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

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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 P_{\rm C} M_I = M_I</math>
<math>\displaystyle P_{\rm C} M_I = M_I</math>


Notice
Notice<ref group="note">It is not known yet why this is the case.</ref>


<math>\displaystyle P_{\rm F} = P_{\rm C}^+P_{\rm C}</math>
<math>\displaystyle P_{\rm F} = P_{\rm C}^+P_{\rm C}</math>
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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>. The minor matrix of the projection map
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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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