Constrained tuning: Difference between revisions

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== Definition ==

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:

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. If the tuning is contrained by the eigenmonzo list B, the CTE tuning is equivalent to the following optimization problem:

Minimize

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

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

The problem is feasible if

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== Computation ==

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

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

<math>

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\begin{bmatrix}

VJ^{\mathsf T}\\

J_0 B

(J_0 B)^{\mathsf T}

\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~~.

which is almost an analytical solution. 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 discarded.

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

@@ Line 4: / Line 4: @@
 == Definition ==
-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:
+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. If the tuning is contrained by the eigenmonzo list B, 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, B the monzo list, and O the zero matrix.
+where G is the generator list, and O the zero matrix.
 The problem is feasible if
@@ Line 21: / Line 21: @@
 == Computation ==
-The tuning can be solved in the [[wikipedia: Lagrange multiplier|method of Lagrange multiplier]]. The solution is almost analytically given by
+The tuning can be solved in the [[wikipedia: Lagrange multiplier|method of Lagrange multiplier]]. The solution is given by
 <math>
@@ Line 36: / Line 36: @@
 \begin{bmatrix}
 VJ^{\mathsf T}\\
-J_0 B
+(J_0 B)^{\mathsf T}
 \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.
+which is almost an analytical solution. 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 discarded.
 Otherwise, as a standard optimization problem, numerous algorithms exist to solve it, such as [[Wikipedia: Sequential quadratic programming|sequential quadratic programming]], to name one.