Mu badness: Difference between revisions

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-Mu is a function for equal tuning badness provided by Vector Graphics.
+Mu (μ) is a function for equal tuning badness provided by Vector Graphics.
 It is defined as:
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 <math>f\left(x,k\right)=\frac{\operatorname{abs}\left(\operatorname{mod}\left(2g\left(k\right)x,2\right)-1\right)}{k^{2}}</math>
+and
+<math>g\left(k\right)=\log_{2}\left(k\right)</math>
+It is derived as follows:
+For each integer k, the relative error on that integer in the continuum of equal tunings follows a zigzag line where 0 is an equal division of k, and 1 is an odd equal division of 2k (which has the largest possible error on k). Such a zigzag line takes the form of:
+<math>\operatorname{abs}\left(\operatorname{mod}\left(2x,2\right)-1\right)</math>
+for k = 2, if integer values of x are edos.
+Equal divisions of any integer k can be found by multiplying 2x by
+<math>g\left(k\right)=\log_{2}\left(k\right)</math>.
+As such, finding our final function is simply a matter of summing up
+<math>\operatorname{abs}\left(\operatorname{mod}\left(2g\left(k\right)x,2\right)-1\right)</math>
+for all integers k. To make the sum finite at all values, we weight each term by 1/(k^2), producing our final formula for f, and thus for μ.
 WIP