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

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This article gives an analytical form of Euclidean-normed [[constrained tuning]]s.  
This article gives an analytical form of Euclidean-normed [[constrained tuning]]s.  


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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 PM = O</math>
<math>\displaystyle PM = O</math>


For any Euclidean aka ''L''<sup>2</sup> tuning without constraints, the weight–skew transformed projection map is
The [[Frobenius tuning|Frobenius projection map]] can be used to characterize a temperament. If ''V'' is the temperament mapping matrix, then its Frobenius projection map ''P''<sub>F</sub> is
 
<math>\displaystyle P_{\rm F} = V^+ V</math>
 
We can generalize that to any other Euclidean a.k.a. ''L''<sup>2</sup> tuning without constraints, so that the weight–skew transformed projection map ''P''<sub>''X''</sub> is


<math>\displaystyle P_X = V_X^+ V_X</math>
<math>\displaystyle P_X = V_X^+ V_X</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 projection map by ''P''. The goal is to work out the constrained projection map ''P''<sub>C</sub>, which 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
VP_{\rm C} = V \\
\begin{align}
P_{\rm C}M = O
VP_{\rm C} &= V \\
P_{\rm C}M &= O
\end{align}
</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>


Since ''P'' is characteristic of the temperament and is independent of the specific tuning, 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 = 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  
P_{\rm C}^+ M_I
P_{\rm C}^+ M_I
= P_{\rm C}^+P_{\rm C} M_I
= P_{\rm C}^+P_{\rm C} M_I
= P M_I
= P_{\rm F} M_I
</math>
</math>


Both ''P''<sub>C</sub><sup>+</sup>''M''<sub>''I''</sub> and ''PM''<sub>''I''</sub> are the same slice of the first ''r'' columns of ''P''.
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 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.  


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


<math>\displaystyle  
<math>\displaystyle  
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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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