User:Ganaram inukshuk/MOS scale: Difference between revisions

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{{User:Ganaram inukshuk/Template:Rewrite draft|MOS scale|compare=https://en.xen.wiki/w/Special:ComparePages?page1=MOS+scale&rev1=&page2=User%3AGanaram+inukshuk%2FMOS+scale&rev2=&action=&diffonly=&unhide=

|changes=general rewrites~~; definition; wrangle different ways~~ to ~~say "mos"~~

|changes=make lead section up-to-date with how mos/MOS is written; general rewrites aimed at the page being beginner page (so some stuff ''may'' need to be moved)

}}A '''moment-of-symmetry scale''' (~~also~~ called '''moment-of-symmetry''', commonly abbreviated as '''MOS scale'''~~, '''MOSS''',~~ or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]] originally invented by [[Erv Wilson]].

}}A '''moment-of-symmetry scale''' (originally called '''moment of symmetry'''; commonly abbreviated as '''MOS scale''' or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''' or '''MOSS''', pronounced "moss"; plural '''moments of symmetry''', '''moment of symmetry scales''', '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]]. The concept of moment of symmetry scales were originally invented by [[Erv Wilson]].

== ~~Definition~~ ==

== An example with the diatonic scale ''(for beginner page)'' ==

''Use sintel's example here.''

== An example with the diatonic scale ''(for advanced page)''==

=== Erv Wilson's original definition ===

~~The concept of MOS scales were invented by~~ Erv Wilson in 1975 in ~~his paper~~ ''Moments of Symmetry''. A moment-of-symmetry scale consists of:

Erv Wilson first described the concept in 1975 in ''Moments of Symmetry''. A moment-of-symmetry scale consists of:

* A generator ~~and~~ an [[equivalence interval]], called ~~the~~ period, which is usually the [[octave~~]].~~

* A generator, an interval that is repeatedly stacked.

** The generator is commonly denoted using a quantity of steps from an [[EDO|equal division of the octave]], where both the edo and generator are coprime, meaning they do not share any common factors greater than 1.

* An equivalence interval, commonly called a period, which is usually the octave.

* Two unique step sizes, called ''large'' and ''small'', commonly denoted using the letters L and s.

* Two unique step sizes, called ''large'' and ''small'', commonly denoted using the letters L and s, respectively.

** The quantities of ~~these~~ steps ~~are also~~ coprime.

* A quantity of large and small steps that is coprime, meaning they have no common factors other than 1.

=== An example with the diatonic scale ===

The prototypical example of a moment-of-symmetry is the common diatonic scale of [[12edo]], which can be produced using a generator of 7 edosteps.

{| class="wikitable"

|+Constructing a scale in 12edo using a generator of 7 edosteps

!Generators added

!Step visualization

!Step pattern

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!Scale produced

|-

|1

|{{Step vis|7 5}}

|7 5

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Line 33:

|

|-

|2

|{{Step vis|2 5 5}}

|2 5 5

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Line 40:

|

|-

|3

|{{Step vis|2 5 2 3}}

|2 5 2 3

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Line 47:

|

|-

|4

|{{Step vis|2 2 3 2 3}}

|2 2 3 2 3

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|The common pentatonic scale, denoted as '''2L 3s'''.

|-

|5

|{{Step vis|2 2 3 2 2 1}}

|2 2 3 2 2 1

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Line 61:

|

|-

|6

|{{Step vis|2 2 2 1 2 2 1}}

|2 2 2 1 2 2 1

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|The next degree is at 18 edosteps. This is reduced (18 mod 12) to 6.

|The common diatonic scale, denoted as '''5L 2s'''. This is the lydian mode, equivalent to WWWHWWH.

|-

|7

|{{Step vis|1 1 2 2 1 2 2 1}}

|1 1 2 2 1 2 2 1

|

| rowspan="5" |The next 5 degrees are located at 1, 8, 3, 10, and 5 edosteps.

| rowspan="5" |The common chromatic scale. At this point, the two step sizes are the same, so the scale structure is no longer valid as a MOS scale.

