Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions
Replace weight W and skew X with a single transformation X to improve readability |
m Adopt "just tuning map" |
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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 prime]]s in terms of [[monzo]]s. The tuning map in the logarithmic scale can be obtained by multiplying the projection map by the [[ | First, it manifests itself as a form of [[tuning map]]. Its columns represent tunings of [[formal prime]]s in terms of [[monzo]]s. The tempered tuning map in the logarithmic scale can be obtained by multiplying the projection map by the [[just tuning map]] on the left. | ||
<math>\displaystyle T = JP</math> | <math>\displaystyle T = JP</math> | ||
where T is the tuning map, J the | 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 matrices|temperament map]] on the left yields its [[Tmonzos and tvals|tempered monzos]]. In particular, if V 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 V is the temperament map of P, then | ||