Operations on MOSes: Difference between revisions

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== Parent MOS ==

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.

Given a MOS pattern ''x''L ''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"

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

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|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.

|}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 ''y''s, x > y), then the parent scale is ''y''L ''(x-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 parent scale is ''x''L ''(x-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 parent scale is ''x''L ''(y-x)''s.

The above definition can be simplified further by adding the min() and abs() functions: given a MOS scale ''x''L ''y''s, its parent is ''z''L ''w''s, 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. ~~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''.

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:

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== Sister MOS ==

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~~).

Given a MOS pattern ''x''L ''y''s, its '''sister''' is obtained by swapping the quantities of large and small steps, thus creating the step pattern ''y''L ''x''s. It is called such because both ''x''L ''y''s and ''y''L ''x''s have the same parent scale of ''z''L ''w''s, where ''z'' = min(''x'', ''y'') and ''w'' = abs(''x'' - ''y'').

{| class="wikitable"

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

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

!MOS

~~{aπ(1)X1 ..~~. a~~π(r)Xr~~ : ~~π a permutation on {1, ..., r}}~~

!Step pattern

!Notes about step sizes

~~the~~ ''~~sisterhood~~'' ~~of a1X1 ... arXr.~~

|-

|5L 2s

~~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~~.

|LLLsLLs

|Large steps are replaced with small steps, and vice-versa.

Examples:

|-

|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 ==

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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 the daughter is (''x''+''y'')L ''x''s.

* 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 ''x''s.

* 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.

* If c is larger than s, then c becomes the new large step and s becomes the new small step. The daughter scale is ''x''L (''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 ~~is not~~ a valid MOS ~~due to~~ the two step sizes ~~being~~ the same, so it's not considered a daughter.

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:

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|After rearranging, the neutralized scale is 3L 4s since:

* Original large step becomes the new scale's large step

* 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.

|}

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Given a MOS pattern ''x''L ''y''s 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'', 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.

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.

{| class="wikitable"

|+Example with 5L 2s in 43edo

@@ Line 2: / Line 2: @@
 == Parent MOS ==
-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.
+Given a MOS pattern ''x''L ''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"
 |+Example with 5L 2s and its parent of 2L 3s
@@ Line 21: / Line 21: @@
 |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.
+|}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 ''y''s, x > y), then the parent scale is ''y''L ''(x-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 parent scale is ''x''L ''(x-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 parent scale is ''x''L ''(y-x)''s.
+The above definition can be simplified further by adding the min() and abs() functions: given a MOS scale ''x''L ''y''s, its parent is ''z''L ''w''s, 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. 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''.
+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:
@@ Line 35: / Line 36: @@
 == Sister MOS ==
-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).
+Given a MOS pattern ''x''L ''y''s, its '''sister''' is obtained by swapping the quantities of large and small steps, thus creating the step pattern ''y''L ''x''s. It is called such because both ''x''L ''y''s and ''y''L ''x''s have the same parent scale of ''z''L ''w''s, where ''z'' = min(''x'', ''y'') and ''w'' = abs(''x'' - ''y'').
+{| class="wikitable"
-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
+|+Example with 5L 2s and its sister of 2L 5s
+!MOS
-{a<sub>π(1)</sub>X<sub>1</sub> ... a<sub>π(r)</sub>X<sub>r</sub> : π a permutation on {1, ..., r}}
+!Step pattern
+!Notes about step sizes
-the ''sisterhood'' of a<sub>1</sub>X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub>.
+|-
+|5L 2s
-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.
+|LLLsLLs
+|Large steps are replaced with small steps, and vice-versa.
-Examples:
+|-
+|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 ==
@@ Line 76: / Line 82: @@
 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 the daughter is (''x''+''y'')L ''x''s.
+* 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 ''x''s.
-* 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.
+* If c is larger than s, then c becomes the new large step and s becomes the new small step. The daughter scale is ''x''L (''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 is not a valid MOS due to the two step sizes being the same, so it's not considered a daughter.
+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:
@@ Line 106: / Line 112: @@
 |After rearranging, the neutralized scale is 3L 4s since:
-* Original large step becomes the new scale's large step
+* 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.
 |}
@@ Line 126: / Line 132: @@
 Given a MOS pattern ''x''L ''y''s 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'', 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.
+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.
 {| class="wikitable"
 |+Example with 5L 2s in 43edo