Constrained tuning: Difference between revisions

Line 4:

== Definition ==

Given a temperament [[mapping]] A, the CTE tuning is equivalent to the following optimization problem:

Given a temperament [[mapping]] A and the [[JIP]] J0, denote the Tenney-weighted temperament mapping by V = AW, and the Tenney-weighted JIP by J = J0W. The CTE tuning is equivalent to the following optimization problem:

Minimize

Line 14:

where G is the generator list, ~~V = AW~~ the ~~Tenney-weighted temperament mapping, J = J0W the Tenney-weighted [[JIP]]~~, and B the ~~monzo list~~.

where G is the generator list, B the monzo list, and O the zero matrix.

The problem is feasible if

Line 21:

== 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.

The tuning can be solved in the [[wikipedia: Lagrange multiplier|method of Lagrange multiplier]]. The solution is almost analytically given by

<math>

\begin{bmatrix}

G^{\mathsf T} \\

\Lambda^{\mathsf T}

\end{bmatrix}

=

\begin{bmatrix}

VV^{\mathsf T} & AB \\

(AB)^{\mathsf T} & O

\end{bmatrix}^{-1}

\begin{bmatrix}

VJ^{\mathsf T}\\

J_0 B

\end{bmatrix}

</math>

Notice we introduced the vector of lagrange multipliers Λ, with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.

Otherwise, as a standard optimization problem, numerous algorithms exist to solve it, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one.

The following [https://www.python.org Python] code is an excerpt from [[Flora Canou]]'s [https://github.com/FloraCanou/te_temperament_measures/blob/54c9ec58acf0ad5adc7c1d96f2deaf1d2a503702/tuning_optimizer.py tuning optimizer]. Note: it depends on [https://scipy.org/ Scipy].

@@ Line 4: / Line 4: @@
 == Definition ==
-Given a temperament [[mapping]] A, the CTE tuning is equivalent to the following optimization problem:
+Given a temperament [[mapping]] A and the [[JIP]] J<sub>0</sub>, denote the Tenney-weighted temperament mapping by V = AW, and the Tenney-weighted JIP by J = J<sub>0</sub>W. The CTE tuning is equivalent to the following optimization problem:
 Minimize
@@ Line 14: / Line 14: @@
 <math>(GA - J_0)B = O</math>
-where G is the generator list, V = AW the Tenney-weighted temperament mapping, J = J<sub>0</sub>W the Tenney-weighted [[JIP]], and B the monzo list.
+where G is the generator list, B the monzo list, and O the zero matrix.
 The problem is feasible if
@@ Line 21: / Line 21: @@
 == 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.
+The tuning can be solved in the [[wikipedia: Lagrange multiplier|method of Lagrange multiplier]]. The solution is almost analytically given by
+<math>
+\begin{bmatrix}
+G^{\mathsf T}  \\
+\Lambda^{\mathsf T}
+\end{bmatrix}
+=
+\begin{bmatrix}
+VV^{\mathsf T} & AB \\
+(AB)^{\mathsf T} & O
+\end{bmatrix}^{-1}
+\begin{bmatrix}
+VJ^{\mathsf T}\\
+J_0 B
+\end{bmatrix}
+</math>
+Notice we introduced the vector of lagrange multipliers Λ, with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be ignored.
+Otherwise, as a standard optimization problem, numerous algorithms exist to solve it, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one.
 The following [https://www.python.org Python] code is an excerpt from [[Flora Canou]]'s [https://github.com/FloraCanou/te_temperament_measures/blob/54c9ec58acf0ad5adc7c1d96f2deaf1d2a503702/tuning_optimizer.py tuning optimizer]. Note: it depends on [https://scipy.org/ Scipy].