Operations on MOSes

Sisterization

Sisterization is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the sister of xL ys. Inverted MOS patterns are often denoted with the prefix anti-.

If xL ys has a generator range between a\x and b\(x+y) (it always holds that a < b), then its inversion yL xs has a generator range between b\(x+y) and (b-a)\y.

Examples:

• 2L 5s is the sister of 5L 2s (anti-diatonic)
• 3L 5s is the sister 5L 3s (anti-oneirotonic)

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.

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.

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.

If x = y the resulting scale will just be (x+y)-edo = 2x-edo. For example 5L 5s becomes 10edo.

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:

• Neutralizing 5L 2s (gen between 4\7 and 3\5) results in 3L 4s, with generator between 2\7 and 3\10.
• 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 2L 5s (gen between 6\11 and 4\7) results in 4L 3s with generator 3\11 to 2\7.