Operations on MOSes

This page describes common operations that can be performed on MOS scales. These operations, with a few exceptions, assume an abstract step pattern – a step pattern for which the step sizes are not specified – but can also apply to a concrete step pattern. Additionally, although the step patterns described here assume an equivalence interval of an octave, these operations apply to any MOS pattern regardless of equivalence interval.

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

Given a MOS pattern xL ys, neutralization is the process in which pairs of large and small steps are each replaced with two neutral mossteps, whose size is defined as n = (L+s) / 2, with respect to the original scale.

Example with 5L 2s neutralized to 3L 4s

MOS	Step pattern	Notes about step sizes
5L 2s	LLLsLLs	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: 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 neutralized MOS 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). This scale has the same number of steps as the original, but with one step size that is different from the original. Since the size of a neutral step is, by definition, between the sizes of a large and small step (as it's the average of the two step sizes), 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 there are more large steps than small steps in the original scale (that is, if in xL ys, x > y), then the neutral step becomes the small step and the original large step becomes the new scale's large step. The neutralized scale is (x-y)L 2ys.
• If there are more small steps than large steps in the original scale (that is, if in xL ys, x < y), then the neutral step becomes the large step and the original small step becomes the new scale's small step. The neutralized scale is 2xL (y-x)s.
• If the number of large and small steps is the same in the original scale, 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.

Examples:

• Neutralizing 5L 2s produces 4 neutral steps with 3 large steps left over, thus producing 4n 3L, or 3L 4s.
• Neutralizing 2L 5s produces 4 neutral steps with 3 small steps left over, thus producing 4n 3s, or 4L 3s.
• Neutralizing 5L 3s produces 6 neutral steps with 2 small steps left over, thus producing 6n 2s, or 6L 2s.
• Neutralizing 5L 4s produces 8 neutral steps with 1 large step left over, thus producing 8n 1L, or 1L 8s.
• Neutralizing 4L 4s produces 8 neutral steps with no large or small steps left over, thus produces 8 equal divisions of the octave.

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