Mu badness: Difference between revisions

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Mu (μ) is a function for equal tuning badness provided by Vector Graphics, and in a slightly different form by [[User:Lériendil]].

Mu (μ) is a function for equal tuning badness provided by Vector Graphics, and in a slightly different form by [[User:Lériendil|Lériendil]].

It is defined as:

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μ always provides a value between 1 and ζ(2) = (π^2)/6 ≈ 1.6449, as such, the final "mu badness" result can be obtained by

<math>\mu_{s}\left(x\right)=1-\frac{\mu\left(x\right)-1}{\left(\frac{\pi^{2}}{6}\right)-1}</math>

<math>\mu_{s}\left(x\right)=\frac{\left(\frac{\pi^{2}}{6}\right) - \mu\left(x\right)}{\left(\frac{\pi^{2}}{6}\right)-1}</math>

This also flips the result so that higher values represent worse tunings, as would be expected from a "badness" function.

Lériendil prefers to set the denominator to <math>\frac{\pi^{2}}{20}</math> instead, as it can be shown that this represents a stricter bound on μ and has the advantage of the maximal possible badness for an [[EDO]] being a rational number, 5/9. This also flips the result so that higher values represent worse tunings, as would be expected from a "badness" function.

{| class="wikitable"

|+Mu badness (μ<sub>s</sub>(x)) for edos, calculated up to k=100

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-Mu (μ) is a function for equal tuning badness provided by Vector Graphics, and in a slightly different form by [[User:Lériendil]].
+Mu (μ) is a function for equal tuning badness provided by Vector Graphics, and in a slightly different form by [[User:Lériendil|Lériendil]].
 It is defined as:
@@ Line 37: / Line 37: @@
 μ always provides a value between 1 and ζ(2) = (π^2)/6 ≈ 1.6449, as such, the final "mu badness" result can be obtained by
-<math>\mu_{s}\left(x\right)=1-\frac{\mu\left(x\right)-1}{\left(\frac{\pi^{2}}{6}\right)-1}</math>
+<math>\mu_{s}\left(x\right)=\frac{\left(\frac{\pi^{2}}{6}\right) - \mu\left(x\right)}{\left(\frac{\pi^{2}}{6}\right)-1}</math>
-This also flips the result so that higher values represent worse tunings, as would be expected from a "badness" function.
+Lériendil prefers to set the denominator to <math>\frac{\pi^{2}}{20}</math> instead, as it can be shown that this represents a stricter bound on μ and has the advantage of the maximal possible badness for an [[EDO]] being a rational number, 5/9. This also flips the result so that higher values represent worse tunings, as would be expected from a "badness" function.
 {| class="wikitable"
 |+Mu badness (μ<sub>s</sub>(x)) for edos, calculated up to k=100