Mu badness: Difference between revisions
m +category |
You know the drill, capitalization of "EDO" |
||
| Line 1: | Line 1: | ||
{{ | {{texops}} | ||
'''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]]. | '''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]]. | ||
| Line 38: | Line 38: | ||
$$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$ | $$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$ | ||
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. | 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|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]] | [[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 | 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 | Below is a table of mu badness (''μ''<sub>''s''</sub>(''x'')) for EDOs, calculated up to {{nowrap|''k'' {{=}} 100}}. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|- | |- | ||
! | ! EDO | ||
! Badness | ! Badness | ||
|- | |- | ||
| Line 110: | Line 110: | ||
|} | |} | ||
One can also define mu peaks, similar to zeta peaks. The mu peak integer | 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 | 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, …. | ||
| Line 120: | Line 120: | ||
== Weighted mu == | == Weighted mu == | ||
In order to more or less strongly favor lower primes, one can generalize the weighting factor 1/''k''<sup>2</sup> to 1/''k''<sup>''s''</sup>, 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. | In order to more or less strongly favor lower primes, one can generalize the weighting factor 1/''k''<sup>2</sup> to 1/''k''<sup>''s''</sup>, 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. | ||