Operations on MOSes

This page assumes the reader is familiar with the concept of MOS scales.

This page describes common operations that can be performed on MOS scales. These operations, unless otherwise specified, 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.

Relationship-based operations

Parent MOS

Given a MOS pattern xL ys, its parent is obtained by merging pairs of large and small steps together into one single step. The unpaired steps, regardless of their size, become the parent scale's small step. This process creates a subset MOS, called so since its scale degrees are a subset of the original scale's degrees.

Example with 5L 2s and its parent of 2L 3s

MOS	Step pattern	Notes about step sizes
5L 2s	LLLsLLs	Large steps and small steps pairs (shown in bold) are each merged into one larger step (2 in total). The remaining 3 large steps are left untouched.
3L 2(Ls)	LL(Ls)L(Ls)	The merged steps, denoted using (Ls), are larger than the large steps.
2L 3s	ssLsL	After denoting (Ls) as the large step and the original large steps as the small steps, the parent scale is 2L 3s.

The number of large steps in the parent is based on whether the original scale has more large steps or more small steps

• If there are more large steps than small steps in the original scale (that is, if in xL ys, x > y), then the parent scale is yL (x-y)s.
• If there are more small steps than large steps in the original scale (that is, if in xL ys, x < y), then the parent scale is xL (y-x)s.

The above definition can be simplified further by adding the min() and abs() functions: given a MOS scale xL ys, its parent is zL ws, where z = min(x, y) and w = abs(x - y).

There is a special case that can occur: if the number of large and small steps is the same in the original scale, then the parent scale is an equal division of the octave with (x+y)/2 divisions. This isn't a valid MOS since every large step and small step are paired with one another, so such scales are said to have no parent.

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 swapping the quantities of large and small steps, thus creating the step pattern yL xs. It is called such because both xL ys and yL xs have the same parent scale of zL ws, where z = min(x, y) and w = abs(x - y).

Example with 5L 2s and its sister of 2L 5s

MOS	Step pattern	Notes about step sizes
5L 2s	LLLsLLs	Large steps are replaced with small steps, and vice-versa.
2L 5s	sssLssL	The resulting pattern is 2L 5s.

There is a special case that can occur: if both x and y are the same quantity, then the MOS scale is said to be its own sister.Examples:

• The sister of 5L 2s is 2L 5s.
• The sister of 5L 3s is 3L 5s.
• The sister of 4L 4s is itself.

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 the size of c is defined as c = L - s. This process creates a superset MOS, called so since the original scale's degrees can be found in the daughter scale.

Example with 5L 2s and its daughters of 5L 7s and 7L 5s

MOS	Step pattern	Notes about step sizes
5L 2s	LLLsLLs	Each large step is split into two smaller steps s and c.
5c 7s	(sc)(sc)(sc)s(sc)(sc)s	The quantity of small steps increases by however many large steps there originally were. Parentheses denote where the large steps were.
5L 7s	LsLsLssLsLss	If the step c is larger than s, then c becomes the new large step.
7L 5s	sLsLsLLsLsLL	If the step s is larger than c, then c becomes the new small step and the s's become the new large step.

The daughters have two forms, depending on whether s or c is larger. Note that when working with abstract step values, it makes sense to talk about both daughters, but if the step sizes L and s are specified, then there will only be one daughter.

• If s is larger than c, then s becomes the new large step and c becomes the new small step. The daughter scale is (x+y)L xs.
• If c is larger than s, then c becomes the new large step and s becomes the new small step. The daughter scale is xL (x+y)s, which is also the sister of (x+y)L xs.

There is a special case that can occur: if s and c are the same size, then the daughter is an equal division of the octave with (x+y) divisions. This isn't a valid MOS pattern since the two step sizes are the same, so it's not considered a daughter.

Examples:

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

Other operations

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. The resulting scale, after rearranging the steps, is also a MOS scale since it has two step sizes: the neutral steps and either the large step or small step, depending on which step size is left over.

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.

There is a special case that can occur: 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. This doesn't produce a valid MOS since every step is the same size.

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

Given a MOS pattern xL ys with concrete step sizes L and s, dualization is the process in which the values of x and L are swapped, the values of y and s are swapped, or both are swapped. This is not to be confused with the sister operation.

Depending on which values are swapped, a different MOS scale is produced; however, the relationship between these scales depends on the sizes of L and s, and since x * L + y * s corresponds to the edo that supports the MOS, these relationships also depends on the edo. Additionally, it's possible for a dual to be itself, if either x and L are the same or y and s are the same.

Example with 5L 2s in 43edo

MOS	Step pattern	Notes about step sizes	Step visualization
5L 2s	LLLsLLs	L = 7, s = 4 Original scale	├──────┼──────┼──────┼───┼──────┼──────┼───┤
7L 2s	LLLLsLLLs	L = 5, s = 4 L-dual, as x and L are swapped	├────┼────┼────┼────┼───┼────┼────┼────┼───┤
5L 4s	LLsLsLsLs	L = 7, s = 2 s-dual, as y and s are swapped	├──────┼──────┼─┼──────┼─┼──────┼─┼──────┼─┤
7L 4s	LLsLLsLLsLs	L = 5, s = 2 Ls-dual, as x and L are swapped, as are y with s	├────┼────┼─┼────┼────┼─┼────┼────┼─┼────┼─┤