Mu badness
Mu (μ) is a function for equal tuning badness provided by Vector Graphics, and in a slightly different form by Lériendil.
For a given edo x, it is defined as:
[math]\displaystyle{ \mu\left(x\right)=\sum_{k=1}^{\infty}f\left(x,k\right) }[/math]
where
[math]\displaystyle{ 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]\displaystyle{ g\left(k\right)=\log_{2}\left(k\right) }[/math].
The function essentially sums up the relative error on all integer harmonics k, weighted by the inverse square of k in order to converge to a finite value.
It is derived as follows:
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]\displaystyle{ \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]\displaystyle{ g\left(k\right)=\log_{2}\left(k\right) }[/math].
As such, finding our final function is simply a matter of summing up
[math]\displaystyle{ \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.
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]\displaystyle{ \mu_{s}\left(x\right)=\frac{\left(\frac{\pi^{2}}{6}\right) - \mu\left(x\right)}{\left(\frac{\pi^{2}}{6}\right)-1} }[/math]
Lériendil prefers to set the denominator to [math]\displaystyle{ \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.
Mu badness for equal-step tunings between 1edo and 121edo using [math]\displaystyle{ \frac{\pi^{2}}{20} }[/math] scaling convention. The blue and orange dotted lines represent the best possible odd ed4 and the worst possible edo, respectively.
Peaks and Valleys
| Edo | Badness |
|---|---|
| 5 | 0.182 |
| 7 | 0.184 |
| 12 | 0.126 |
| 13 | 0.311 |
| 15 | 0.227 |
| 16 | 0.278 |
| 17 | 0.191 |
| 19 | 0.175 |
| 22 | 0.163 |
| 23 | 0.369 |
| 24 | 0.147 |
| 25 | 0.278 |
| 26 | 0.239 |
| 27 | 0.253 |
| 29 | 0.177 |
| 31 | 0.139 |
| 34 | 0.170 |
| 41 | 0.108 |
| 53 | 0.086 |
One can also define mu peaks, similar to zeta peaks. The mu peak integer edos (ignoring zero) calculated up to k=100 include 1, 2, 3, 5, 12, 41, 53, 441, 494, 612, 2460, 3125, 6079... Note that this may differ slightly from the true list, because I am using only the first 100 terms of μ.
The mu valley edos calculated up to k=100 include 1, 8, 11, 18, 23, 76, 194, 247...
The "Parker mu peak integers" are 4, 7, 19, 22, 24, 31, 65, 94, 118, 171, 665...
It is of note that, compared to zeta, this badness metric strongly favors low prime limits.
Weighted mu
In order to more or less strongly favor lower primes, one can generalize the weighting factor 1/(k^2) to 1/(k^s), where s is a number greater than 1. Note that this requires many more iterations to reasonably converge on a value the closer s is to 1.