Neutral and interordinal intervals in MOS scales: Difference between revisions

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+Given a tuning of a primitive (single-period) [[mos]] pattern aLbs with a > b, we may define two types of notes "in the cracks of" interval categories defined by aLbs:
+# Given 1 <= ''k'' <= a + b - 1, the ''neutral'' ''k''-step (abbrev. n''k''s) is an 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 be called a ''semiperfect'' ''k''-step (abbrev. sP''k''ms or 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 &minus; 2, and assuming that the larger ''k''-steps < the smaller (''k'' + 1)-step, the ''interordinal'' between ''k''-steps and (''k'' + 1)-steps, denoted ''k''x(''k'' + 1) or ''k''X(''k''+1), is the interval exactly halfway between the larger k-step and the smaller (''k'' + 1)-step. The name comes from the fact that in conventional diatonic theory, ''k''-steps are called "(''k'' + 1)ths".
 == The Interordinal Theorem ==
 Recall that the “impropriety number” (b - 1) of a primitive (i.e. single-period) mos aLbs satisfies: