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== Sistering ==
{{interwiki
'''Sistering''' 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. It is called thus because a MOS pattern and its sister share the same MOS as a subset (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).
| en = Operations on MOSes
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| ro = Operații asupra G2S
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: ''This page assumes the reader is familiar with the concept of [[MOS scales]].''


The ''sisterhood'' of xL ys is the set {xL ys, yL xs}. More generally, given an r-step 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 a<sub>π(1)</sub>X<sub>1</sub> ... a<sub>π(r)</sub>X<sub>r</sub> over all permutations π on {1, ..., r} the ''sisterhood'' of a<sub>1</sub>X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub>.
This page describes common operations that can be performed on [[MOS scale|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 [[Octave equivalence|an equivalence interval of an octave]], these operations apply to any MOS pattern regardless of [[equivalence interval]].


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.
== Relationship-based operations ==
=== Parent MOS ===
Given a MOS pattern ''x''L&nbsp;''y''s, 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.
 
{| class="wikitable"
|+ style="font-size: 105%;" | Example with 5L&nbsp;2s and its parent of 2L&nbsp;3s
|-
! MOS
! Step pattern
! Notes about step sizes
|-
| 5L&nbsp;2s
| LL'''Ls'''L'''Ls'''
| Large steps and small steps pairs (shown in '''bold''') are each merged into one larger step (2 in total).<br />The remaining 3 large steps are left untouched.
|-
| 3L&nbsp;2(Ls)
| LL'''(Ls)'''L'''(Ls)'''
| The merged steps, denoted using '''(Ls)''', are larger than the large steps.
|-
| 2L&nbsp;3s
| ssLsL
| After denoting (Ls) as the large step and the original large steps as the small steps, the parent scale is 2L&nbsp;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 ''x''L&nbsp;''y''s, {{nowrap|''x'' &gt; ''y''}}), then the parent scale is {{nowrap|''y''L (''x'' − ''y'')s}}.
* If there are more small steps than large steps in the original scale (that is, if in ''x''L&nbsp;''y''s, {{nowrap|''x'' &lt; ''y''}}), then the parent scale is {{nowrap|''x''L (''x'' − ''y'')s}}.
 
The above definition can be simplified further by adding the minimum and absolute value functions: given a MOS scale ''x''L&nbsp;''y''s, its parent is ''z''L&nbsp;''w''s, where {{nowrap|''z'' {{=}} min(''x'', ''y'')}} and {{nowrap|''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 {{sfrac|''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&nbsp;2s is 2L&nbsp;3s.
* The parent of 2L&nbsp;5s is 2L&nbsp;3s.
* The parent of 5L&nbsp;3s is 3L&nbsp;2s.
 
=== Sister MOS ===
Given a MOS pattern ''x''L&nbsp;''y''s, its '''sister''' is obtained by swapping the quantities of large and small steps, thus creating the step pattern ''y''L&nbsp;''x''s. It is called such because both ''x''L&nbsp;''y''s and ''y''L&nbsp;''x''s have the same parent scale of ''z''L&nbsp;''w''s, where {{nowrap|''z'' {{=}} min(''x'', ''y'')}} and {{nowrap|''w'' {{=}} {{abs|''x'' − ''y''}}}}).
 
{| class="wikitable"
|+ style="font-size: 105%;" | Example with 5L&nbsp;2s and its sister of 2L&nbsp;5s
|-
! MOS
! Step pattern
! Notes about step sizes
|-
| 5L&nbsp;2s
| LLLsLLs
| Large steps are replaced with small steps, and vice-versa.
|-
| 2L&nbsp;5s
| sssLssL
| The resulting pattern is 2L&nbsp;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&nbsp;2s is 2L&nbsp;5s.
* The sister of 5L&nbsp;3s is 3L&nbsp;5s.
* The sister of 4L&nbsp;4s is itself.
 
=== Daughter MOS ===
Given a MOS pattern ''x''L&nbsp;''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 {{nowrap|''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"
|+ style="font-size: 105%;" | Example with 5L&nbsp;2s and its daughters of 5L&nbsp;7s and 7L&nbsp;5s
|-
! MOS
! Step pattern
! Notes about step sizes
|-
| 5L&nbsp;2s
| LLLsLLs
| Each large step is split into two smaller steps ''s'' and ''c''.
|-
| 5c&nbsp;7s
| (sc)(sc)(sc)s(sc)(sc)s
| The quantity of small steps increases by however many large steps there originally were.<br />Parentheses denote where the large steps were.
|-
| 5L&nbsp;7s
| LsLsLssLsLss
| If the step ''c'' is larger than ''s'', then ''c'' becomes the new large step.
|-
| 7L&nbsp;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 {{nowrap|(''x'' + ''y'')L ''x''s}}.
* If ''c'' is larger than ''s'', then ''c'' becomes the new large step and ''s'' becomes the new small step. The daughter scale is {{nowrap|''x''L (''x'' + ''y'')s}}, which is also the sister of {{nowrap|(''x'' + ''y'')L ''x''s}}.
 
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 {{nowrap|(''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:
Examples:
* [[2L 5s]] is the sister of [[5L 2s]] (and is often called the anti-diatonic).
* [[3L 5s]] is the sister of [[5L 3s]].


== Neutralization ==
* The daughters of 5L&nbsp;2s are 7L&nbsp;5s and 5L&nbsp;7s.
'''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 daughters of 5L&nbsp;3s are 8L&nbsp;5s and 5L&nbsp;8s.


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.
== Other operations ==
=== Neutralization ===
Given a MOS pattern ''x''L&nbsp;''y''s, '''neutralization''' is the process in which pairs of large and small steps are each replaced with two neutral mossteps, whose size is defined as {{nowrap|''n'' {{=}} {{sfrac|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.


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.
{| class="wikitable"
|+ style="font-size: 105%;" | Example with 5L&nbsp;2s neutralized to 3L&nbsp;4s
|-
! MOS
! Step pattern
! Notes about step sizes
|-
| 5L&nbsp;2s
| LL'''Ls'''L'''Ls'''
| Large steps and small steps pairs (shown in '''bold''') are each replaced with two neutral steps (4 in total).<br />The remaining 3 large steps are left untouched.
|-
| 4n&nbsp;3L
| LL'''nn'''L'''nn'''
| Replacing adjacent L's and s's doesn't produce a valid MOS, but the steps can be rearranged to produce one.
|-
| 3L&nbsp;4s
| LsLsLss
| After rearranging, the neutralized scale is 3L 4s since:<br />
* 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.
|}


If x = y the resulting scale will just be (x+y)-edo = 2x-edo. For example 5L 5s becomes 10edo.
The neutralized MOS has a quantity of neutral mossteps that is twice that of min(''x'',&nbsp;''y''), and a quantity of remaining large or small steps that is {{nowrap|{{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:


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.
* If there are more large steps than small steps in the original scale (that is, if in ''x''L&nbsp;''y''s, {{nowrap|''x'' &gt; ''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 {{nowrap|(''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&nbsp;''y''s, {{nowrap|''x'' &lt; ''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 {{nowrap|2''x''L (''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 {{nowrap|''x'' + ''y''}} divisions. This doesn't produce a valid MOS since every step is the same size.


Examples:
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 ==
* Neutralizing 5L&nbsp;2s produces 4 neutral steps with 3 large steps left over, thus producing 4n&nbsp;3L, or 3L&nbsp;4s.
'''Dualization''' creates new MOS patterns from a MOS pattern in a specific EDO by swapping step sizes with step frequencies.  
* Neutralizing 2L&nbsp;5s produces 4 neutral steps with 3 small steps left over, thus producing 4n&nbsp;3s, or 4L&nbsp;3s.
* Neutralizing 5L&nbsp;3s produces 6 neutral steps with 2 small steps left over, thus producing 6n&nbsp;2s, or 6L&nbsp;2s.
* Neutralizing 5L&nbsp;4s produces 8 neutral steps with 1 large step left over, thus producing 8n&nbsp;1L, or 1L&nbsp;8s.
* Neutralizing 4L&nbsp;4s produces 8 neutral steps with no large or small steps left over, thus produces 8 equal divisions of the octave.
 
=== Dualization ===
Unlike all the other operations, this operation only applies to a MOS tuned to an edo. The resulting MOS varies depending on the specific edo.
 
Given a MOS pattern ''x''L&nbsp;''y''s with whole-number 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.


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''').
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 {{nowrap|''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.


For example, take 5L 2s in 43 EDO, with L=7 and s=4:
{| class="wikitable"
* The L-dual is 7L 2s with L=5 and s=4
|+ style="font-size: 105%;" | Example with 5L&nbsp;2s in 43edo
* The s-dual is 5L 4s with L=7 and s=2
|-
* The Ls-dual is 7L 4s with L=5 and s=2
! MOS
! Step pattern
! Notes about step sizes
! Step visualization
|-
| 5L&nbsp;2s
| LLLsLLs
| {{nowrap|L {{=}} 7|s {{=}} 4}}<br />Original scale
| {{Step vis|7 7 7 4 7 7 4}}
|-
| 7L&nbsp;2s
| LLLLsLLLs
| {{nowrap|L {{=}} 5|s {{=}} 4}}<br />L-dual, as ''x'' and L are swapped
| {{Step vis|5 5 5 5 4 5 5 5 4}}
|-
| 5L&nbsp;4s
| LLsLsLsLs
| {{nowrap|L {{=}} 7|s {{=}} 2}}<br />s-dual, as ''y'' and s are swapped
| {{Step vis|7 7 2 7 2 7 2 7 2}}
|-
| 7L&nbsp;4s
| LLsLLsLLsLs
| {{nowrap|L {{=}} 5|s {{=}} 2}}<br />Ls-dual, as ''x'' and L are swapped, as are ''y'' with s
| {{Step vis|5 5 2 5 5 2 5 5 2 5 2}}
|}


[[Category:MOS]]
[[Category:MOS scale]]
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