Ringer scale: Difference between revisions

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:

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.