User:Sintel/CTE tuning: Difference between revisions

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 == Definition ==
-Given a temperament [[mapping]] A, the CTE tuning is equivalent to the following optimization problem:
+Given a temperament [[mapping]] M, the CTE tuning is equivalent to the following optimization problem:
 <math>
 \begin{align}
-& \underset{g}{\text{minimize}}   && \left\|   g\mathrm{AW} - j\mathrm{W}   \right\|^2   \\
+& \underset{g}{\text{minimize}}   && \left\|   g\mathrm{MW} - j\mathrm{W}   \right\|^2   \\
-& \text{subject to} &&  g\mathrm{AB}  = j\mathrm{B} \\
+& \text{subject to} &&  g\mathrm{MV}  = j\mathrm{V} \\
 \end{align}
 </math>
-where ''g'' is the (unknown) generator list, W the diagonal Tenney-Euclidean weighting matrix, ''j'' is the [[JIP]], and B is a matrix obtained by stacking the monzos that we want to be pure. This problem can be solved using the method of lagrange multipliers:
+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:
 <math>
-\begin{bmatrix}
+\begin{align}
-\mathrm{AW^2A}^{\mathsf T} & \mathrm{AB} \\
+\mathrm{A} &= (\mathrm{MW})^{\mathsf T}
-\mathrm{(AB)}^{\mathsf T} & 0
+&b &= (j\mathrm{W})^{\mathsf T} \\
-\end{bmatrix}
+\mathrm{C} &= (\mathrm{MV})^{\mathsf T}
+&d &= (j\mathrm{V})^{\mathsf T} \\
+\end{align}
+</math>
+The problem then becomes:
-\begin{bmatrix}
+<math>
-g^{\mathsf T}  \\
+\begin{align}
-\lambda^{\mathsf T}
+& \underset{g}{\text{minimize}}   && \left\|   \mathrm{A}g^{\mathsf T} - b   \right\|^2   \\
-\end{bmatrix}
+& \text{subject to} &&  \mathrm{C}g^{\mathsf T} = d \\
-=
+\end{align}
-\begin{bmatrix}
-\mathrm{AW}^2j^{\mathsf T} \\
-\mathrm{B}^{\mathsf T}j^{\mathsf T}
-\end{bmatrix}
 </math>
-or equivalently:
+The solution can be found by solving the dual problem:
 <math>
 \begin{bmatrix}
-g & \lambda
+\mathrm{A}^{\mathsf T}\mathrm{A} & \mathrm{C}^{\mathsf T} \\
-\end{bmatrix}
+\mathrm{C} & 0
+\end{bmatrix}
 \begin{bmatrix}
-\mathrm{A}^{\mathsf T}\mathrm{W^2A} & \mathrm{(AB)}^{\mathsf T} \\
+g^{\mathsf T}  \\
-\mathrm{AB} & 0
+\lambda^{\mathsf T}
 \end{bmatrix}
 =
 \begin{bmatrix}
-j\mathrm{W}^2 \mathrm{A}^{\mathsf T} &
+\mathrm{A}^{\mathsf T} b\\
-j\mathrm{B}
+d
 \end{bmatrix}
 </math>
-The problem is feasible if
+The problem is feasible if rank (V) ≤ rank (M).
-# rank (B) ≤ rank (A), and
-# A has full rank.
 == Computation ==