Mu badness: Difference between revisions

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'''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 {{u|VectorGraphics}}, and in a slightly different form by {{u|Lériendil}}.

For a given ~~edo~~ ''x'', it is defined as:

For a given EDO ''x'', it is defined as:

$$ \mu \left( x \right) = \sum_{k=1}^{\infty}f \left( x, k \right) $$

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$$ \abs{\operatorname{mod} \left( 2x, 2 \right) - 1} $$

for {{nowrap|''k'' {{=}} 2}}, if integer values of ''x'' are ~~edos~~.

for {{nowrap|''k'' {{=}} 2}}, if integer values of ''x'' are EDOs.

Equal divisions of any integer ''k'' can be found by multiplying 2''x'' by

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$$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$

Lériendil prefers to set the denominator to {{frac|π2|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.

Lériendil prefers to set the denominator to {{frac|π2|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|π2|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|π2|20}} scaling convention. The blue and orange dotted lines represent the best possible odd ED4 and the worst possible EDO, respectively.

== Peaks and valleys ==

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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 peaks proper (at the same fidelity) include 1, 2, 3, 5, 12, 41, 53 (52.99916), 171 (170.98894), 441 (441,0088), 494, 612, 2460, 3125...

The mu valley EDOs calculated up to {{nowrap| ''k'' {{=}} 100 }} include 1, 8, 11, 18, 23, 76, 194, 247, ….

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== 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.

In order to more or less strongly favor lower primes, one can generalize the weighting factor {{frac|1|''k''2}} to {{frac|1|''k''''σ''}}, where ''σ'' is a number greater than 1. Note that this requires many more iterations to reasonably converge on a value the closer ''σ'' is to 1.

== Alternative relative error function ==

If the cosine function is used as the relative error function as opposed to the zigzag, the result is the real part of the [[zeta]] function at {{nowrap|''s'' {{=}} σ + ''ix''}}.

[[Category:Badness]]

@@ Line 1: / Line 1: @@
-{{texops}}
+{{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 {{u|VectorGraphics}}, and in a slightly different form by {{u|Lériendil}}.
-For a given edo ''x'', it is defined as:
+For a given EDO ''x'', it is defined as:
 $$ \mu \left( x \right) = \sum_{k=1}^{\infty}f \left( x, k \right) $$
@@ Line 22: / Line 22: @@
 $$ \abs{\operatorname{mod} \left( 2x, 2 \right) - 1} $$
-for {{nowrap|''k'' {{=}} 2}}, if integer values of ''x'' are edos.
+for {{nowrap|''k'' {{=}} 2}}, if integer values of ''x'' are EDOs.
 Equal divisions of any integer ''k'' can be found by multiplying 2''x'' by
@@ Line 38: / Line 38: @@
 $$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$
-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.
+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 ==
@@ Line 111: / Line 111: @@
 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 peaks proper (at the same fidelity) include 1, 2, 3, 5, 12, 41, 53 (52.99916), 171 (170.98894), 441 (441,0088), 494, 612, 2460, 3125...
 The mu valley EDOs calculated up to {{nowrap| ''k'' {{=}} 100 }} include 1, 8, 11, 18, 23, 76, 194, 247, ….
@@ Line 119: / Line 121: @@
 == 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 {{frac|1|''k''<sup>2</sup>}} to {{frac|1|''k''<sup>''σ''</sup>}}, where ''σ'' is a number greater than 1. Note that this requires many more iterations to reasonably converge on a value the closer ''σ'' is to 1.
+== Alternative relative error function ==
+If the cosine function is used as the relative error function as opposed to the zigzag, the result is the real part of the [[zeta]] function at {{nowrap|''s'' {{=}} σ + ''ix''}}.
+[[Category:Badness]]