User:Sintel/CTE tuning: Difference between revisions
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<math> | <math> | ||
\begin{align} | \begin{align} | ||
\underset{g}{\text{minimize}} & \quad \| | \underset{g}{\text{minimize}} & \quad \| gMW - jW \|^2 \\ | ||
\text{subject to} & \quad | \text{subject to} & \quad ( gM - j )V = 0 \\ | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
| Line 21: | Line 21: | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
A &= (MW)^{\mathsf T} | |||
&b &= ( | &b &= (jW)^{\mathsf T} \\ | ||
C &= (MV)^{\mathsf T} | |||
&d &= ( | &d &= (jV)^{\mathsf T} \\ | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
| Line 32: | Line 32: | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
\underset{g}{\text{minimize}} & \quad \left\| | \underset{g}{\text{minimize}} & \quad \left\| Ag^{\mathsf T} - b \right\|^2 \\ | ||
\text{s.t.} & \quad \phantom{\|} | \text{s.t.} & \quad \phantom{\|} Cg^{\mathsf T} - d = 0 \\ | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
| Line 40: | Line 40: | ||
<math> | <math> | ||
\begin{bmatrix} | \begin{bmatrix} | ||
g^{\mathsf T} \\ | g^{\mathsf T} \\ | ||
| Line 51: | Line 46: | ||
= | = | ||
\begin{bmatrix} | \begin{bmatrix} | ||
\ | A^{\mathsf T}A & C^{\mathsf T} \\ | ||
C & 0 | |||
\end{bmatrix}^{-1} | |||
\begin{bmatrix} | |||
A^{\mathsf T} b\\ | |||
d | d | ||
\end{bmatrix} | \end{bmatrix} | ||
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It can be speculated that POTE tends to result in biased tunings whereas CTE less so. | It can be speculated that POTE tends to result in biased tunings whereas CTE less so. | ||
[[Category:Regular temperament | [[Category:Regular temperament tuning]] | ||
[[Category:Terms]] | [[Category:Terms]] | ||
[[Category:Math]] | [[Category:Math]] | ||