Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions
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<math>\displaystyle P = WV^+VW^{-1} = W(AW)^+A</math> | <math>\displaystyle P = WV^+VW^{-1} = W(AW)^+A</math> | ||
== | == CFE tuning == | ||
Let us start with CFE tuning (<u>c</u>onstrained <u>F</u>rob<u>e</u>nius tuning) since its weighter is the identity matrix and the constraint is simply the octave. | Let us start with CFE tuning (<u>c</u>onstrained <u>F</u>rob<u>e</u>nius tuning) since its weighter is the identity matrix and the constraint is simply the octave. | ||
Denote the constraint by B<sub>C</sub>. In the case of CFE, it is the octave: | |||
Denote the constraint by B<sub>C</sub>. In | |||
<math>\displaystyle B_{\rm C} = [ \begin{matrix} 1 & 0 & 0 & 0 \end{matrix} \rangle</math> | <math>\displaystyle B_{\rm C} = [ \begin{matrix} 1 & 0 & 0 & 0 \end{matrix} \rangle</math> | ||
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but it works as long as it is the first ''r'' elements of the subgroup basis. | but it works as long as it is the first ''r'' elements of the subgroup basis. | ||
We will | We will denote the projection map by P. The goal is to work out the constrained projection map P<sub>C</sub>, which also satisfies | ||
The goal is to work out the constrained projection map P<sub>C</sub>, which also satisfies | |||
<math>\displaystyle | <math>\displaystyle | ||
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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 | 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 | ||
<math>\displaystyle | <math>\displaystyle P_{\rm M} = A_{\rm M}^+ A_{\rm M} </math> | ||
P_{\rm | |||
</math> | |||
forms an orthogonal projection map filling the bottom-right section of P<sub>C</sub><sup>+</sup>. | |||
< | |||
</ | |||
In general, if B<sub>C</sub> is the first ''r'' elements of the subgroup basis, then P<sub>C</sub> is of the form | In general, if B<sub>C</sub> is the first ''r'' elements of the subgroup basis, then P<sub>C</sub> is of the form | ||
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</math> | </math> | ||
If there is a weight, such as CTE tuning, the weight should be applied to the map and the constraint: | == Otherwise normed tuning == | ||
If there is a weight, such as CTE tuning, the weight should be applied to the map and the constraint first: | |||
<math>\displaystyle | <math>\displaystyle | ||
V = AW \\ | \begin{align} | ||
M_{\rm C} = W^{-1}B_{\rm C} | V &= AW \\ | ||
M_{\rm C} &= W^{-1}B_{\rm C} | |||
\end{align} | |||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
A_S = AS \\ | \begin{align} | ||
(B_{\rm C})_S = S^{-1}B_{\rm C} | A_S &= AS \\ | ||
(B_{\rm C})_S &= S^{-1}B_{\rm C} | |||
\end{align} | |||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
V = AW \\ | \begin{align} | ||
M_{\rm C} = W^{-1}B_{\rm C} | V &= AW \\ | ||
M_{\rm C} &= W^{-1}B_{\rm C} | |||
\end{align} | |||
</math> | </math> | ||
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<math>\displaystyle | <math>\displaystyle | ||
V_S = VS_W \\ | \begin{align} | ||
(M_{\rm C})_S = S_W^{-1}M_{\rm C} | V_S &= VS_W \\ | ||
(M_{\rm C})_S &= S_W^{-1}M_{\rm C} | |||
\end{align} | |||
</math> | </math> | ||
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<math>\displaystyle P_{\rm C} = WS (P_{\rm C})_{WS} S^{-1} W^{-1}</math> | <math>\displaystyle P_{\rm C} = WS (P_{\rm C})_{WS} S^{-1} W^{-1}</math> | ||
== Example == | |||
Let us try tuning septimal meantone to CFE. | |||
Its mapping is | |||
<math>\displaystyle | |||
A = | |||
\begin{bmatrix} | |||
1 & 0 & -4 & -13 \\ | |||
0 & 1 & 4 & 10 | |||
\end{bmatrix} | |||
</math> | |||
The projection map is | |||
<math>\displaystyle | |||
\begin{align} | |||
P &= A^+A \\ | |||
&= \frac{1}{446} | |||
\begin{bmatrix} | |||
117 & 146 & 116 & -61 \\ | |||
146 & 186 & 160 & -38 \\ | |||
116 & 160 & 176 & 92 \\ | |||
-61 & -38 & 92 & 413 | |||
\end{bmatrix} | |||
\end{align} | |||
</math> | |||
The minor of the mapping is | |||
<math>\displaystyle A_{\rm M} = \begin{bmatrix} 1 & 4 & 10\end{bmatrix}</math> | |||
and the minor projection map is | |||
<math>\displaystyle | |||
\begin{align} | |||
P_{\rm M} &= A_{\rm M}^+A_{\rm M} \\ | |||
&= \frac{1}{117} | |||
\begin{bmatrix} | |||
1 & 4 & 10 \\ | |||
4 & 16 & 40 \\ | |||
10 & 40 & 100 | |||
\end{bmatrix} | |||
\end{align} | |||
</math> | |||
In fact, | |||
<math>\displaystyle | |||
P_{\rm C}^+ = | |||
\begin{bmatrix} | |||
117/446 & 0 & 0 & 0 \\ | |||
146/446 & 1/117 & 4/117 & 10/117 \\ | |||
116/446 & 4/117 & 16/117 & 40/117 \\ | |||
-61/446 & 10/117 & 40/117 & 100/117 | |||
\end{bmatrix} | |||
</math> | |||
Hence, | |||
<math>\displaystyle | |||
P_{\rm C} = \frac{1}{117} | |||
\begin{bmatrix} | |||
117 & 146 & 116 & -61 \\ | |||
0 & 1 & 4 & 10 \\ | |||
0 & 4 & 16 & 40 \\ | |||
0 & 10 & 40 & 100 | |||
\end{bmatrix} | |||
</math> |