Ringer scale: Difference between revisions

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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 ~~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 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. {{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 66: / Line 66: @@
 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 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 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. {{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: