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> |