Ringer scale: Difference between revisions

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== Proof of CS by linearity ==

~~NOTE: This section~~ is ~~a work in progress~~.

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.

~~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 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.

~~Consider an~~ ''n''~~-note [[periodic scale]] with period an octave as being defined by a function~~ '''f('''''k'''''~~) : Z -> Q>0~~''' ~~with~~ '''f('''''nk'''''~~) = 2~~'''~~~~''k''</~~sup>~~.

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.

~~Then consider a~~ [[~~val]] [[map~~]] '''m('''''k''''') : Q>0 ~~-> Z~~'''~~. The CS property would guarantee that~~ '''m(f('''''a'''''~~)f('''''b''''')~~) =''' ''a'' '''+'~~'' ''b'' and '''m(f('''''a''''')~~/~~f('''''b''''')) =''' ''a'' '''-''' ''b'' for all ''a''''',''' ''b'' in '''Z''' but we cannot yet assume this~~.

=== Sketch of the proof ===

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

~~Instead~~ assume ~~we find~~ some val map '''m''' 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's worth of ~~single~~-~~step~~ intervals.)

Then consider a [[val]] [[map]] '''m('''''k''''') : Q>0 -> 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.

== Ringer scales ==

@@ Line 58: / Line 58: @@
 == Proof of CS by linearity ==
-NOTE: This section is a work in progress.
+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.
-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 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.
-Consider an ''n''-note [[periodic scale]] with period an octave as being defined by a function '''f('''''k''''') : Z -> Q<sub>>0</sub>''' with '''f('''''nk''''') = 2'''<sup>''k''</sup>.
+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.
-Then consider a [[val]] [[map]] '''m('''''k''''') : Q<sub>>0</sub> -> Z'''. The CS property would guarantee that '''m(f('''''a''''')f('''''b''''')) =''' ''a'' '''+''' ''b'' and '''m(f('''''a''''')/f('''''b''''')) =''' ''a'' '''-''' ''b'' for all ''a''''',''' ''b'' in '''Z''' but we cannot yet assume this.
+=== 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>.
-Instead assume we find some val map '''m''' 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's worth of single-step intervals.)
+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.
 == Ringer scales ==