Mu badness - Revision history

ArrowHead294 at 13:55, 14 April 2025

2025-04-14T13:55:43Z

← Older revision		Revision as of 13:55, 14 April 2025
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	{{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:
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	$$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$		$$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$

	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]]		[[File:Mu badness.png\|alt=Mu badness.png\|1024x107px]]
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	== 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>''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.		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 ==		== 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 s = σ+ix.		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]]		[[Category:Badness]]

VectorGraphics at 00:49, 10 April 2025

2025-04-10T00:49:12Z

← Older revision		Revision as of 00:49, 10 April 2025
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	== 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>''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 s = σ+ix.
	[[Category:Badness]]		[[Category:Badness]]

VectorGraphics at 08:54, 25 March 2025

2025-03-25T08:54:34Z

← Older revision		Revision as of 08:54, 25 March 2025
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.		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, ….		The mu valley EDOs calculated up to {{nowrap\| ''k'' {{=}} 100 }} include 1, 8, 11, 18, 23, 76, 194, 247, ….

VectorGraphics: capitalization of EDO

2025-03-13T09:27:11Z

capitalization of EDO

FloraC: I don't think absolute value function needs explanation. Re-install cleanup and category

2025-03-13T08:43:08Z

I don't think absolute value function needs explanation. Re-install cleanup and category

← Older revision		Revision as of 08:43, 13 March 2025
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	{{~~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 [[User:VectorGraphics\|Vector Graphics]], and in a slightly different form by [[User:Lériendil\|Lériendil]].

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	$$ f \left( x, k \right) = \frac{\abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1}}{k^{2}} $$		$$ f \left( x, k \right) = \frac{\abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1}}{k^{2}} $$

	~~(where \|....\| represents the absolute value function)~~ and		and

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

	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]]		[[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 center-all"		{\| class="wikitable center-all"
	\|-		\|-
	! ~~EDO~~		! Edo
	! Badness		! Badness
	\|-		\|-
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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.		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, ….
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	== 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.

			[[Category:Badness]]

VectorGraphics at 01:59, 13 March 2025

2025-03-13T01:59:16Z

← Older revision		Revision as of 01:59, 13 March 2025
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	$$ f \left( x, k \right) = \frac{\abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1}}{k^{2}} $$		$$ f \left( x, k \right) = \frac{\abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1}}{k^{2}} $$

	and		(where \|....\| represents the absolute value function) and

	$$ g \left( k \right) = \log_{2} \left( k \right) $$.		$$ g \left( k \right) = \log_{2} \left( k \right) $$.

VectorGraphics: You know the drill, capitalization of "EDO"

2025-03-13T01:58:39Z

You know the drill, capitalization of "EDO"

← Older revision		Revision as of 01:58, 13 March 2025
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	{{~~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 [[User:VectorGraphics\|Vector Graphics]], and in a slightly different form by [[User:Lériendil\|Lériendil]].

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

	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 ~~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 center-all"		{\| class="wikitable center-all"
	\|-		\|-
	! ~~Edo~~		! EDO
	! Badness		! Badness
	\|-		\|-
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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.		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, ….
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	== 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.

	~~[[Category:Badness]]~~

FloraC: +category

2025-03-12T08:53:58Z

+category

← Older revision		Revision as of 08:53, 12 March 2025
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	{{~~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 [[User:VectorGraphics\|Vector Graphics]], and in a slightly different form by [[User:Lériendil\|Lériendil]].

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	== 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.

			[[Category:Badness]]

FloraC: Cleanup and use display-style math

2025-03-12T08:53:12Z

Cleanup and use display-style math

← Older revision		Revision as of 08:53, 12 March 2025
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	{{texops}}		{{texops}}
	Mu ~~(μ)~~ is a ~~function~~ for equal tuning ~~badness~~ provided by 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]].

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

	~~<math>~~\mu\left(x\right)=\sum_{k=1}^{\infty}f\left(x,k\right)~~</math>~~		$$ \mu \left( x \right) = \sum_{k=1}^{\infty}f \left( x, k \right) $$

	where		where

	~~<math>~~f\left(x,k\right)=\frac{\abs{\operatorname{mod}\left(2g\left(k\right)x,2\right)-1}}{k^{2}}~~</math>~~		$$ f \left( x, k \right) = \frac{\abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1}}{k^{2}} $$

	and		and

	~~<math>~~g\left(k\right)=\log_{2}\left(k\right)~~</math>~~.		$$ g \left( k \right) = \log_{2} \left( k \right) $$.

	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:

	~~<math>~~\abs{\operatorname{mod}\left(2x,2\right)-1}~~</math>~~		$$ \abs{\operatorname{mod} \left( 2x, 2 \right) - 1} $$

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

	~~<math>~~g\left(k\right)=\log_{2}\left(k\right)~~</math>.~~		$$ g \left( k \right) = \log_{2} \left( k \right) $$


	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

	~~<math>~~\abs{\operatorname{mod}\left(2g\left(k\right)x,2\right)-1}~~</math>~~		$$ \abs{\operatorname{mod} \left( 2g \left( k \right) x, 2 \right) - 1} $$

	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.

	~~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~~		[[File:Mu badness.png\|alt=Mu badness.png\|1024x107px]]

	~~<math>\mu_{s}\left(x\right)=\frac{\left(\frac{\pi^{2}}{6}\right) - \mu\left(x\right)}{\left(\frac{\pi^{2}}{6}\right)-1}</math>~~

	~~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]]<br />~~
	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
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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, ~~because I am using~~ only the first 100 terms of μ.		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.

ArrowHead294 at 23:58, 11 March 2025

2025-03-11T23:58:45Z

Show changes

← Older revision		Revision as of 09:27, 13 March 2025
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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 [[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:

	$$ \mu \left( x \right) = \sum_{k=1}^{\infty}f \left( x, k \right) $$		$$ \mu \left( x \right) = \sum_{k=1}^{\infty}f \left( x, k \right) $$
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	$$ \abs{\operatorname{mod} \left( 2x, 2 \right) - 1} $$		$$ \abs{\operatorname{mod} \left( 2x, 2 \right) - 1} $$

	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		Equal divisions of any integer ''k'' can be found by multiplying 2''x'' by
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	$$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$		$$ \mu_{s} \left( x \right) = \frac{\left( \frac{\pi^{2}}{6} \right) - \mu \left( x \right)}{\left( \frac{\pi^{2}}{6} \right) - 1} $$

	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 ~~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 center-all"		{\| class="wikitable center-all"
	\|-		\|-
	! ~~Edo~~		! EDO
	! Badness		! Badness
	\|-		\|-
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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.		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, ….