which can be plugged back into

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

	to obtain the least-squares linear error.		to obtain the least-squares linear error.