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

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Clarify on some issues
→CEE tuning: I don't think these are ever gonna be solved, so I'm gonna discuss them as plain observations
 
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where ''T'' is the tempered tuning map, ''J'' the just tuning map, and ''P'' the projection map.  
where ''T'' is the tempered tuning map, ''J'' the just tuning map, and ''P'' the projection map.  


The projection map multipled by a [[temperament mapping matrix]] on the left yields its [[tmonzos and tvals|tempered monzos]]. In particular, if ''V'' is the temperament mapping matrix of ''P'', then
The projection map multipled by a [[temperament mapping matrix]] on the left yields its [[tempered monzos and vals|tempered monzos]]. In particular, if ''V'' is the temperament mapping matrix of ''P'', then


<math>\displaystyle VP = V</math>
<math>\displaystyle VP = V</math>
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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>


but it works as long as it is the first ''r'' elements of the [[subgroup basis matrix|subgroup basis]].  
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
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<math>\displaystyle P_{\rm C} M_I = M_I</math>
<math>\displaystyle P_{\rm C} M_I = M_I</math>


Notice
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>


That makes the pseudoinverse of ''P''<sub>C</sub> easier to work with than ''P''<sub>C</sub> itself, as
and


<math>\displaystyle  
<math>\displaystyle  
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</math>
</math>


Both ''P''<sub>C</sub><sup>+</sup>''M''<sub>''I''</sub> and ''P''<sub>F</sub>''M''<sub>''I''</sub> are the same slice of the first ''r'' columns of ''P''<sub>F</sub>.
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>. The minor matrix of the projection map
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>
</math>


In fact,
The pseudoinverse of the CEE projection map can be composed as


<math>\displaystyle  
<math>\displaystyle  
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\end{align}
\end{align}
</math>
</math>
== Notes ==
<references group="note"/>


[[Category:Math]]
[[Category:Math]]
[[Category:Pages with open problems]]
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