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 in a specific tuning (i.e. with a specific [[hardness]] value for L/s), 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 in a specific tuning (i.e. with a specific [[hardness]] value for L/s), 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. The following always holds for a given interval class ''k'' between 1 and a + b − 1, inclusive: | # 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. The following always holds for a given interval class ''k''-steps between 1-steps and (a + b − 1)-steps, inclusive: | ||
#: neutral ''k''-step = smaller ''k''-step + c/2 = larger ''k''-step − c/2, where c = L − s is the chroma of the mos. | #: neutral ''k''-step = smaller ''k''-step + c/2 = larger ''k''-step − c/2, where c = L − s is the chroma of the mos. | ||
# 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" or "''k'' inter (''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 generalizable to non-diatonic mosses, the term comes from the fact that ''k''-steps in the diatonic mos are conventionally called "(''k'' + 1)ths".) The following always holds for a given interordinal interval ''k''-inter-(''k'' + 1)-step: | # 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" or "''k'' inter (''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 generalizable to non-diatonic mosses, the term comes from the fact that ''k''-steps in the diatonic mos are conventionally called "(''k'' + 1)ths".) The following always holds for a given interordinal interval ''k''-inter-(''k'' + 1)-step: | ||