Mu badness: Difference between revisions

Line 1:

Mu (μ) is a function for equal tuning badness provided by 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>

Line 7:

Line 8:

where

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

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

and

Line 13:

Line 14:

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

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:

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

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

for 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 2x 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>.

Line 30:

Line 31:

As such, finding our final function is simply a matter of summing up

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

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

for all integers k. To make the sum finite at all values, we weight each term by 1/(k^2), 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''2, producing our final formula for ''f'', and thus for ''μ''.

μ always provides a value between 1 and ζ(2) = (π^2)/6 ≈ 1.6449, as such, the final "mu badness" result can be obtained by

μ always provides a value between 1 and {{nowrap|ζ(2) {{=}} π2/6 ≈ 1.6449}}, as such, the final "mu badness" result can be obtained by

<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 <~~math~~>~~\frac{\pi^{~~2~~}}{20}~~</~~math~~> 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|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]]

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

Mu badness for equal-step tunings between 1edo and 121edo using <~~math~~>~~\frac{\pi^{~~2~~}}{20}~~</~~math~~> 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~~ ==

== Peaks and valleys ==

Below is a table of mu badness (μs(x)) for edos, calculated up to {{nowrap|''k'' {{=}} 100}}.

{| class="wikitable"

~~|+Mu badness (μs(x)) for edos, calculated up to k=100~~

~~!Edo~~

~~!Badness~~

|-

|5

! Edo

~~|0.182~~

! Badness

|-

|7

| 5

|0.~~184~~

| 0.182

|-

|12

| 7

|0.~~126~~

| 0.184

|-

|13

| 12

|0.~~311~~

| 0.126

|-

|15

| 13

|0.~~227~~

| 0.311

|-

|16

| 15

|0.~~278~~

| 0.227

|-

|17

| 16

|0.~~191~~

| 0.278

|-

|19

| 17

|0.~~175~~

| 0.191

|-

|22

| 19

|0.~~163~~

| 0.175

|-

|23

| 22

|0.~~369~~

| 0.163

|-

|24

| 23

|0.~~147~~

| 0.369

|-

|25

| 24

|0.~~278~~

| 0.147

|-

|26

| 25

|0.~~239~~

| 0.278

|-

|27

| 26

|0.~~253~~

| 0.239

|-

|29

| 27

|0.~~177~~

| 0.253

|-

|31

| 29

|0.~~139~~

| 0.177

|-

|34

| 31

|0.~~170~~

| 0.139

|-

|41

| 34

|0.~~108~~

| 0.170

|-

|53

| 41

|0.086

| 0.108

|-

| 53

| 0.086

|}

One can also define mu peaks, similar to zeta peaks. The mu peak integer edos (ignoring zero) calculated up to 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.

The mu valley edos calculated up to k=100 include 1, 8, 11, 18, 23, 76, 194, 247...

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.

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...

Line 119:

Line 122:

== 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 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.

@@ Line 1: / Line 1: @@
+{{texops}}
 Mu (μ) is a function for equal tuning badness provided by 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>
@@ Line 7: / Line 8: @@
 where
-<math>f\left(x,k\right)=\frac{\operatorname{abs}\left(\operatorname{mod}\left(2g\left(k\right)x,2\right)-1\right)}{k^{2}}</math>
+<math>f\left(x,k\right)=\frac{\abs{\operatorname{mod}\left(2g\left(k\right)x,2\right)-1}}{k^{2}}</math>
 and
@@ Line 13: / Line 14: @@
 <math>g\left(k\right)=\log_{2}\left(k\right)</math>.
-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:
-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 2k (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>\operatorname{abs}\left(\operatorname{mod}\left(2x,2\right)-1\right)</math>
+<math>\abs{\operatorname{mod}\left(2x,2\right)-1}</math>
-for 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 2x 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>.
@@ Line 30: / Line 31: @@
 As such, finding our final function is simply a matter of summing up
-<math>\operatorname{abs}\left(\operatorname{mod}\left(2g\left(k\right)x,2\right)-1\right)</math>
+<math>\abs{\operatorname{mod}\left(2g\left(k\right)x,2\right)-1}</math>
-for all integers k. To make the sum finite at all values, we weight each term by 1/(k^2), 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 ζ(2) = (π^2)/6 ≈ 1.6449, as such, the final "mu badness" result can be obtained by
+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
 <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 <math>\frac{\pi^{2}}{20}</math> 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]] 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 />
+[[File:Mu badness.png|alt=Mu badness.png|1024x107px]]<br />
-Mu badness for equal-step tunings between 1edo and 121edo using <math>\frac{\pi^{2}}{20}</math> 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}}.
 {| class="wikitable"
-|+Mu badness (μ<sub>s</sub>(x)) for edos, calculated up to k=100
-!Edo
-!Badness
 |-
-|5
+! Edo
-|0.182
+! Badness
 |-
-|7
+| 5
-|0.184
+| 0.182
 |-
-|12
+| 7
-|0.126
+| 0.184
 |-
-|13
+| 12
-|0.311
+| 0.126
 |-
-|15
+| 13
-|0.227
+| 0.311
 |-
-|16
+| 15
-|0.278
+| 0.227
 |-
-|17
+| 16
-|0.191
+| 0.278
 |-
-|19
+| 17
-|0.175
+| 0.191
 |-
-|22
+| 19
-|0.163
+| 0.175
 |-
-|23
+| 22
-|0.369
+| 0.163
 |-
-|24
+| 23
-|0.147
+| 0.369
 |-
-|25
+| 24
-|0.278
+| 0.147
 |-
-|26
+| 25
-|0.239
+| 0.278
 |-
-|27
+| 26
-|0.253
+| 0.239
 |-
-|29
+| 27
-|0.177
+| 0.253
 |-
-|31
+| 29
-|0.139
+| 0.177
 |-
-|34
+| 31
-|0.170
+| 0.139
 |-
-|41
+| 34
-|0.108
+| 0.170
 |-
-|53
+| 41
-|0.086
+| 0.108
+|-
+| 53
+| 0.086
 |}
-One can also define mu peaks, similar to zeta peaks. The mu peak integer edos (ignoring zero) calculated up to 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.
-The mu valley edos calculated up to k=100 include 1, 8, 11, 18, 23, 76, 194, 247...
+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.
+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...
@@ Line 119: / Line 122: @@
 == 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 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.