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 '''[[#Origin of the term Ringer|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]].

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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== 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 numbers 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):

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

Ringer 2: 2:3:4

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Ringer 7: 7:8:9:10:11:12:13:14

Notice how all of these do not skip any harmonics ''while'' representing the harmonic series ''completely'' up to some [[odd-limit]].

Notice how all of these do not skip any harmonics while representing the harmonic series ''completely'' up to some [[odd-limit]].

== Origin of the term Ringer ==

The name "Ringer" was chosen by tuning theorist Scott Dakota to refer to the property of these scales to [[Harmonic_entropy#Background|"ring"]] extremely and about as much as might be possible for a [[JI]] scale because the [[odd-limit]] complexity of the intervals in such scales is near-minimal meaning they consume as much of the early harmonic series as possible. It is worth noting however that the appearance of a [[virtual fundamental]] depends strongly on which notes of the scale you play - an observation important to [[primodality]]. The concept of Ringer scales was additionally further developed by tuning theorists Praveen Venkataramana and later [[user:Godtone]].

== Example: Ringer 15 ==

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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 '''[[#Origin of the term Ringer|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]].
 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.
@@ Line 7: / Line 7: @@
 == 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 numbers 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):
+A perfect Ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics (up to [[octave equivalence]]) to distinct numbers of steps. It is likely that only a small finite number of perfect Ringer scales exist. Here are the known ones so far (to be expanded as/if more are found):
 Ringer 2: 2:3:4
@@ Line 19: / Line 19: @@
 Ringer 7: 7:8:9:10:11:12:13:14
-Notice how all of these do not skip any harmonics ''while'' representing the harmonic series ''completely'' up to some [[odd-limit]].
+Notice how all of these do not skip any harmonics while representing the harmonic series ''completely'' up to some [[odd-limit]].
+== Origin of the term Ringer ==
+The name "Ringer" was chosen by tuning theorist Scott Dakota to refer to the property of these scales to [[Harmonic_entropy#Background|"ring"]] extremely and about as much as might be possible for a [[JI]] scale because the [[odd-limit]] complexity of the intervals in such scales is near-minimal meaning they consume as much of the early harmonic series as possible. It is worth noting however that the appearance of a [[virtual fundamental]] depends strongly on which notes of the scale you play - an observation important to [[primodality]]. The concept of Ringer scales was additionally further developed by tuning theorists Praveen Venkataramana and later [[user:Godtone]].
 == Example: Ringer 15 ==