Ringer scale: Difference between revisions

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A '''ringer ''n'' scale''' is a [[detempering]] of an [[edo]] to a minimal complexity [[harmonic series]] [[scale]] with the goals of having the [[constant structure]] (CS) property while having as many consecutive [[harmonic]]s (starting from 1) as possible, meaning that the set of all [[interval]]s present in the scale should have the maximal [[odd limit]] possible under that restriction, with remaining notes being given "filler harmonics" chosen subjectively based on taste/preference. ~~The fact that it has~~ a ~~constant structure implies there is at least one~~ [[val]] – corresponding to ''n''-[[edo]] – ~~that~~ will [[map]] every [[interval]] present to the same number of abstract "scale steps". This means [[2/1]] ''must'' be mapped to ''n'' scale steps. Note that the val is not required to be [[patent val|patent]] and that the most [[consistent]] val is not always the [[patent val]] and usually depends on the tendency towards sharpness or flatness of the corresponding [[edo]]. The name ''ringer'' comes from the tendency of these JI scales to "ring" a lot because of them generally being as low complexity as possible in the sense of [[odd-limit]].

A '''ringer ''n'' scale''' is a [[detempering]] of an [[edo]] to a minimal complexity [[harmonic series]] [[scale]] with the goals of having the [[constant structure]] (CS) property while having as many consecutive [[harmonic]]s (starting from 1) as possible, meaning that the set of all [[interval]]s present in the scale should have the maximal [[odd limit]] possible under that restriction, with remaining notes being given "filler harmonics" chosen subjectively based on taste/preference. Ringer scales are constructed using a specific [[val]] corresponding to ''n''-[[edo]] – which will [[map]] every [[interval]] present to the same number of abstract "scale steps". This means [[2/1]] ''must'' be mapped to ''n'' scale steps. Note that the val is not required to be [[patent val|patent]] and that the most [[consistent]] val is not always the [[patent val]] and usually depends on the tendency towards sharpness or flatness of the corresponding [[edo]]. The name ''ringer'' comes from the tendency of these JI scales to "ring" a lot because of them generally being as low complexity as possible in the sense of [[odd-limit]].

A ringer scale can be thought of as testing the very limits of what the [[constant structure]] property (and the corresponding [[val]] by proxy) is capable of for the harmonic series. Note that as the maximum number of consecutive harmonics that are possible to fit for a given edo is not always clear, we informally often call something we think is likely to be the maximum a ringer scale. If we suspect it might not be maximal we can say it might not be a proper ringer scale. If we know it is not maximal we can say it is an improper ringer scale. Improper ringer scales are often desirable as a result of user preference/customisation, but are not ringer scales because they do not achieve the goal of approximating as much of the low end of the harmonic series (without exclusion) as mathematically possible while preserving CS. These can be called "pseudoringer" scales if they still very much go for the aesthetic and complexity of a ringer scale while deviating from the corresponding ringer scale in a small number of ways.

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'''Non-monotonic''' (otherwise-)perfect Ringer 9: 9:10:11:12:13:14:'''16:15''':17:18

The [[17-limit]] [[val]] that confirms this scale is CS is {{val|9 15 22 26 32 34 38}}, which written as [[wart]]s is 9bccdefgg. (Note that in this case, where there is two warts this corresponds to the patent val mapping for the prime already being sharp and being warted to be a step sharper. If we assume that every wart means "sharpen by one step from patent val" this val can be written rather curiously as 9bcdefg, which shows that this val is the one sharpening every applicable prime by one step above the [[patent val]] mapping.) One can confirm that the above is CS because if one traverses it step by step, every one-step interval is mapped to one EDOstep which by [[wikipedia:linearity|linearity]] [[#Proof of CS of by linearity|implies CS]]. Note that it is important to preserve the order of these intervals. 14:16 = 16/14 = 8/7 is mapped to one positive step, as is 16:15 = 15/16, as is 15:17 = 17/15. Similarly (or thus/by linearity), 14:15 = 15/14 is mapped to 2 steps, as is 16:17 = 17/16, as is 15:18 = 18/15 = 6/5.

