Ringer scale: Difference between revisions

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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]].

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.

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b means 3 gets a next-best mapping, c means 5 gets a next-best mapping, d means 7 gets a next-best mapping and so on: e means 11, f means 13, g means 17, h means 19, i means 23, j means 29, k means 31, l means 37, m means 41. (2 = a is not used as it must always be patent.)

[[Category:Just intonation scales]]

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-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]].
+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.
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 b means 3 gets a next-best mapping, c means 5 gets a next-best mapping, d means 7 gets a next-best mapping and so on: e means 11, f means 13, g means 17, h means 19, i means 23, j means 29, k means 31, l means 37, m means 41. (2 = a is not used as it must always be patent.)
+[[Category:Just intonation scales]]