User:Sintel/CTE tuning - Revision history

Fredg999 category edits: Categories

2023-03-10T21:07:34Z

Sintel at 23:35, 16 December 2021

2021-12-16T23:35:02Z

← Older revision		Revision as of 23:35, 16 December 2021
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	<math>		<math>
	\begin{align}		\begin{align}
	\underset{g}{\text{minimize}} & \quad \\| ~~g\mathrm{MW}~~ - ~~j\mathrm{W}~~ \\|^2 \\		\underset{g}{\text{minimize}} & \quad \\| gMW - jW \\|^2 \\
	\text{subject to} & \quad ( ~~g\mathrm{M}~~ - j )~~\mathrm{~~V} = 0 \\		\text{subject to} & \quad ( gM - j )V = 0 \\
	\end{align}		\end{align}
	</math>		</math>
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	<math>		<math>
	\begin{align}		\begin{align}
	~~\mathrm{~~A} &= (~~\mathrm{~~MW})^{\mathsf T}		A &= (MW)^{\mathsf T}
	&b &= (~~j\mathrm{W}~~)^{\mathsf T} \\		&b &= (jW)^{\mathsf T} \\
	~~\mathrm{~~C} &= (~~\mathrm{~~MV})^{\mathsf T}		C &= (MV)^{\mathsf T}
	&d &= (~~j\mathrm{V}~~)^{\mathsf T} \\		&d &= (jV)^{\mathsf T} \\
	\end{align}		\end{align}
	</math>		</math>
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	<math>		<math>
	\begin{align}		\begin{align}
	\underset{g}{\text{minimize}} & \quad \left\\| ~~\mathrm{A}g~~^{\mathsf T} - b \right\\|^2 \\		\underset{g}{\text{minimize}} & \quad \left\\| Ag^{\mathsf T} - b \right\\|^2 \\
	\text{s.t.} & \quad \phantom{\\|} ~~\mathrm{C}g~~^{\mathsf T} - d = 0 \\		\text{s.t.} & \quad \phantom{\\|} Cg^{\mathsf T} - d = 0 \\
	\end{align}		\end{align}
	</math>		</math>
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	=		=
	\begin{bmatrix}		\begin{bmatrix}
	~~\mathrm{~~A}^{\mathsf T}~~\mathrm{~~A} & ~~\mathrm{~~C}^{\mathsf T} \\		A^{\mathsf T}A & C^{\mathsf T} \\
	~~\mathrm{~~C} & 0		C & 0
	\end{bmatrix}^{-1}		\end{bmatrix}^{-1}

	\begin{bmatrix}		\begin{bmatrix}
	~~\mathrm{~~A}^{\mathsf T} b\\		A^{\mathsf T} b\\
	d		d
	\end{bmatrix}		\end{bmatrix}

Sintel at 22:58, 16 December 2021

2021-12-16T22:58:33Z

← Older revision		Revision as of 22:58, 16 December 2021
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	\begin{align}		\begin{align}
	\underset{g}{\text{minimize}} & \quad \left\\| \mathrm{A}g^{\mathsf T} - b \right\\|^2 \\		\underset{g}{\text{minimize}} & \quad \left\\| \mathrm{A}g^{\mathsf T} - b \right\\|^2 \\
	\text{s.t.} & \quad ( \mathrm{C}g^{\mathsf T} - d ) = 0 \\		\text{s.t.} & \quad \phantom{\\|} \mathrm{C}g^{\mathsf T} - d = 0 \\
	\end{align}		\end{align}
	</math>		</math>

Sintel at 22:57, 16 December 2021

2021-12-16T22:57:38Z

← Older revision		Revision as of 22:57, 16 December 2021
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	\begin{align}		\begin{align}
	\underset{g}{\text{minimize}} & \quad \\| g\mathrm{MW} - j\mathrm{W} \\|^2 \\		\underset{g}{\text{minimize}} & \quad \\| g\mathrm{MW} - j\mathrm{W} \\|^2 \\
	\text{subject to} & \quad ~~\phantom{\\|}~~ g\mathrm{MV} = j\mathrm{V} \\		\text{subject to} & \quad ( g\mathrm{M} - j )\mathrm{V} = 0 \\
	\end{align}		\end{align}
	</math>		</math>
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	\begin{align}		\begin{align}
	\underset{g}{\text{minimize}} & \quad \left\\| \mathrm{A}g^{\mathsf T} - b \right\\|^2 \\		\underset{g}{\text{minimize}} & \quad \left\\| \mathrm{A}g^{\mathsf T} - b \right\\|^2 \\
	\text{s.t.} & \quad ~~\phantom{\\|}~~ \mathrm{C}g^{\mathsf T} = d \\		\text{s.t.} & \quad ( \mathrm{C}g^{\mathsf T} - d ) = 0 \\
	\end{align}		\end{align}
	</math>		</math>
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	<math>		<math>
	~~\begin{bmatrix}~~
	~~\mathrm{A}^{\mathsf T}\mathrm{A} & \mathrm{C}^{\mathsf T} \\~~
	~~\mathrm{C} & 0~~
	~~\end{bmatrix}~~

	\begin{bmatrix}		\begin{bmatrix}
	g^{\mathsf T} \\		g^{\mathsf T} \\
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	\end{bmatrix}		\end{bmatrix}
	=		=
			\begin{bmatrix}
			\mathrm{A}^{\mathsf T}\mathrm{A} & \mathrm{C}^{\mathsf T} \\
			\mathrm{C} & 0
			\end{bmatrix}^{-1}

	\begin{bmatrix}		\begin{bmatrix}
	\mathrm{A}^{\mathsf T} b\\		\mathrm{A}^{\mathsf T} b\\

