Ringer scale: Difference between revisions

Line 1:

A '''[[#Origin of Ringer scales|Ringer]] ''n'' scale''' is a minimal complexity* [[constant structure]] (CS) [[~~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]].

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.

== Minimal complexity ==

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.

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.

== Perfect Ringer scale ==

@@ Line 1: / Line 1: @@
-A '''[[#Origin of Ringer scales|Ringer]] ''n'' scale''' is a minimal complexity* [[constant structure]] (CS) [[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]].
+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.
 == Minimal complexity ==
-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.
+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.
 == Perfect Ringer scale ==