Ringer scale: Difference between revisions

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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 <math>f: \mathbb{Z} \to \mathbb{Q}_{>0}</math> with <math>f(Nk) = P^k.</math>~~

By the construction 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 (see proof in the article [[epimorphic scale]]). {{qed}}

== Ringer scales ==

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 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 <math>f: \mathbb{Z} \to \mathbb{Q}_{>0}</math> with <math>f(Nk) = P^k.</math>
-By the construction 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 (see proof in the article [[epimorphic scale]]). {{qed}}
 == Ringer scales ==