Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions
More explanations and mark the open problem |
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<math>\displaystyle M_I = [ \begin{matrix} 1 & 0 & \ldots & 0 \end{matrix} \rangle</math> | <math>\displaystyle M_I = [ \begin{matrix} 1 & 0 & \ldots & 0 \end{matrix} \rangle</math> | ||
The following observations work as long as the constraint is the first ''r'' elements of the [[subgroup basis matrix|subgroup basis]]. | |||
We will denote the Frobenius projection map by ''P''<sub>F</sub>. The goal is to work out the constrained projection map ''P''<sub>C</sub>, which, like ''P''<sub>F</sub>, also satisfies | We will denote the Frobenius projection map by ''P''<sub>F</sub>. The goal is to work out the constrained projection map ''P''<sub>C</sub>, which, like ''P''<sub>F</sub>, also satisfies | ||
<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
VP_{\rm C} &= V \\ | VP_{\rm C} &= V \\ | ||
P_{\rm C}M &= O | P_{\rm C}M &= O | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math>\displaystyle P_{\rm C} M_I = M_I</math> | <math>\displaystyle P_{\rm C} M_I = M_I</math> | ||
For an arbitrary projection map ''P'' of the same temperament, notice | |||
<math>\displaystyle P_{\rm F} = P^+ P</math> | |||
so if we substitute ''P''<sub>C</sub> for ''P'', we have | |||
<math>\displaystyle P_{\rm F} = P_{\rm C}^+P_{\rm C}</math> | <math>\displaystyle P_{\rm F} = P_{\rm C}^+P_{\rm C}</math> | ||
and | |||
<math>\displaystyle | <math>\displaystyle | ||
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Since ''PM''<sub>''I''</sub> is the tuning of ''M''<sub>''I''</sub> in terms of monzos, which is just the slice of the first ''r'' columns of ''P'' in this case, it follows that {{subsup|''P''|C|+}} and ''P''<sub>F</sub> share the first ''r'' columns. | Since ''PM''<sub>''I''</sub> is the tuning of ''M''<sub>''I''</sub> in terms of monzos, which is just the slice of the first ''r'' columns of ''P'' in this case, it follows that {{subsup|''P''|C|+}} and ''P''<sub>F</sub> share the first ''r'' columns. | ||
With the first ''r'' rows and columns removed, the remaining part in the mapping is another invariant of the temperament, which will be dubbed the minor matrix, denoted ''V''<sub>M</sub>. | With the first ''r'' rows and columns removed, the remaining part in the mapping is another invariant of the temperament, which will be dubbed the minor matrix, denoted ''V''<sub>M</sub>. We observe that the minor matrix of the projection map | ||
<math>\displaystyle P_{\rm M} = V_{\rm M}^+ V_{\rm M} </math> | <math>\displaystyle P_{\rm M} = V_{\rm M}^+ V_{\rm M} </math> | ||
forms an orthogonal projection map filling the bottom-right section of ''P''<sub>C</sub><sup>+</sup>. | forms an orthogonal projection map filling the bottom-right section of ''P''<sub>C</sub><sup>+</sup>, and the top-right section comprises only zeros. | ||
Therefore, in general, if ''M''<sub>''I''</sub> is the first ''r'' elements of the subgroup basis, then ''P''<sub>C</sub> is of the form | Therefore, in general, if ''M''<sub>''I''</sub> is the first ''r'' elements of the subgroup basis, then ''P''<sub>C</sub> is of the form | ||
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<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
V_X &= VX \\ | V_X &= VX \\ | ||
(M_I)_X &= X^+ M_I | (M_I)_X &= X^+ M_I | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
V_S &= VS \\ | V_S &= VS \\ | ||
(M_I)_S &= S^{-1}M_I | (M_I)_S &= S^{-1}M_I | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
V_X &= VX \\ | V_X &= VX \\ | ||
(M_I)_X &= X^+ M_I | (M_I)_X &= X^+ M_I | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
V_{XS} &= VXS \\ | V_{XS} &= VXS \\ | ||
(M_I)_{XS} &= (XS)^+ M_I | (M_I)_{XS} &= (XS)^+ M_I | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
P &= V^+V \\ | P &= V^+V \\ | ||
&= \frac{1}{446} | &= \frac{1}{446} | ||
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-61 & -38 & 92 & 413 | -61 & -38 & 92 & 413 | ||
\end{bmatrix} | \end{bmatrix} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
P_{\rm M} &= V_{\rm M}^+V_{\rm M} \\ | P_{\rm M} &= V_{\rm M}^+V_{\rm M} \\ | ||
&= \frac{1}{117} | &= \frac{1}{117} | ||
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10 & 40 & 100 | 10 & 40 & 100 | ||
\end{bmatrix} | \end{bmatrix} | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
\begin{ | \begin{aligned} | ||
T_{\rm C} &= JP_{\rm C} \\ | T_{\rm C} &= JP_{\rm C} \\ | ||
&= \langle \begin{matrix} 1200 & 1896.8843 & 2787.5374 & 3368.8435 \end{matrix} ] | &= \langle \begin{matrix} 1200 & 1896.8843 & 2787.5374 & 3368.8435 \end{matrix} ] | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||