Ringer scale: Difference between revisions

A '''ringer ''n'' scale''' is a minimal complexity* [[constant structure]] [[neji]] [[periodic scale]] with a [[period]] of an [[octave]] which has ''n'' notes (AKA scale degrees) per octave. (*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 possible.

== Perfect ringer scale ==

A perfect ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics (up to [[octave equivalence]]) to distinct number of steps. It is likely that only a small finite number of perfect ringer s. Here are some known ones (to be expanded as more are found):

If we change the mapping of prime 7 to second best (using the 15d val), which is our only other option for trying to untemper [[28/27]], then we get [[21/20]] tempered instead. Note that of the primes present in the [[prime factorization]] of [[21/20]], 2 is fixed, 3 we deduced must be fixed to patent and 7 is what we are changing, leaving only changing the mapping of prime 5 as a way out, which if we then use the 15cd val we get [[15/14]] tempered with no options left [[#Problem of warts|unless we use try to use a third-, fourth-, etc. mapping for primes]], which we will assume for simplicity does not lead anywhere. If we change the mapping of both (using the 15bd val) we also get [[15/14]] tempered. Therefore it seems like the patent val gives the best performance, so we will continue the construction of ringer 15 assuming that the 25-odd-limit is the highest it is capable of. The corresponding [[harmonic mode|mode]] of the harmonic series is mode 13, giving us:

Then we can notice we are missing two notes (as 26-13=13) to make it a 15-note scale and thus a ringer 15 scale, so we need to add two odd harmonics above 27 to complete it. Here there are multiple choices based on the [[val]] used and one's preference. The patent val way to complete the scale, which seems to be the lowest complexity and thus arguably the canonical one, is:

Another ringer 15 scale, if one prefers to get a 33rd harmonic instead of a 35th, is:

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.

This section will detail known ringers for edos smaller than 100. Because [[wart]]s are limited when it comes to large primes, any primes past 41 are explicitly listed in the form [p, q, r, ...] rather than abbreviated (rather cryptically) as letters. A quick summary of all the warts up to 41 is:

There should be at least two forms listed. One will be in the form used for the example of ringer 15. One will be in the minimum mode of the harmonic series that contains all harmonics. The latter can be pasted directly into scale workshop using the enumerate chord feature or into other programs like scala.