Neutral and interordinal intervals in MOS scales: Difference between revisions
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Given a tuning of a primitive (i.e. single-period) [[mos]] pattern aLbs<E> with arbitrary [[equave]] E, we may define two types of notes "in the cracks of" interval categories defined by aLbs<E>: | Given a tuning of a primitive (i.e. single-period) [[mos]] pattern aLbs<E> with arbitrary [[equave]] E, we may define two types of notes "in the cracks of" interval categories defined by aLbs<E>: | ||
# Given 1 ≤ ''k'' ≤ a + b - 1, the '''neutral''' ''k''-step (abbrev. n''k''s) is the interval exactly halfway between the smaller ''k''-step and the larger ''k''-step of the mos. When the mos is generated by a (perfect) ''k''-step, this may instead be called the '''semiperfect''' ''k''-step (abbrev. sP''k''s), since it is halfway between the perfect and imperfect (either diminished or augmented, depending on whether the generator is bright or dark) ''k''-step. | # Given 1 ≤ ''k'' ≤ a + b - 1, the '''neutral''' ''k''-step (abbrev. n''k''s) is the interval exactly halfway between the smaller ''k''-step and the larger ''k''-step of the mos. When the mos is generated by a (perfect) ''k''-step, this may instead be called the '''semiperfect''' ''k''-step (abbrev. sP''k''s), since it is halfway between the perfect and imperfect (either diminished or augmented, depending on whether the generator is bright or dark) ''k''-step. | ||
# Assume a > b. Given 1 ≤ ''k'' ≤ a + b − 2, and assuming that the larger ''k''-step < the smaller (''k'' + 1)-step, the '''interordinal''' between ''k''-steps and (''k'' + 1)-steps, denoted ''k''x(''k'' + 1)s or ''k''X(''k'' + 1)s (read "''k'' cross (''k'' + 1) step"), is the interval exactly halfway between the larger ''k''-step and the smaller (''k'' + 1)-step. (Though the term "interordinal" is intended to be JI-agnostic and generalize to non-diatonic mosses, the term ''interordinal'' comes from the fact that k-steps in the diatonic mos are conventionally called "(k + 1)ths".) | # Assume a > b. Given 1 ≤ ''k'' ≤ a + b − 2, and assuming that the larger ''k''-step < the smaller (''k'' + 1)-step, the '''interordinal''' between ''k''-steps and (''k'' + 1)-steps, denoted ''k''x(''k'' + 1)s or ''k''X(''k'' + 1)s (read "''k'' cross (''k'' + 1) step"), is the interval exactly halfway between the larger ''k''-step and the smaller (''k'' + 1)-step. (Though the term "interordinal" is intended to be JI-agnostic and generalize to non-diatonic mosses, the term ''interordinal'' comes from the fact that ''k''-steps in the diatonic mos are conventionally called "(''k'' + 1)ths".) | ||
Given such a mos, it's easy to observe the following properties of the simplest equal tunings for the mos, due to the way they divide the small step (s) and the chroma (L − s): | Given such a mos, it's easy to observe the following properties of the simplest equal tunings for the mos, due to the way they divide the small step (s) and the chroma (L − s): | ||