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>


== Trivially constrained tuning ==
== 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.  


As a concrete example, consider septimal meantone, whose mapping is
Denote the constraint by B<sub>C</sub>. In the case of CFE, it is the octave:  
 
<math>\displaystyle
A =
\begin{bmatrix}
1 & 0 & -4 & -13 \\
0 & 1 & 4 & 10
\end{bmatrix}
</math>
 
Denote the constraint by B<sub>C</sub>. In this case, it is the octave:  


<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 also denote the projection map by P. For septimal meantone,
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
 
<math>\displaystyle
P = \frac{1}{446}
\begin{bmatrix}
117 & 146 & 116 & -61 \\
146 & 186 & 160 & -38 \\
116 & 160 & 176 & 92 \\
-61 & -38 & 92 & 413
\end{bmatrix}
</math>
 
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 of A, denoted A<sub>M</sub>. For septimal meantone, it is
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 A_{\rm M} = \begin{bmatrix} 1 & 4 & 10\end{bmatrix}</math>
 
That forms a projection map filling the bottom-right section of P<sub>C</sub><sup>+</sup> as
 
<math>\displaystyle
A_{\rm M}^+A_{\rm M} = \frac{1}{117}
\begin{bmatrix}
1 & 4 & 10 \\
4 & 16 & 40 \\
10 & 40 & 100
\end{bmatrix}
</math>
 
In fact,


<math>\displaystyle  
<math>\displaystyle P_{\rm M} = A_{\rm M}^+ A_{\rm M} </math>
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,
forms an orthogonal projection map filling the bottom-right section of P<sub>C</sub><sup>+</sup>.
 
<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>


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>