Constrained tuning/Analytical solution to constrained Euclidean tunings: Difference between revisions
+intro; correct wording (the "minor" is actually the minor matrix, not its determinant); +final tuning map of the example |
m Frobenius -> equilateral Euclidean |
||
| Line 26: | Line 26: | ||
<math>\displaystyle P = WV^+VW^{-1} = W(AW)^+A</math> | <math>\displaystyle P = WV^+VW^{-1} = W(AW)^+A</math> | ||
== | == CEE tuning == | ||
Let us start with | Let us start with CEE tuning (constrained equilateral-Euclidean tuning): there is no weight or skew, and the constraint is the octave. | ||
Denote the constraint by B<sub>C</sub>. In the case of | Denote the constraint by B<sub>C</sub>. In the case of CEE, it is the octave: | ||
<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> | ||
| Line 171: | Line 171: | ||
== Example == | == Example == | ||
Let us try tuning septimal meantone to | Let us try tuning septimal meantone to CEE. | ||
Its mapping is | Its mapping is | ||