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}
-& \text{minimize}   && \left\|   g\mathrm{V} - \mathrm{J}   \right\|   \\[8pt]
+\underset{g}{\text{minimize}}   & \quad \|   gMW - jW   \|^2   \\
-& \text{subject to} &&  (g\mathrm{A} - \mathrm{J}_0)b_i = 0, \quad i = 1, \dots, r
+\text{subject to} & \quad (  gM - j )V  = 0 \\
 \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 B the monzo list.
+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).
-The problem is feasible if
+== Computation ==
-# rank (B) ≤ rank (A), and
-# Each column in B and N (A) are [[Wikipedia:linear independence|linearly independent]].
+Since this is a convex problem, it can be solved using the method of lagrange multipliers. Let's first simplify:
+<math>
+\begin{align}
+A &= (MW)^{\mathsf T}
+&b &= (jW)^{\mathsf T} \\
+C &= (MV)^{\mathsf T}
+&d &= (jV)^{\mathsf T} \\
+\end{align}
+</math>
+The problem then becomes:
+<math>
+\begin{align}
+\underset{g}{\text{minimize}}   & \quad  \left\|   Ag^{\mathsf T} - b   \right\|^2   \\
+\text{s.t.} & \quad \phantom{\|} Cg^{\mathsf T} - d  = 0 \\
+\end{align}
+</math>
+The solution can be found by solving the dual problem:
+<math>
+\begin{bmatrix}
+g^{\mathsf T}  \\
+\lambda^{\mathsf T}
+\end{bmatrix}
+=
+\begin{bmatrix}
+A^{\mathsf T}A & C^{\mathsf T} \\
+C & 0
+\end{bmatrix}^{-1}
+\begin{bmatrix}
+A^{\mathsf T} b\\
+d
+\end{bmatrix}
+</math>
+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.
-== 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.
@@ Line 96: / Line 134: @@
 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:Math]]
-[[Category:Tuning]]
-[[Category:Tuning technique]]