Ringer scale: Difference between revisions

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== Problem of warts ==

When trying to find a maximal odd-limit for a Ringer scale, there is a problem of a [[wikipedia:combinatorial explosion|combinatorial explosion]] if we insist on checking every possible val to try to increase the odd-limit. (Note that using a second-, third-, etc. -best mapping of a prime is called "[[wart]]ing" that prime.) This is a difficult problem to solve as it means it is unclear whether a scale is as high odd-limit as it could possibly be while maintaining the constant structure property. A potential solution to this problem is to insist that we do not use a val that uses more than one wart for a prime in order to try to keep the val as accurate and faithful to the structure of JI as possible. This makes checking all vals computationally possible. However, there are serious cases, for example [[167edo]], where the "tendency" towards sharpness or flatness of an edo is so strong that we need more than one wart for a prime in order to fit the pattern and therefore potentially achieve a higher odd-limit, so this is only really a serious solution for smaller edos, and is a partial solution for larger edos that prefers edos that do not have any "tendency". This solution works for edos as big as [[80edo]], resulting in scales like [[User:Godtone#RINGER_80|Ringer 80]], which is an important example as [[80edo]] has a strong sharp tendency for its size, to the extent that it does not map [[21/16]] or [[27/16]] consistently. It also tends to work well for edos that are relatively "well-tuned" in the traditional [[LCJI]]-focused [[RTT]] sense.

An example of this problem is that there is in some sense a "perfect Ringer 9" scale but that it is not quite [[Periodic_scale#Definition|monotonic]] in that in order for the CS property to apply, you need to consider the harmonics as being in a specific order that is different from being ordered simply by size. Consider:

'''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.

== 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. The proof of this is as follows.

For convenience and clarity I will use some specific terminology here, even if the terminology is not fully standard or accurate. This is just to assist in communicating clearly.

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'''.

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.

== Ringer scales ==

@@ Line 43: / Line 43: @@
 == Problem of warts ==
 When trying to find a maximal odd-limit for a Ringer scale, there is a problem of a [[wikipedia:combinatorial explosion|combinatorial explosion]] if we insist on checking every possible val to try to increase the odd-limit. (Note that using a second-, third-, etc. -best mapping of a prime is called "[[wart]]ing" that prime.) This is a difficult problem to solve as it means it is unclear whether a scale is as high odd-limit as it could possibly be while maintaining the constant structure property. A potential solution to this problem is to insist that we do not use a val that uses more than one wart for a prime in order to try to keep the val as accurate and faithful to the structure of JI as possible. This makes checking all vals computationally possible. However, there are serious cases, for example [[167edo]], where the "tendency" towards sharpness or flatness of an edo is so strong that we need more than one wart for a prime in order to fit the pattern and therefore potentially achieve a higher odd-limit, so this is only really a serious solution for smaller edos, and is a partial solution for larger edos that prefers edos that do not have any "tendency". This solution works for edos as big as [[80edo]], resulting in scales like [[User:Godtone#RINGER_80|Ringer 80]], which is an important example as [[80edo]] has a strong sharp tendency for its size, to the extent that it does not map [[21/16]] or [[27/16]] consistently. It also tends to work well for edos that are relatively "well-tuned" in the traditional [[LCJI]]-focused [[RTT]] sense.
+An example of this problem is that there is in some sense a "perfect Ringer 9" scale but that it is not quite [[Periodic_scale#Definition|monotonic]] in that in order for the CS property to apply, you need to consider the harmonics as being in a specific order that is different from being ordered simply by size. Consider:
+'''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.
+== 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. The proof of this is as follows.
+For convenience and clarity I will use some specific terminology here, even if the terminology is not fully standard or accurate. This is just to assist in communicating clearly.
+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'''.
+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.
 == Ringer scales ==