Mu badness: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
m Cleanup and use display-style math |
||
| Line 1: | Line 1: | ||
{{texops}} | {{texops}} | ||
Mu | '''Mu badness''' is a [[badness]] for [[equal tuning]]s provided by [[User:VectorGraphics|Vector Graphics]], and in a slightly different form by [[User:Lériendil|Lériendil]]. | ||
For a given edo ''x'', it is defined as: | For a given edo ''x'', it is defined as: | ||
$$ \mu \left( x \right) = \sum_{k=1}^{\infty}f \left( x, k \right) $$ | |||
where | where | ||
$$ f \left( x, k \right) = \frac{\abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1}}{k^{2}} $$ | |||
and | and | ||
$$ g \left( k \right) = \log_{2} \left( k \right) $$. | |||
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. | 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 | 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 2''k'' (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 2''k'' (which has the largest possible error on ''k''). Such a zigzag line takes the form of: | ||
$$ \abs{\operatorname{mod} \left( 2x, 2 \right) - 1} $$ | |||
for {{nowrap|''k'' {{=}} 2}}, if integer values of ''x'' are edos. | for {{nowrap|''k'' {{=}} 2}}, if integer values of ''x'' are edos. | ||
| Line 26: | Line 26: | ||
Equal divisions of any integer ''k'' can be found by multiplying 2''x'' by | Equal divisions of any integer ''k'' can be found by multiplying 2''x'' by | ||
$$ g \left( k \right) = \log_{2} \left( k \right) $$ | |||
As such, finding our final function is simply a matter of summing up | As such, finding our final function is simply a matter of summing up | ||
$$ \abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1} $$ | |||
for all integers ''k''. To make the sum finite at all values, we weight each term by 1/''k''<sup>2</sup>, producing our final formula for ''f'', and thus for ''μ''. | for all integers ''k''. To make the sum finite at all values, we weight each term by 1/''k''<sup>2</sup>, producing our final formula for ''f'', and thus for ''μ''. | ||
''μ'' always provides a value between 1 and {{nowrap| ζ(2) {{=}} π<sup>2</sup>/6 ≈ 1.6449 }}, as such, the final "mu badness" result can be obtained by | |||
$$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$ | |||
Lériendil prefers to set the denominator to {{frac|π<sup>2</sup>|20}} 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. | |||
[[File:Mu badness.png|alt=Mu badness.png|1024x107px]] | |||
Mu badness for equal-step tunings between 1edo and 121edo using {{frac|π<sup>2</sup>|20}} scaling convention. The blue and orange dotted lines represent the best possible odd ed4 and the worst possible edo, respectively. | Mu badness for equal-step tunings between 1edo and 121edo using {{frac|π<sup>2</sup>|20}} scaling convention. The blue and orange dotted lines represent the best possible odd ed4 and the worst possible edo, respectively. | ||
== Peaks and valleys == | == Peaks and valleys == | ||
Below is a table of mu badness (μ<sub>s</sub>(x)) for edos, calculated up to {{nowrap|''k'' {{=}} 100}}. | Below is a table of mu badness (''μ''<sub>''s''</sub>(''x'')) for edos, calculated up to {{nowrap|''k'' {{=}} 100}}. | ||
{| class="wikitable" | {| class="wikitable center-all" | ||
|- | |- | ||
! Edo | ! Edo | ||
| Line 113: | Line 110: | ||
|} | |} | ||
One can also define mu peaks, similar to zeta peaks. The mu peak integer edos (ignoring zero) calculated up to {{nowrap|''k'' {{=}} 100}} include 1, 2, 3, 5, 12, 41, 53, 441, 494, 612, 2460, 3125, 6079 | One can also define mu peaks, similar to zeta peaks. The mu peak integer edos (ignoring zero) calculated up to {{nowrap|''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, only the first 100 terms of ''μ'' are calculated. | ||
The mu valley edos calculated up to {{nowrap|''k'' {{=}} 100}} include 1, 8, 11, 18, 23, 76, 194, 247 | The mu valley edos calculated up to {{nowrap| ''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 | 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 limit]]s. | It is of note that, compared to [[zeta]], this badness metric strongly favors low [[prime limit]]s. | ||