Error measures for DR chords - Revision history

Inthar: /* Fully DR, rooted linear */ Correct analytical solution for x

2026-02-28T20:58:08Z

Fully DR, rooted linear: Correct analytical solution for x

← Older revision		Revision as of 20:58, 28 February 2026
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	Setting the derivative to 0 gives us the closed-form solution		Setting the derivative to 0 gives us the closed-form solution

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

	which can be plugged back into		which can be plugged back into

Inthar: /* External links */ Update GItHub link

2026-01-03T16:30:55Z

External links: Update GItHub link

← Older revision		Revision as of 16:30, 3 January 2026
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	== External links ==		== External links ==
	* [http://~~inthar-raven~~.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]		* [http://turbofishcrow.github.io/delta/ Inthar's DR chord explorer (includes least-squares error calculation in both domains and multiple error models)]

	[[Category:Math]]		[[Category:Math]]

Inthar at 04:55, 26 December 2025

2025-12-26T04:55:49Z

← Older revision		Revision as of 04:55, 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.

Inthar: /* Conventions and introduction */

2025-12-26T04:55:27Z

Conventions and introduction

← Older revision		Revision as of 04:55, 26 December 2025
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	with root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap\|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>}} written cumulatively.		with root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap\|+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'' (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 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.

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

Inthar: /* Domain and error model */

2025-12-21T06:00:16Z

Domain and error model

← Older revision		Revision as of 06:00, 21 December 2025
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	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| ~~objective~~ function for various modes and error models		\|+ style="font-size: 105%;" \| Objective function for various modes and error models
	\|-		\|-
	! Domain		! Domain

Inthar: /* Domain and error model */

2025-12-21T05:59:56Z

Domain and error model

← Older revision		Revision as of 05:59, 21 December 2025
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	! Domain		! Domain
	! Error model		! Error model
	! ~~objective~~ function		! Objective function
	\|-		\|-
	! rowspan="3" \| Linear		! rowspan="3" \| Linear

Inthar at 05:59, 21 December 2025

2025-12-21T05:59:38Z

← Older revision		Revision as of 05:59, 21 December 2025
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	with root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap\|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>}} written cumulatively.		with root real-valued harmonic ''x''. Let <math>D_0 = 0, D_i = \sum_{k=1}^i \delta_k</math> be the delta signature {{nowrap\|+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'' (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 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 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 ==
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	** ''All-steps'': Only intervals between adjacent notes are compared.		** ''All-steps'': Only intervals between adjacent notes are compared.

	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:		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:

	<math>		<math>
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	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| ~~Error~~ function for various modes and error models		\|+ style="font-size: 105%;" \| objective function for various modes and error models
	\|-		\|-
	! Domain		! Domain
	! Error model		! Error model
	! ~~Error~~ function		! objective function
	\|-		\|-
	! rowspan="3" \| Linear		! rowspan="3" \| Linear
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	BFGS-B is a quasi-Newton optimization method (based on BFGS) particularly suited for this problem:		BFGS-B is a quasi-Newton optimization method (based on BFGS) particularly suited for this problem:
	* The ~~error~~ function is smooth, allowing use of gradients		* The objective function is smooth, allowing use of gradients
	* Fast convergence, requiring at worst 20 iterations for accuracy		* Fast convergence, requiring at worst 20 iterations for accuracy
	* Naturally deals with the {{nowrap\|''x'' > 0}} constraint using a log barrier and minimizing the transformed function using the unconstrained BFGS method		* Naturally deals with the {{nowrap\|''x'' > 0}} constraint using a log barrier and minimizing the transformed function using the unconstrained BFGS method
	* Acceptable memory usage given a realistic number of parameters for practical DR chords (up to 3 interior free delta segments, thus 4 parameters).		* Acceptable memory usage given a realistic number of parameters for practical DR chords (up to 3 interior free delta segments, thus 4 parameters).

	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.		It is a ''quasi-Newton method'' because it uses an approximation of the Hessian (matrix of mixed second partial derivatives) of the objective 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.		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="python">		<syntaxhighlight lang="python">

Inthar: /* Conventions and introduction */

2025-12-17T18:52:40Z

Conventions and introduction

← Older revision		Revision as of 18:52, 17 December 2025
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	== Conventions and introduction ==		== Conventions and introduction ==
	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)~~:		The idea motivating least-squares error measures on a chord as an approximation to a given delta signature is the following:

	We want the error of a chord {{nowrap\|1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub>}} (in increasing order, we also take {{nowrap\|''r''<sub>0</sub> {{=}} 1}}), with {{nowrap\|''n'' > 1}}, in the linear domain as an approximation to a ~~fully~~ delta-rational chord with signature {{nowrap\|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>}}, i.e. a chord		We want the error of a chord {{nowrap\|1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub>}} (in increasing order, we also take {{nowrap\|''r''<sub>0</sub> {{=}} 1}}), with {{nowrap\|''n'' > 1}}, in the linear domain as an approximation to a delta-rational chord with signature {{nowrap\|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>}} (possibly with some deltas free), i.e. a chord

	<math>x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l</math>		<math>x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l</math>

Inthar: /* Conventions and introduction */

2025-12-17T18:41:14Z

Conventions and introduction

← Older revision		Revision as of 18:41, 17 December 2025
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	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):		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 {{nowrap\|1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub>}} (in increasing order), with {{nowrap\|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap\|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>}}, i.e. a chord		We want the error of a chord {{nowrap\|1:''r''<sub>1</sub>:''r''<sub>2</sub>:...:''r''<sub>''n''</sub>}} (in increasing order, we also take {{nowrap\|''r''<sub>0</sub> {{=}} 1}}), with {{nowrap\|''n'' > 1}}, in the linear domain as an approximation to a fully delta-rational chord with signature {{nowrap\|+δ<sub>1</sub> +δ<sub>2</sub> ... +δ<sub>''n''</sub>}}, i.e. a chord

	<math>x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l</math>		<math>x : x + \delta_1 : \cdots : x + \sum_{l=1}^n \delta_l</math>

Inthar: /* BFGS-B (one related delta set, arbitrary free deltas) */

2025-12-17T18:22:02Z

BFGS-B (one related delta set, arbitrary free deltas)

← Older revision		Revision as of 18:22, 17 December 2025
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	=== BFGS-B (one related delta set, arbitrary free deltas) ===		=== BFGS-B (one related delta set, arbitrary free deltas) ===
	We let {{nowrap\|''x''<sub>1</sub> {{=}} ''x''}} and include additional free variables ''x''<sub>2</sub>, ..., ''x''<sub>n</sub>, one for every additional +?, after coalescing strings of consecutive +?'s into one +? and after trimming leading and trailing free delta segments.		We let {{nowrap\|''x''<sub>1</sub> {{=}} ''x''}} and include additional free variables ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>, one for every additional +?, after coalescing strings of consecutive +?'s into one +? and after trimming leading and trailing free delta segments.

	BFGS-B is a quasi-Newton optimization method (based on BFGS) particularly suited for this problem:		BFGS-B is a quasi-Newton optimization method (based on BFGS) particularly suited for this problem: