Ringer scale: Difference between revisions

A '''[[#Origin of Ringer scales|Ringer]] ''n'' scale''' is a [[detempering]] of an [[edo]] to a minimal complexity* [[harmonic series]] [[scale]] with the goals of having the [[constant structure]] (CS) property while having as many consecutive [[harmonic]]s (starting from 1) as possible, meaning that the set of all [[interval]]s present in the scale should have the maximal [[odd limit]] possible under that restriction, with remaining notes being given "filler harmonics" chosen subjectively based on taste/preference. (*What "minimal complexity" means is discussed [[#Minimal complexity|later in this article]].) The fact that it has a constant structure implies there is at least one [[val]] - corresponding to ''n'' [[edo]] - that will [[map]] every [[interval]] present to the same number of abstract "scale steps". (This means [[2/1]] ''must'' be mapped to ''n'' (scale) steps.) Note that the val is not required to be [[patent val|patent]] and that the most [[consistent]] val is not always the [[patent val]] and usually depends on the tendency towards sharpness or flatness of the corresponding [[edo]]. The name, "Ringer", comes from the tendency of these JI scales to "ring" a lot because of them generally being as low complexity as possible in the sense of [[odd-limit]].

An important consideration when building a Ringer ''n'' scale is what odd harmonics to add once you have reached the maximum [[odd-limit]]. To figure out where to place odd harmonics imbetween simpler odd harmonics already present, you need to use a choice of [[val]] to see what adjacent harmonics in the scale are mapped to more than 1 abstract scale step. The goal then is to make it so that every adjacent pair of harmonics in the Ringer ''n'' scale is mapped by the [[val]] to 1 scale degree.

The most striking feature of a Ringer scale is that it is "minimal complexity" in the sense that the maximum full [[odd-limit]] must be achieved, meaning as many odd harmonics (up to [[octave equivalence]]) must be present in the scale as possible ''without missing any'', which can be thought of as testing the very limits of what the [[constant structure]] property (and the corresponding [[val]] by proxy) is capable of for the harmonic series. As this maximum is not always easy to find, often informally we call something that we think is likely to be the maximum a "Ringer scale". If we suspect it might not be maximal we can say it might not be a proper Ringer scale. If we know it is not maximal we can say it is an improper Ringer scale. Improper Ringer scales are often desirable as a result of user preference/customisation, but are not Ringer scales because they do not achieve the goal of approximating as much of the low end of the harmonic series as mathematically possible while preserving CS.