Ringer scale: Difference between revisions
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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]]. | 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 | == Perfect Ringer scales == | ||
A perfect Ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics to distinct numbers of steps up to [[octave equivalence]]. It is conjectured that these are the only perfect Ringer scales: | A perfect Ringer ''n'' scale is one that by some val can map the first ''n'' odd harmonics to distinct numbers of steps up to [[octave equivalence]]. It is conjectured that these are the only perfect Ringer scales: | ||