Ringer scale: Difference between revisions

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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~~. The proof of this is as follows.~~

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:

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''.

Then consider a [[val]] [[map]] '''m('''''k''''') : Q>0 -> Z'''. The CS property ~~guarantees~~ that '''m(f('''''a''''')f('''''b''''')) =''' ''a'' '''+''' ''b'' and '''m(f('''''a''''')/f('''''b''''')) =''' ''a'' '''-''' ''b'' for all ''a''''',''' ''b'' in '''Z'''.

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.

~~Therefore if~~ '''m(f('''''k'''''+1)/f('''''k''''')) = 1''' for all ''k'' in '''Z''', 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.

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.)

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.

== Ringer scales ==

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 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. The proof of this is as follows.
+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:
 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>.
-Then consider a [[val]] [[map]] '''m('''''k''''') : Q<sub>>0</sub> -> Z'''. The CS property guarantees that '''m(f('''''a''''')f('''''b''''')) =''' ''a'' '''+''' ''b'' and '''m(f('''''a''''')/f('''''b''''')) =''' ''a'' '''-''' ''b'' for all ''a''''',''' ''b'' in '''Z'''.
+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.
-Therefore if '''m(f('''''k'''''+1)/f('''''k''''')) = 1''' for all ''k'' in '''Z''', 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.
+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.)
+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.
 == Ringer scales ==