Operations on MOSes

This page describes common operations that can be performed on MOS scales.

Parent MOS

Given a MOS pattern xL ys, its parent is obtained by merging pairs of large and small steps together. This process creates a subset MOS. Mathematically, the parent MOS of zL ws is found by finding the values of z and w:

• Calculate z to be the smaller value of x and y, or min(x, y).
• Calculate w to be the absolute difference between x and y, or abs(x, y).

Examples:

• The parent of 5L 2s is 2L 3s.
• The parent of 2L 5s is 2L 3s.
• The parent of 5L 3s is 3L 2s.

Sister MOS

Given a MOS pattern xL ys, its sister is obtained by reversing the roles of large and small steps, thus creating a yL xs pattern. It is called thus because a MOS pattern and its sister share the same parent (for example, 5L 2s and 2L 5s both have 2L 3s subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the scale tree, via taking generator ranges).

The sisterhood of xL ys is the set {xL ys, yL xs}. More generally, given a scale pattern a₁X₁ ... a_rX_r with r step sizes X₁ > ... > X_r, we call the set of patterns

{a_π(1)X₁ ... a_π(r)X_r : π a permutation on {1, ..., r}}

the sisterhood of a₁X₁ ... a_rX_r.

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

Examples:

• The sister of 5L 2s is 2L 5s.
• The sister of 5L 3s is 3L 5s.

Daughter MOS

Given a MOS pattern xL ys, its daughters are obtained by splitting its large steps into two more smaller steps s and c, where s is equal to the original small step and c (also called the chroma) is the difference between a large step and small step. This process creates a superset MOS. The daughters have two forms:

• (x+y)L xs, where splitting the original large step results in s being larger than c. Here, s and c become the large and small steps, respectively.
• xL (x+y)s, where splitting the original large step results in c being larger than s. Here, c and s become the large and small steps, respectively. This is also the sister of (x+y)L xs.

Examples:

• The daughters of 5L 2s are 7L 5s and 5L 7s.
• The daughters of 5L 3s are 8L 5s and 5L 8s.

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.

Dualization

Dualization creates new MOS patterns from a MOS pattern in a specific EDO by swapping step sizes with step frequencies.

xL ys can be read as a formula: x * L + y * s = edo-size. From this formula it is clear we can swap for example x (the number of L-steps) with L (the size of the L-step) to get a new MOS scale in the same EDO, this is called the L-dual. Similarly we have the s-dual and when swapping both we get the Ls-dual (or just the dual).

For example, take 5L 2s in 43 EDO, with L=7 and s=4:

• The L-dual is 7L 2s with L=5 and s=4
• The s-dual is 5L 4s with L=7 and s=2
• The Ls-dual is 7L 4s with L=5 and s=2