The [[17-limit]] [[val]] that confirms this scale is CS is {{val|9 15 22 26 32 34 38}}, which written as [[wart]]s is 9bccdefgg. (Note that in this case, where there is two warts this corresponds to the patent val mapping for the prime already being sharp and being warted to be a step sharper. If we assume that every wart means "sharpen by one step from patent val" this val can be written rather curiously as 9bcdefg, which shows that this val is the one sharpening every applicable prime by one step above the [[patent val]] mapping.) One can confirm that the above is CS because if one traverses it step by step, every one-step interval is mapped to one EDOstep which by [[wikipedia:linearity|linearity]] (more precisely, [[epimorphic]]ity) [[#Proof of CS of by linearity|implies CS]]. Note that it is important to preserve the order of these intervals. 14:16 = 16/14 = 8/7 is mapped to one positive step, as is 16:15 = 15/16, as is 15:17 = 17/15. Similarly (or thus/by linearity), 14:15 = 15/14 is mapped to 2 steps, as is 16:17 = 17/16, as is 15:18 = 18/15 = 6/5.

== Proof of CS by linearity ==

== Proof of CS by linearity of the epimorphic val ==

Because the CS property means that every occurrence of an interval must occur with the same number of steps, it suffices to show that every one-step interval is mapped to one step by the [[val]] that the Ringer scale is constructed with. (This val shows that the Ringer scale is [[epimorphic]].)

~~Because~~ the ~~CS property means that~~ every ~~occurrence of an~~ interval ~~must occur with~~ the ~~same number~~ of ~~steps, it suffices to show that every one-step interval~~ is ~~mapped by an appropriate~~ [[~~val~~]] ~~to one step~~.

In other words, if the Ringer scale's val maps every 1-scalestep interval to 1\''N'' (where ''N'' is the notes-per-period) then by linearity of the val the scale is CS and the corresponding [[rank|rank-1]] temperament resulting from tempering the differences between all the 1-scalestep intervals is the 'logic' that it obeys.

In other words, if you can find a val that maps every 1-scalestep interval to 1\''N'' (where ''N'' is the notes-per-period) then by induction the scale is CS and the corresponding [[rank|rank-1]] temperament resulting from tempering the differences between all the 1-scalestep intervals is the 'logic' that it obeys.

However, conversely, a scale being CS does not imply that such a val exists! In almost all observed practical cases if a scale is CS there is some val, but it is possible to construct scales where, for example, one 1-scalestep interval is equal to the product of more than one other 1-scalestep intervals; that is, if we have 1-scalestep intervals {''a'', ''b'', ''c'', ...} then we can choose ''ab'' as a 1-scalestep interval as long as ''ab'' doesn't occur as a 2-scalestep interval anywhere in the scale, which is why at least one extra 1-scalestep interval ''c'' is necessary to separate instances of ''a'' and ''b''. You can even choose ''b'' = ''a'' but you need to be careful to avoid CS-violating contradictions. For a concrete example, you can use {[[5/4]], [[9/8]], [[45/32]], ...} as 1-scalestep intervals to generate a nonlinear CS scale as long as [[45/32]] does not occur as a 2-scalestep interval anywhere in your scale.

~~HOWEVER~~, conversely, a scale being CS does not imply that such a val exists! In almost all observed practical cases if a scale is CS there is some val, but it is possible to construct scales where, for example, one 1-scalestep interval is equal to the product of more than one other 1-scalestep intervals; that is, if we have 1-scalestep intervals {''a'', ''b'', ''c'', ...} then we can choose ''ab'' as a 1-scalestep interval as long as ''ab'' doesn't occur as a 2-scalestep interval anywhere in the scale, which is why at least one extra 1-scalestep interval ''c'' is necessary to separate instances of ''a'' and ''b''. You can even choose ''b'' = ''a'' but you need to be careful to avoid CS-violating contradictions. For a concrete example, you can use {[[5/4]], [[9/8]], [[45/32]], ...} as 1-scalestep intervals to generate a nonlinear CS scale as long as [[45/32]] does not occur as a 2-scalestep interval anywhere in your scale.

=== Sketch of the proof ===

Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function ~~'''~~f~~('''''k''''')~~ : Z -> Q<~~sub~~>>0<~~/sub~~>~~''' with '''~~f(~~'''''~~Nk~~'''''~~) =~~''' ''~~P~~''''~~k~~''~~.

Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function <math>f: \mathbb{Z} \to \mathbb{Q}_0</math> with <math>f(Nk) = P^k.</math>

~~Then consider a [[val]] [[map]] '''m('''''k''''') : Q>0~~</~~sub> -~~> ~~Z'''.~~

We find (i.e. are given/assume) some val map '''m''' exists such that '''m(f('''''k'''''+1)/f('''''k''''')) = 1''' for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period ''P'''s worth of 1-scalestep intervals.)

By induction it implies '''m(f('''''k'''''+'''''s''''')/f('''''k''''')) =''' ''s'' because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together.

~~This then implies linearity because for two values of ''s'', which we will call ''a'' and ''b'', we obtain (again by induction):~~

~~'''m(f('''''a''''')f('''''b''''')) =''' ''a'' '''+''' ''b'' and '''m(f('''''a''''')/f('''''b''''')) =''' ''a'' '''-''' ''b'' for all ''a''''',''' ''b'' in '''Z'''. {{qed}}~~

~~Making rigorous~~ the ~~last part~~ of ~~this proof is symmetric to the derivation of the properties of addition via the~~ [~~https~~://~~en.wikipedia.org/wiki~~/~~Peano_axioms#Addition:~~''~~Peano axioms~~''~~] for natural numbers~~.

By the definition of a ringer scale, we are given some [[val]] [[map]] <math>m : \mathbb{Q}_0 \to \mathbb{Z}.</math> that satisfies <math>m(f(k+1)/f(k)) = 1</math> for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period Ps worth of 1-scalestep intervals.)

By induction this implies <math>m(f(k+s)/f(k)) = s</math> because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together. This also implies <math>m(f(k))=k,</math> proving ''f'' to be [[epimorphic]], therefore CS. {{qed}}

== Ringer scales ==

This section will detail known ringers for edos smaller than 100. Because [[wart]]s are limited when it comes to large primes, any primes past 43 are explicitly listed in the form [p, q, r, ...] rather than abbreviated (rather cryptically) as letters. A quick summary of all the warts up to 43 is:

