Neutral and interordinal intervals in MOS scales: Difference between revisions
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# Every interordinal interval of the parent MOS bL(a − b)s{{angbr|E}} of basic aLbs{{angbr|E}} excluding 0×1ms and (a+b−1)×(a+b)ms is a neutral or semiperfect interval of basic aLbs{{angbr|E}}. | # Every interordinal interval of the parent MOS bL(a − b)s{{angbr|E}} of basic aLbs{{angbr|E}} excluding 0×1ms and (a+b−1)×(a+b)ms is a neutral or semiperfect interval of basic aLbs{{angbr|E}}. | ||
# Except the neutral/semiperfect 1-step and the neutral/semiperfect (a + b − 1)-step, every neutral or semiperfect interval of basic aLbs{{angbr|E}} is a proper interordinal of bL(a − b)s{{angbr|E}}. The number (b − 1) counts the places in 2(2a + b)edE (twice the basic MOS tuning for aLbs{{angbr|E}}) where the parent's interordinal is improper, being two steps away, instead of one step away, from each of the adjacent ordinal categories. | # Except the neutral/semiperfect 1-step and the neutral/semiperfect (a + b − 1)-step, every neutral or semiperfect interval of basic aLbs{{angbr|E}} is a proper interordinal of bL(a − b)s{{angbr|E}}. The number (b − 1) counts the places in 2(2a + b)edE (twice the basic MOS tuning for aLbs{{angbr|E}}) where the parent's interordinal is improper, being two steps away, instead of one step away, from each of the adjacent ordinal categories. | ||
# | # aLbs{{angbr|E}} has a + 1 proper interordinals and b − 1 improper interordinals. | ||
=== Proof === | === Proof === | ||
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(b − 1) = |(brightest mode of basic aLbs, ignoring equaves) ∩ (darkest mode of basic aLbs, ignoring equaves)| | (b − 1) = |(brightest mode of basic aLbs, ignoring equaves) ∩ (darkest mode of basic aLbs, ignoring equaves)| | ||
= #{k : 0 < k < a + b and larger k-step of basic aLbs = smaller (k + 1)-step of basic aLbs} = # of | = #{k : 0 < k < a + b and larger k-step of basic aLbs = smaller (k + 1)-step of basic aLbs} = # of potential improprieties, | ||
where ''potential improprieties'' are pairs of interval classes that witness the impropriety of a hard-of-basic tuning of the MOS. | |||
Also recall that the following are equivalent for a MOS aLbs: | Also recall that the following are equivalent for a MOS aLbs: | ||
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* To show that these actually occur in bL(a − b)s, consider smaller and larger j-steps (1 ≤ j ≤ a − 1) in the parent MOS. These intervals also occur in the MOS aLbs separated by s, and the number of j’s (“junctures”) that correspond to these places in aLbs is exactly a − 1. These j's correspond to values of k such that larger k-step < smaller (k + 1)-step. Note that we are considering “junctures” between k-steps and (k + 1)-steps in aLbs, excluding k = 0 and k = a + b − 1, so the total number of “junctures” to consider is finite, namely a + b − 2. This proves parts (1) and (2). | * To show that these actually occur in bL(a − b)s, consider smaller and larger j-steps (1 ≤ j ≤ a − 1) in the parent MOS. These intervals also occur in the MOS aLbs separated by s, and the number of j’s (“junctures”) that correspond to these places in aLbs is exactly a − 1. These j's correspond to values of k such that larger k-step < smaller (k + 1)-step. Note that we are considering “junctures” between k-steps and (k + 1)-steps in aLbs, excluding k = 0 and k = a + b − 1, so the total number of “junctures” to consider is finite, namely a + b − 2. This proves parts (1) and (2). | ||
Part (3) is also immediate now: when larger k-step = smaller (k + 1)-step, larger (k + 1)-step − smaller k-step = 2(L − s) = 2s = L. The step L is 4 steps in 2n-edo. Part (4) | Part (3) is also immediate now: when larger k-step = smaller (k + 1)-step, larger (k + 1)-step − smaller k-step = 2(L − s) = 2s = L. The step L is 4 steps in 2n-edo. Part (4) follows since the parent MOS has a notes, corresponding to a − 1 interval classes that can be neutralized and the improper interordinals correspond to the potential improprieties of the MOS. {{qed}} | ||
[[Category:MOS scale]] | [[Category:MOS scale]] | ||
[[Category:Pages with proofs]] | [[Category:Pages with proofs]] | ||