Mu badness
Mu (μ) is a function for equal tuning badness provided by Vector Graphics, and in a slightly different form by User:Lériendil.
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]
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:
[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)=1-\frac{\mu\left(x\right)-1}{\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.
| 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...