@@ Line 1: / Line 1: @@
-A '''ringer ''n'' scale''' is a [[detempering]] of an [[edo]] to a minimal complexity [[harmonic series]] [[scale]] with the goals of having the [[constant structure]] (CS) property while having as many consecutive [[harmonic]]s (starting from 1) as possible, meaning that the set of all [[interval]]s present in the scale should have the maximal [[odd limit]] possible under that restriction, with remaining notes being given "filler harmonics" chosen subjectively based on taste/preference. The fact that it has a constant structure implies there is at least one [[val]] – corresponding to ''n''-[[edo]] – that will [[map]] every [[interval]] present to the same number of abstract "scale steps". This means [[2/1]] ''must'' be mapped to ''n'' scale steps. Note that the val is not required to be [[patent val|patent]] and that the most [[consistent]] val is not always the [[patent val]] and usually depends on the tendency towards sharpness or flatness of the corresponding [[edo]]. The name ''ringer'' comes from the tendency of these JI scales to "ring" a lot because of them generally being as low complexity as possible in the sense of [[odd-limit]].
+A '''ringer ''n'' scale''' is a [[detempering]] of an [[edo]] to a minimal complexity [[harmonic series]] [[scale]] with the goals of having the [[constant structure]] (CS) property while having as many consecutive [[harmonic]]s (starting from 1) as possible, meaning that the set of all [[interval]]s present in the scale should have the maximal [[odd limit]] possible under that restriction, with remaining notes being given "filler harmonics" chosen subjectively based on taste/preference. Ringer scales are constructed using a specific [[val]] corresponding to ''n''-[[edo]] – which will [[map]] every [[interval]] present to the same number of abstract "scale steps". This means [[2/1]] ''must'' be mapped to ''n'' scale steps. Note that the val is not required to be [[patent val|patent]] and that the most [[consistent]] val is not always the [[patent val]] and usually depends on the tendency towards sharpness or flatness of the corresponding [[edo]]. The name ''ringer'' comes from the tendency of these JI scales to "ring" a lot because of them generally being as low complexity as possible in the sense of [[odd-limit]].
 A ringer scale can be thought of as testing the very limits of what the [[constant structure]] property (and the corresponding [[val]] by proxy) is capable of for the harmonic series. Note that as the maximum number of consecutive harmonics that are possible to fit for a given edo is not always clear, we informally often call something we think is likely to be the maximum a ringer scale. If we suspect it might not be maximal we can say it might not be a proper ringer scale. If we know it is not maximal we can say it is an improper ringer scale. Improper ringer scales are often desirable as a result of user preference/customisation, but are not ringer scales because they do not achieve the goal of approximating as much of the low end of the harmonic series (without exclusion) as mathematically possible while preserving CS. These can be called "pseudoringer" scales if they still very much go for the aesthetic and complexity of a ringer scale while deviating from the corresponding ringer scale in a small number of ways.
@@ Line 54: / Line 54: @@
 '''Non-monotonic''' (otherwise-)perfect Ringer 9: 9:10:11:12:13:14:'''16:15''':17:18
-The [[17-limit]] [[val]] that confirms this scale is CS is  {{val|9 15 22 26 32 34 38}}, which written as [[wart]]s is 9bccdefgg. (Note that in this case, where there is two warts this corresponds to the patent val mapping for the prime already being sharp and being warted to be a step sharper. If we assume that every wart means "sharpen by one step from patent val" this val can be written rather curiously as 9bcdefg, which shows that this val is the one sharpening every applicable prime by one step above the [[patent val]] mapping.) One can confirm that the above is CS because if one traverses it step by step, every one-step interval is mapped to one EDOstep which by [[wikipedia:linearity|linearity]] [[#Proof of CS of by linearity|implies CS]]. Note that it is important to preserve the order of these intervals. 14:16 = 16/14 = 8/7 is mapped to one positive step, as is 16:15 = 15/16, as is 15:17 = 17/15. Similarly (or thus/by linearity), 14:15 = 15/14 is mapped to 2 steps, as is 16:17 = 17/16, as is 15:18 = 18/15 = 6/5.
+The [[17-limit]] [[val]] that confirms this scale is CS is  {{val|9 15 22 26 32 34 38}}, which written as [[wart]]s is 9bccdefgg. (Note that in this case, where there is two warts this corresponds to the patent val mapping for the prime already being sharp and being warted to be a step sharper. If we assume that every wart means "sharpen by one step from patent val" this val can be written rather curiously as 9bcdefg, which shows that this val is the one sharpening every applicable prime by one step above the [[patent val]] mapping.) One can confirm that the above is CS because if one traverses it step by step, every one-step interval is mapped to one EDOstep which by [[wikipedia:linearity|linearity]] (more precisely, [[epimorphic]]ity) [[#Proof of CS of by linearity|implies CS]]. Note that it is important to preserve the order of these intervals. 14:16 = 16/14 = 8/7 is mapped to one positive step, as is 16:15 = 15/16, as is 15:17 = 17/15. Similarly (or thus/by linearity), 14:15 = 15/14 is mapped to 2 steps, as is 16:17 = 17/16, as is 15:18 = 18/15 = 6/5.
