Operations on MOSes: Difference between revisions

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! Notes about step sizes

|-

| 5L 2s

| 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)

| 3L 2(Ls)

| LL'''(Ls)'''L'''(Ls)'''

| The merged steps, denoted using '''(Ls)''', are larger than the large steps.

|-

| 2L 3s

| 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.

| After denoting (Ls) as the large step and the original large steps as the small steps, the parent scale is 2L 3s.

|}

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* If there are more small steps than large steps in the original scale (that is, if in ''x''L ''y''s, {{nowrap|''x'' < ''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 ''y''s, its parent is ''z''L ''w''s, where {{nowrap|''z'' {{=}} min(''x'', ''y'')}} and {{nowrap|''w'' {{=}} {{~~!}}~~''x'' − ''y''~~{{!~~}}}}).

The above definition can be simplified further by adding the minimum and absolute value functions: given a MOS scale ''x''L ''y''s, its parent is ''z''L ''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''.

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=== Sister MOS ===

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 {{nowrap|''z'' {{=}} min(''x'', ''y'')}} and {{nowrap|''w'' {{=}} {{~~!}}~~''x'' − ''y''~~{{!~~}}}}).

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 {{nowrap|''z'' {{=}} min(''x'', ''y'')}} and {{nowrap|''w'' {{=}} {{abs|''x'' − ''y''}}}}).

{| class="wikitable"

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! Notes about step sizes

|-

| 5L 2s

| 5L 2s

| LLLsLLs

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

|-

| 2L 5s

| 2L 5s

| sssLssL

| The resulting pattern is 2L 5s.

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=== Daughter MOS ===

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 {{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 2s and its daughters of 5L 7s and 7L 5s

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! Notes about step sizes

|-

| 5L 2s

| 5L 2s

| LLLsLLs

| Each large step is split into two smaller steps ''s'' and ''c''.

|-

| 5c 7s

| 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

| 5L 7s

| LsLsLssLsLss

| If the step ''c'' is larger than ''s'', then ''c'' becomes the new large step.

|-

| 7L 5s

| 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.

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! Notes about step sizes

|-

| 5L 2s

| 5L 2s

| LL'''Ls'''L'''Ls'''

| 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

| 4n 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 4s

| 3L 4s

| LsLsLss

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

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|}

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 {{nowrap|{{~~!}}~~''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:

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 {{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:

* If there are more large steps than small steps in the original scale (that is, if in ''x''L ''y''s, {{nowrap|''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 {{nowrap|(''x'' − ''y'')L 2''y''s}}.

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! Step visualization

|-

| 5L 2s

| 5L 2s

| LLLsLLs

| L = 7, s = 4 Original scale

| {{nowrap|L {{=}} 7|s {{=}} 4}} Original scale

| {{Step vis|7 7 7 4 7 7 4}}

|-

| 7L 2s

| 7L 2s

| LLLLsLLLs

| L = 5, s = 4 L-dual, as ''x'' and L are swapped

| {{nowrap|L {{=}} 5|s {{=}} 4}} L-dual, as ''x'' and L are swapped

| {{Step vis|5 5 5 5 4 5 5 5 4}}

|-

| 5L 4s

| 5L 4s

| LLsLsLsLs

| L = 7, s = 2 s-dual, as ''y'' and s are swapped

| {{nowrap|L {{=}} 7|s {{=}} 2}} s-dual, as ''y'' and s are swapped

| {{Step vis|7 7 2 7 2 7 2 7 2}}

|-

| 7L 4s

| 7L 4s

| LLsLLsLLsLs

| L = 5, s = 2 Ls-dual, as ''x'' and L are swapped, as are ''y'' with s

| {{nowrap|L {{=}} 5|s {{=}} 2}} 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 scale]]

@@ Line 22: / Line 22: @@
 ! Notes about step sizes
 |-
-| 5L 2s
+| 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 2(Ls)
+| 3L&nbsp;2(Ls)
 | LL'''(Ls)'''L'''(Ls)'''
 | The merged steps, denoted using '''(Ls)''', are larger than the large steps.
 |-
-| 2L 3s
+| 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 3s.
+| After denoting (Ls) as the large step and the original large steps as the small steps, the parent scale is 2L&nbsp;3s.
 |}
@@ Line 40: / Line 40: @@
 * 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'' {{=}} {{!}}''x'' − ''y''{{!}}}}).
+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''.
@@ Line 51: / Line 51: @@
 === 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'' {{=}} {{!}}''x'' − ''y''{{!}}}}).
+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"
@@ Line 60: / Line 60: @@
 ! Notes about step sizes
 |-
-| 5L 2s
+| 5L&nbsp;2s
 | LLLsLLs
 | Large steps are replaced with small steps, and vice-versa.
 |-
-| 2L 5s
+| 2L&nbsp;5s
 | sssLssL
 | The resulting pattern is 2L&nbsp;5s.
@@ Line 76: / Line 76: @@
 === 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
@@ Line 83: / Line 84: @@
 ! Notes about step sizes
 |-
-| 5L 2s
+| 5L&nbsp;2s
 | LLLsLLs
 | Each large step is split into two smaller steps ''s'' and ''c''.
 |-
-| 5c 7s
+| 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 7s
+| 5L&nbsp;7s
 | LsLsLssLsLss
 | If the step ''c'' is larger than ''s'', then ''c'' becomes the new large step.
 |-
-| 7L 5s
+| 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.
@@ Line 123: / Line 124: @@
 ! Notes about step sizes
 |-
-| 5L 2s
+| 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 3L
+| 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 4s
+| 3L&nbsp;4s
 | LsLsLss
 | After rearranging, the neutralized scale is 3L 4s since:<br />
@@ Line 138: / Line 139: @@
 |}
-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|{{!}}''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:
+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:
 * 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}}.
@@ Line 165: / Line 166: @@
 ! Step visualization
 |-
-| 5L 2s
+| 5L&nbsp;2s
 | LLLsLLs
-| L = 7, s = 4<br />Original scale
+| {{nowrap|L {{=}} 7|s {{=}} 4}}<br />Original scale
 | {{Step vis|7 7 7 4 7 7 4}}
 |-
-| 7L 2s
+| 7L&nbsp;2s
 | LLLLsLLLs
-| L = 5, s = 4<br />L-dual, as ''x'' and L are swapped
+| {{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 4s
+| 5L&nbsp;4s
 | LLsLsLsLs
-| L = 7, s = 2<br />s-dual, as ''y'' and s are swapped
+| {{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 4s
+| 7L&nbsp;4s
 | LLsLLsLLsLs
-| L = 5, s = 2<br />Ls-dual, as ''x'' and L are swapped, as are ''y'' with s
+| {{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 scale]]