Neutral and interordinal intervals in MOS scales: Difference between revisions
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The Interordinal Theorem relates the neutral (resp. interordinal) intervals of aLbs (with a > b) with the interordinal (resp. neutral) intervals of its parent mos, bL(a − b)s. | The Interordinal Theorem relates the neutral (resp. interordinal) intervals of aLbs (with a > b) with the interordinal (resp. neutral) intervals of its parent mos, bL(a − b)s. | ||
=== Statement === | === Statement === | ||
Suppose a > b and gcd(a, b) = 1. | Suppose a > b and gcd(a, b) = 1. (Below we assume that the equave is 2/1, but the proof generalizes to any equave.) | ||
# Every interordinal in basic aLbs (an interval that is exactly halfway between the larger k-step and the smaller (k+1)-step) is a neutral or semiperfect interval in the parent mos bL(a-b)s. | # Every interordinal in basic aLbs (an interval that is exactly halfway between the larger k-step and the smaller (k+1)-step) is a neutral or semiperfect interval in the parent mos bL(a-b)s. | ||
# Every interordinal interval in the parent mos bL(a-b)s of basic aLbs is a neutral or semiperfect interval in basic aLbs. The number (b - 1) counts the places in 2(2a+b)edo (twice the basic mos tuning for aLbs) where the parent's interordinal is two steps away from the ordinal categories. | # Every interordinal interval in the parent mos bL(a-b)s of basic aLbs is a neutral or semiperfect interval in basic aLbs. The number (b - 1) counts the places in 2(2a+b)edo (twice the basic mos tuning for aLbs) where the parent's interordinal is two steps away from the ordinal categories. | ||