Sintel at 22:38, 16 December 2021

2021-12-16T22:38:29Z

← Older revision		Revision as of 22:38, 16 December 2021
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	</math>		</math>

	where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weight matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:		where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weight matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. This problem is feasible if rank (V) ≤ rank (M).

			== Computation ==

			Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:

	<math>		<math>
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	Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.		Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.

	~~The problem is feasible if rank (V) ≤ rank (M).~~

	~~== Computation ==~~
	As a standard optimization problem, numerous algorithms exist to solve for this tuning, such as [[Wikipedia: Sequential quadratic programming\|sequential quadratic programming]], to name one.		As a standard optimization problem, numerous algorithms exist to solve for this tuning, such as [[Wikipedia: Sequential quadratic programming\|sequential quadratic programming]], to name one.

Sintel at 22:34, 16 December 2021

2021-12-16T22:34:15Z

← Older revision		Revision as of 22:34, 16 December 2021
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	</math>		</math>

	Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with ~~legnth~~ equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.		Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.

	The problem is feasible if rank (V) ≤ rank (M).		The problem is feasible if rank (V) ≤ rank (M).

Sintel at 22:32, 16 December 2021

2021-12-16T22:32:57Z

← Older revision		Revision as of 22:32, 16 December 2021
Line 52:		Line 52:
	</math>		</math>

	Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with ~~size~~ equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.		Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with legnth equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.

	The problem is feasible if rank (V) ≤ rank (M).		The problem is feasible if rank (V) ≤ rank (M).

Sintel at 22:30, 16 December 2021

2021-12-16T22:30:05Z

← Older revision		Revision as of 22:30, 16 December 2021
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	</math>		</math>

	where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean ~~weighting~~ matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:		where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weight matrix, ''j'' is the [[JIP]], and V is a matrix obtained by stacking the monzos that we want to be pure. Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:

	<math>		<math>

Sintel at 22:27, 16 December 2021

2021-12-16T22:27:48Z

← Older revision		Revision as of 22:27, 16 December 2021
Line 8:		Line 8:
	<math>		<math>
	\begin{align}		\begin{align}
	& \underset{g}{\text{minimize}} && \~~left~~\\| g\mathrm{MW} - j\mathrm{W} ~~\right~~\\|^2 \\		\underset{g}{\text{minimize}} & \quad \\| g\mathrm{MW} - j\mathrm{W} \\|^2 \\
	& \text{subject to} && g\mathrm{MV} = j\mathrm{V} \\		\text{subject to} & \quad \phantom{\\|} g\mathrm{MV} = j\mathrm{V} \\
	\end{align}		\end{align}
	</math>		</math>
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	<math>		<math>
	\begin{align}		\begin{align}
	& \underset{g}{\text{minimize}} && \left\\| \mathrm{A}g^{\mathsf T} - b \right\\|^2 \\		\underset{g}{\text{minimize}} & \quad \left\\| \mathrm{A}g^{\mathsf T} - b \right\\|^2 \\
	& \text{~~subject to~~} && \mathrm{C}g^{\mathsf T} = d \\		\text{s.t.} & \quad \phantom{\\|} \mathrm{C}g^{\mathsf T} = d \\
	\end{align}		\end{align}
	</math>		</math>
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	</math>		</math>

	Where we introduced the vector of lagrange multipliers ~~<nowiki>~~<math>\lambda</math~~></nowiki~~>, with size equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.		Where we introduced the vector of lagrange multipliers <math>\lambda</math>, with size equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.

	The problem is feasible if rank (V) ≤ rank (M).		The problem is feasible if rank (V) ≤ rank (M).

Sintel at 22:19, 16 December 2021

2021-12-16T22:19:19Z

← Older revision		Revision as of 22:19, 16 December 2021
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	\end{bmatrix}		\end{bmatrix}
	</math>		</math>

			Where we introduced the vector of lagrange multipliers <nowiki><math>\lambda</math></nowiki>, with size equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.

	The problem is feasible if rank (V) ≤ rank (M).		The problem is feasible if rank (V) ≤ rank (M).

← Older revision		Revision as of 21:07, 10 March 2023
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	It can be speculated that POTE tends to result in biased tunings whereas CTE less so.		It can be speculated that POTE tends to result in biased tunings whereas CTE less so.

	[[Category:Regular temperament ~~theory~~]]		[[Category:Regular temperament tuning]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]
	~~[[Category:Tuning]]~~
	~~[[Category:Tuning technique]]~~