-== Proof of CS by linearity ==
+== Proof of CS by linearity of the epimorphic val ==
+Because the CS property means that every occurrence of an interval must occur with the same number of steps, it suffices to show that every one-step interval is mapped to one step by the [[val]] that the Ringer scale is constructed with. (This val shows that the Ringer scale is [[epimorphic]].)
-Because the CS property means that every occurrence of an interval must occur with the same number of steps, it suffices to show that every one-step interval is mapped by an appropriate [[val]] to one step.
+In other words, if the Ringer scale's val maps every 1-scalestep interval to 1\''N'' (where ''N'' is the notes-per-period) then by linearity of the val the scale is CS and the corresponding [[rank|rank-1]] temperament resulting from tempering the differences between all the 1-scalestep intervals is the 'logic' that it obeys.
-In other words, if you can find a val that maps every 1-scalestep interval to 1\''N'' (where ''N'' is the notes-per-period) then by induction the scale is CS and the corresponding [[rank|rank-1]] temperament resulting from tempering the differences between all the 1-scalestep intervals is the 'logic' that it obeys.
+However, conversely, a scale being CS does not imply that such a val exists! In almost all observed practical cases if a scale is CS there is some val, but it is possible to construct scales where, for example, one 1-scalestep interval is equal to the product of more than one other 1-scalestep intervals; that is, if we have 1-scalestep intervals {''a'', ''b'', ''c'', ...} then we can choose ''ab'' as a 1-scalestep interval as long as ''ab'' doesn't occur as a 2-scalestep interval anywhere in the scale, which is why at least one extra 1-scalestep interval ''c'' is necessary to separate instances of ''a'' and ''b''. You can even choose ''b'' = ''a'' but you need to be careful to avoid CS-violating contradictions. For a concrete example, you can use {[[5/4]], [[9/8]], [[45/32]], ...} as 1-scalestep intervals to generate a nonlinear CS scale as long as [[45/32]] does not occur as a 2-scalestep interval anywhere in your scale.
-HOWEVER, conversely, a scale being CS does not imply that such a val exists! In almost all observed practical cases if a scale is CS there is some val, but it is possible to construct scales where, for example, one 1-scalestep interval is equal to the product of more than one other 1-scalestep intervals; that is, if we have 1-scalestep intervals {''a'', ''b'', ''c'', ...} then we can choose ''ab'' as a 1-scalestep interval as long as ''ab'' doesn't occur as a 2-scalestep interval anywhere in the scale, which is why at least one extra 1-scalestep interval ''c'' is necessary to separate instances of ''a'' and ''b''. You can even choose ''b'' = ''a'' but you need to be careful to avoid CS-violating contradictions. For a concrete example, you can use {[[5/4]], [[9/8]], [[45/32]], ...} as 1-scalestep intervals to generate a nonlinear CS scale as long as [[45/32]] does not occur as a 2-scalestep interval anywhere in your scale.
 === Sketch of the proof ===
-Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function '''f('''''k''''') : Z -> Q<sub>>0</sub>''' with '''f('''''Nk''''') =''' ''P''<sup>''k''</sup>.
+Consider an ''N''-note [[periodic scale]] with period ''P'' as being defined by a function <math>f: \mathbb{Z} \to \mathbb{Q}_0</math> with <math>f(Nk) = P^k.</math>
-Then consider a [[val]] [[map]] '''m('''''k''''') : Q<sub>>0</sub> -> Z'''.
-We find (i.e. are given/assume) some val map '''m''' exists such that '''m(f('''''k'''''+1)/f('''''k''''')) = 1''' for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period ''P'''s worth of 1-scalestep intervals.)
-By induction it implies '''m(f('''''k'''''+'''''s''''')/f('''''k''''')) =''' ''s'' because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together.
-This then implies linearity because for two values of ''s'', which we will call ''a'' and ''b'', we obtain (again by induction):
-'''m(f('''''a''''')f('''''b''''')) =''' ''a'' '''+''' ''b'' and '''m(f('''''a''''')/f('''''b''''')) =''' ''a'' '''-''' ''b'' for all ''a''''',''' ''b'' in '''Z'''. {{qed}}
-Making rigorous the last part of this proof is symmetric to the derivation of the properties of addition via the [https://en.wikipedia.org/wiki/Peano_axioms#Addition:''Peano axioms''] for natural numbers.
+By the definition of a ringer scale, we are given some [[val]] [[map]] <math>m : \mathbb{Q}_0 \to \mathbb{Z}.</math> that satisfies <math>m(f(k+1)/f(k)) = 1</math> for all ''k'' in '''Z'''. (This can be checked by hand or by computer as we only need to check one period <i>P</i>s worth of 1-scalestep intervals.)
+By induction this implies <math>m(f(k+s)/f(k)) = s</math> because the intervals from ''k'' to ''k''+1, from ''k''+1 to ''k''+2, ..., from ''k''+''s''-1 to ''k''+''s'' all multiply together. This also implies <math>m(f(k))=k,</math> proving ''f'' to be [[epimorphic]], therefore CS. {{qed}}
 == Ringer scales ==
 This section will detail known ringers for edos smaller than 100. Because [[wart]]s are limited when it comes to large primes, any primes past 43 are explicitly listed in the form [p, q, r, ...] rather than abbreviated (rather cryptically) as letters. A quick summary of all the warts up to 43 is: