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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<math>\displaystyle PM = O</math> | <math>\displaystyle PM = O</math> | ||
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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but it works as long as it is the first ''r'' elements of the [[subgroup basis matrix|subgroup basis]]. | but it works as long as it 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> | ||
Notice | |||
<math>\displaystyle | <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 | That makes the pseudoinverse of ''P''<sub>C</sub> easier to work with than ''P''<sub>C</sub> itself, as | ||
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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_{\rm F} M_I | ||
</math> | </math> | ||
Both ''P''<sub>C</sub><sup>+</sup>''M''<sub>''I''</sub> and '' | 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>. | ||
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>. 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> | ||
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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>. | ||
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 | ||