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Operations on MOSes: Difference between revisions

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→Neutralization: Wrote a definition of neutralization that doesn't require knowledge of the mos generators
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== Neutralization ==
== Neutralization ==
'''Neutralization''' is the operation of taking a MOS pattern and creating a new MOS pattern with the same number of notes, but with some of the steps replaced with what would be "neutral seconds" according to the original MOS pattern.
Given a MOS pattern ''x''L ''y''s, '''neutralization''' is the process in which pairs of large and small steps are each replaced with two neutral mossteps, defined as N = (L+s) / 2, with respect to the original mos.
{| class="wikitable"
|+Example with 5L 2s neutralized to 3L 4s
!MOS
!Step pattern
!Notes about step sizes
|-
|5L 2s
|LL'''Ls'''L'''Ls'''
|Large steps and small steps pairs (shown in '''bold''') are each replaced with two neutral steps (4 in total).
The remaining 3 large steps are left untouched.
|-
|4N 3L
|LLnnLnn
|Replacing adjacent L's and s's doesn't produce a valid MOS, but the steps can be rearranged to produce one.
|-
|3L 4s
|LsLsLss
|After rearranging, the neutralized scale is 3L 4s since:


The input to the operation of neutralization is really (MOS pattern, generator range), not just (MOS pattern). MOS pattern alone implies a generator range, but the range is the widest possible generator range that generates the pattern. For example, 4\7 to 3\5 for 5L 2S.
* Original large step becomes the new scale's large step
* Neutral step becomes the small step as it's smaller than the original large step.
|}
The resulting MOS pattern therefore has a quantity of neutral mossteps that is twice that of min(''x'', ''y''), and a quantity of remaining large or small steps that is abs(''x''-y).


When you neutralize a MOS pattern xL yS, you turn whatever step the MOS pattern has less of (let's say that's y, the same thing will work for x if x < y), and replace the y of that step size and y of the other step size into a neutral MOSsecond (i.e. half of Ls). The remaining scale steps (which are all L or all S, depending on whether x > y or x < y) are kept the same.  (Note: The input to this operation is not a temperament; different moses of the same temperament can have different neutralizations that suggest different temperaments.) Finally, the resulting scale steps are arranged in a MOS pattern. The resulting pattern is (x-y)L 2yS if x >= y, and 2xL (y-x)S if x <= y.
Since the size of a neutral step is, by definition, between the sizes of a large and small step, whether the neutral step becomes the new large or small steps solely depends on the number of large or small steps in the original scale:


If x = y the resulting scale will just be (x+y)-edo = 2x-edo. For example 5L 5s becomes 10edo.
* If there are more large steps than small steps (that is, if in ''x''L ''y''s, x > y), then the neutral step becomes the small step and the original large step becomes the new scale's large step.
* If there are more small steps than large steps (that is, if in ''x''L ''y''s, y < x), then the neutral step becomes the large step and the original small step becomes the new scale's small step.
* If the number of large and small steps is the same, the the neutralized scale is an equal division of the octave with ''x''+''y'' divisions. In other words, the large and small steps are [[equalized]].


When a scale is neutralized there would be restrictions on the resulting generator size and step sizes; i.e. a neutralized scale would be more than just the MOS pattern itself. For example, a 3L 4s with generator > 3\10 could not result from neutralizing 5L 2s, because the fifth would get too big for a 5L 2s MOS if the generator is > 3\10.
Examples:


Examples:
* Neutralizing 5L 2s produces 4 neutral steps with 3 large steps left over, thus producing 4N 3L, or 3L 4s.
* Neutralizing 5L 2s (gen between 4\7 and 3\5) results in 3L 4s, with generator between 2\7 and 3\10.
* Neutralizing 2L 5s produces 4 neutral steps with 3 small steps left over, thus producing 4N 3s, or 4L 3s.
* Neutralizing 5L 3s (gen between 3\8 and 3\5) results in 2L 6s with period 1\2 (!) and generator between 1\8 and 1\10 (sinaic to flat neutral 2nd).
* Neutralizing 5L 3s produces 6 neutral steps with 3 large steps left over, thus producing 6N 2s, or 6L 2s.
* Neutralizing 2L 5s (gen between 6\11 and 4\7) results in 4L 3s with generator 3\11 to 2\7.
* Neutralizing 5L 4s produces 8 neutral steps with 1 large step left over, thus producing 1N 8s, or 1L 8s.


== Dualization ==
== Dualization ==
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