Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions
m FloraC moved page Analytical solution to constrained Euclidean tunings to Constrained tuning/Analytical solution to constrained Euclidean tunings without leaving a redirect: Highly related articles |
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== Preliminaries == | == Preliminaries == | ||
The projection map is useful in a lot of ways. We will work extensively with the projection map in the course of solving constrained tunings. | The [[projection map]] is useful in a lot of ways. We will work extensively with the projection map in the course of solving constrained tunings. | ||
First, it manifests itself as a form of tuning map. Its columns represent tunings of formal | First, it manifests itself as a form of [[tuning map]]. Its columns represent tunings of [[formal prime]]s in terms of [[monzo]]s. The tuning map in the logarithmic scale can be obtained by multiplying the projection map by the [[JIP]] on the left. | ||
<math>\displaystyle T = JP</math> | <math>\displaystyle T = JP</math> | ||
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where T is the tuning map, J the JIP, and P the projection map. | where T is the tuning map, J the JIP, and P the projection map. | ||
The projection map multipled by a temperament map on the left yields its tempered monzos. In particular, if A is the temperament map of P, then | The projection map multipled by a [[Temperament mapping matrices|temperament map]] on the left yields its [[Tmonzos and tvals|tempered monzos]]. In particular, if A is the temperament map of P, then | ||
<math>\displaystyle AP = A</math> | <math>\displaystyle AP = A</math> | ||
Second, the projection map multipled by a monzo list on the right yields the tunings of the list in terms of monzos. In particular, if B is the comma list of P, then | Second, the projection map multipled by a monzo list on the right yields the tunings of the list in terms of monzos. In particular, if B is the [[comma list]] of P, then | ||
<math>\displaystyle PB = O</math> | <math>\displaystyle PB = O</math> | ||
For any Euclidean aka ''L''< | For any Euclidean aka ''L''<sup>2</sup> tunings, the weighted projection map is | ||
<math>\displaystyle P_W = V^+V</math> | <math>\displaystyle P_W = V^+V</math> | ||
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<math>\displaystyle B_{\rm C} = [ \begin{matrix} 1 & 0 & \ldots & 0 \end{matrix} \rangle</math> | <math>\displaystyle B_{\rm C} = [ \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. | but it works as long as it is the first ''r'' elements of the [[Subgroup basis matrices|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 projection map by P. The goal is to work out the constrained projection map P<sub>C</sub>, which also satisfies | ||
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<math>\displaystyle P = P_{\rm C}^+P_{\rm C}</math> | <math>\displaystyle P = P_{\rm C}^+P_{\rm C}</math> | ||
where <sup>+</sup> is the pseudoinverse. That makes the pseudoinverse of P<sub>C</sub> easier to work with than P<sub>C</sub> itself, as | where <sup>+</sup> is the [[Wikipedia:Moore–Penrose inverse|pseudoinverse]]. That makes the pseudoinverse of P<sub>C</sub> easier to work with than P<sub>C</sub> itself, as | ||
<math>\displaystyle | <math>\displaystyle | ||
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<math>\displaystyle P_{\rm C} = S (P_{\rm C})_S S^{-1}</math> | <math>\displaystyle P_{\rm C} = S (P_{\rm C})_S S^{-1}</math> | ||
Similarly, if there is a weight and/or a skew X, it should be applied to the map and the constraint first: | Similarly, if there is a weight W and/or a skew X, it should be applied to the map and the constraint first: | ||
<math>\displaystyle | <math>\displaystyle | ||