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Operations on MOSes: Difference between revisions

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== Parent MOS ==
== 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:
Given a MOS pattern ''x''L ''y''s, its '''parent''' is obtained by merging pairs of large and small steps together into a larger 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.
{| class="wikitable"
|+Example with 5L 2s and its parent of 2L 3s
!MOS
!Step pattern
!Notes about step sizes
|-
|5L 2s
|LL'''Ls'''L'''Ls'''
|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, also called the ''child'' or ''daughter'', has more large steps or more small steps.


* Calculate z to be the smaller value of x and y, or min(x, y).
* If there are more large steps than small steps in the original scale (that is, if in ''x''L ''y''s, x > y), then the parent scale is ''y''L ''(x-y)''s.
* Calculate w to be the absolute difference between x and y, or abs(x, y).
* If there are more small steps than large steps in the original scale (that is, if in ''x''L ''y''s, x < y), then the parent scale is ''x''L ''(x-y)''s.
 
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. Since this is not a valid MOS since every large step and small step are paired with one another, such MOS scales are said to ''have no parent''.


Examples:
Examples:
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== Sister MOS ==
== 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).
Given a MOS pattern ''x''L ''y''s, 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 a<sub>1</sub>X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub> with r step sizes X<sub>1</sub> > ... > X<sub>r</sub>, we call the set of patterns  
The ''sisterhood'' of xL ys is the set {xL ys, yL xs}. More generally, given a scale pattern a<sub>1</sub>X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub> with r step sizes X<sub>1</sub> > ... > X<sub>r</sub>, we call the set of patterns  
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== Daughter MOS ==
== 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:
Given a MOS pattern ''x''L ''y''s, 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
{| class="wikitable"
|+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.


* (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.
* If s is larger than c, then the daughter is (''x''+''y'')L ''x''s.
* 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.
* If c is larger than s, then the daughter is ''x''L (''x''+''y'')s. This 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 is not a valid MOS due to the two step sizes being the same, so it's not considered a daughter.


Examples:
Examples:
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* If there are more large steps than small steps in the original scale (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. The neutralized scale is ''(x-y)''L 2''y''s.
* If there are more large steps than small steps in the original scale (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. The neutralized scale is ''(x-y)''L 2''y''s.
* If there are more small steps than large steps in the original scale (that is, if in ''x''L ''y''s, 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 2''x''L ''(y''-''x)''s.
* If there are more small steps than large steps in the original scale (that is, if in ''x''L ''y''s, 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 2''x''L ''(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]].
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:
Examples:
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