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

Polishing
Line 1: Line 1:
== 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 primes in terms of monzos. The tuning map in the logarithmic scale can be obtained by multiplying the projection map by the JIP on the left.  
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>
Line 8: Line 8:
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''<sub>2</sub> tunings, the weighted projection map is
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>
Line 31: Line 31:
<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
Line 48: Line 48:
<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  
Line 140: Line 140:
<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