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:

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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]]

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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]]

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 {{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:
@@ 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]]
@@ Line 121: / 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]]