Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions

Polishing
+intro; correct wording (the "minor" is actually the minor matrix, not its determinant); +final tuning map of the example
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This article gives an analytical form of Euclidean-normed [[constrained tuning]]s.
== Preliminaries ==
== Preliminaries ==
The [[projection map]] is useful in a lot of ways. We will work extensively with the projection map in the course of solving constrained tunings.  
The [[projection map]] is useful in a lot of ways. We will work extensively with the projection map in the course of solving constrained tunings.  
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Both P<sub>C</sub><sup>+</sup>B<sub>C</sub> and PB<sub>C</sub> are the same slice of the first ''r'' columns of P.
Both P<sub>C</sub><sup>+</sup>B<sub>C</sub> and PB<sub>C</sub> are the same slice of the first ''r'' columns of P.


With the first ''r'' rows and columns removed, the remaining part in the mapping will be dubbed the minor, denoted A<sub>M</sub>. The minor of the projection map
With the first ''r'' rows and columns removed, the remaining part in the mapping will be dubbed the minor matrix, denoted A<sub>M</sub>. The minor matrix of the projection map


<math>\displaystyle P_{\rm M} = A_{\rm M}^+ A_{\rm M} </math>
<math>\displaystyle P_{\rm M} = A_{\rm M}^+ A_{\rm M} </math>
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</math>
</math>


The minor of the mapping is
The minor matrix of the mapping is


<math>\displaystyle A_{\rm M} = \begin{bmatrix} 1 & 4 & 10\end{bmatrix}</math>
<math>\displaystyle A_{\rm M} = \begin{bmatrix} 1 & 4 & 10\end{bmatrix}</math>


and the minor projection map is
and the minor matrix of the projection map is


<math>\displaystyle  
<math>\displaystyle  
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0 & 10 & 40 & 100
0 & 10 & 40 & 100
\end{bmatrix}
\end{bmatrix}
</math>
The tuning map T<sub>C</sub> is
<math>\displaystyle
\begin{align}
T_{\rm C} &= JP_{\rm C} \\
&= \langle \begin{matrix} 1200 & 1896.8843 & 2787.5374 & 3368.8435 \end{matrix} ]
\end{align}
</math>
</math>