Mu badness: Difference between revisions
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<math>\mu\left(x\right)=\sum_{k=1}^{\infty}f\left(x,k\right)</math> | <math>\mu\left(x\right)=\sum_{k=1}^{\infty}f\left(x,k\right)</math> | ||
where | for a given edo x, where | ||
<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> | <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> | ||
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It is derived as follows: | 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 1 is an equal division of k, and 0 is an odd equal division of 2k (which has the largest possible error on k). Such a zigzag line takes the form of: | For each integer harmonic k, the relative error on that integer in the continuum of equal tunings follows a zigzag line where 1 is an equal division of k, and 0 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> | <math>\operatorname{abs}\left(\operatorname{mod}\left(2x,2\right)-1\right)</math> | ||