User:Sintel/CTE tuning: Difference between revisions

@@ Line 8: / Line 8: @@
 <math>
 \begin{align}
-& \underset{g}{\text{minimize}}   && \left\|   g\mathrm{V} - j   \right\|^2   \\
+& \underset{g}{\text{minimize}}   && \left\|   g\mathrm{AW} - j\mathrm{W}   \right\|^2   \\
-& \text{subject to} &&  (g\mathrm{A} - j_0)b_i = j_0, \quad i = 1, \dots, m \\
+& \text{subject to} &&  g\mathrm{AB}  = j\mathrm{B} \\
 \end{align}
 </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 <math>b_i</math> is the ''i''-th monzo. If we stack all the ''b''s into a matrix B, then the problem can be solved using he method of lagrange multipliers.
+where ''g'' is the generator list, V = AW the Tenney-weighted temperament mapping, ''j'' is the [[JIP]], and <math>\mathrm{B}</math> is a matrix obtained by stacking the monzos that we need to be just. This problem can be solved using the method of lagrange multipliers:
+<math>
+\begin{bmatrix}
+\mathrm{AW^2A}^{\mathsf T} & \mathrm{AB} \\
+\mathrm{(AB)}^{\mathsf T} & 0
+\end{bmatrix}
+\begin{bmatrix}
+g^{\mathsf T}  \\
+\lambda^{\mathsf T}
+\end{bmatrix}
+=
+\begin{bmatrix}
+\mathrm{AW}^2j^{\mathsf T} \\
+\mathrm{B}^{\mathsf T}j^{\mathsf T}
+\end{bmatrix}
+</math>
 The problem is feasible if