Ringer scale: Difference between revisions

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A Ringer scale 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. Note that as the maximum number of consecutive harmonics that are possible to fit for a given edo is not always clear, we informally often call something 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 (without exclusion) as mathematically possible while preserving CS. These can be called "pseudoringer" scales if they still very much go for the aesthetic and complexity of a Ringer scale while deviating from the corresponding Ringer scale in a small number of ways.

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 abstract scale step. Note that a Ringer scale is completely described by the set of odd harmonics present because of [[octave equivalence]] because Ringer scales are [[periodic ~~scales~~]] with [[period]] equal to an [[octave]].

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 abstract scale step. Note that a Ringer scale is completely described by the set of odd harmonics present because of [[octave equivalence]] because Ringer scales are [[periodic scale]]s with [[period]] equal to an [[octave]].

== Perfect Ringer scale ==

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 A Ringer scale 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. Note that as the maximum number of consecutive harmonics that are possible to fit for a given edo is not always clear, we informally often call something 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 (without exclusion) as mathematically possible while preserving CS. These can be called "pseudoringer" scales if they still very much go for the aesthetic and complexity of a Ringer scale while deviating from the corresponding Ringer scale in a small number of ways.
-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 abstract scale step. Note that a Ringer scale is completely described by the set of odd harmonics present because of [[octave equivalence]] because Ringer scales are [[periodic scales]] with [[period]] equal to an [[octave]].
+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 abstract scale step. Note that a Ringer scale is completely described by the set of odd harmonics present because of [[octave equivalence]] because Ringer scales are [[periodic scale]]s with [[period]] equal to an [[octave]].
 == Perfect Ringer scale ==