|-

|8

|{{Step vis|1 1 2 2 1 1 1 2 1}}

|1 1 2 2 1 1 1 2 1

|

|-

|9

|{{Step vis|1 1 1 1 2 1 1 1 2 1}}

|1 1 1 1 2 1 1 1 2 1

|

|-

|10

|{{Step vis|1 1 1 1 2 1 1 1 1 1 1}}

|1 1 1 1 2 1 1 1 1 1 1

|

|-

|11

|{{Step vis|1 1 1 1 1 1 1 1 1 1 1 1}}

|1 1 1 1 1 1 1 1 1 1 1 1

|0 1 2 ... 11 12

|}

With the above example, valid MOS scales are produced at 2L 3s (the common pentatonic scale) and 5L 2s (the common diatonic scale).

A familiar property with the diatonic scale is that every interval – seconds, thirds, etc – has two sizes of major and minor. With the perfect 4th, these sizes are perfect and augmented, and with the perfect 5th, these sizes are perfect and diminsihed. These different sizes are accessed through the scale's different modes: lydian, ionian, mixolydian, dorian, aeolian, phrygian, and locrian. This property holds for all MOS scales, ''regardless of how many large and small steps there are''.

It should be noted that the intermediate steps (adding generators 7 through 10) suggest that they are also MOS scales, as there are two unique step sizes of 2 and 1, but this is not the case. Looking at 2L 3s and 5L 2s, a pattern can be observed in which the large step of the preceding scale splits into both a large and small step of the next scale. This observation allows for this construction to be simplified further, and disallows the intermediate scales (7 to 10 generators added) from being counted as MOS scales.

{| class="wikitable"

|+Constructing a scale in 12edo using a generator of 7 edosteps, simplified

!Generators added

!Step visualization

!Step pattern

!Scale degrees

!Added scale degrees

!Scale produced

|-

|1

|{{Step vis|7 5}}

|7 5

|0 7 12

|The first scale degree is at 7 edosteps from the root.

|'''1L 1s'''. Included for completeness.

|-

|2

|{{Step vis|2 5 5}}

|2 5 5

|0 2 7 12

|The next MOS scale is reached by adding one scale degree at 2 edosteps.

|'''2L 1s'''. Included for completeness.

|-

|5

|{{Step vis|2 2 3 2 3}}

|2 2 3 2 3

|0 2 4 7 9 12

|The next MOS scale is reached by adding two scale degrees at 4 and 9 edosteps.

|The common pentatonic scale, denoted as '''2L 3s'''.

|-

|7

|{{Step vis|2 2 2 1 2 2 1}}

|2 2 2 1 2 2 1

|0 2 4 6 7 9 11 12

|The next MOS scale is reached by adding two scale degrees at 6 and 11 edosteps.

|The common diatonic scale, denoted as '''5L 2s'''.

|-

|12

|{{Step vis|1 1 1 1 1 1 1 1 1 1 1 1}}

|1 1 1 1 1 1 1 1 1 1 1 1

|0 1 2 ... 11 12

|Adding the remaining scale degrees.

|The common chromatic scale.

|}

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== Notation and naming==

A moment-of-symmetry scale of ''x'' large steps and ''y'' small steps, where ''x'' and ''y'' are whole numbers, is denoted using the [[scale signature]] ''x''L ''y''s. In cases where one does not wish to distinguish between step sizes, the notation ''x''A ''y''B can be used instead, which can either refer to ''x''L ''y''s or ''y''L ''x''s.

A moment-of-symmetry scale of ''x'' large steps and ''y'' small steps, where ''x'' and ''y'' are whole numbers, is denoted using the [[scale signature]] ''x''L ''y''s. In cases where one does not wish to distinguish between step sizes, the notation ''x''A ''y''B can be used instead, which can either refer to ''x''L ''y''s or ''y''L ''x''s. Other notations may use different symbols for ''x''L ''y''s, such as ''a''L ''b''s, but these notations are identical.

By default, the [[Equave|equivalence interval]], or equave, of a MOS scale is assumed to be the [[octave]]. In discussions regarding MOS scales with [[non-octave]] equivalence intervals, the equivalence interval can be enclosed in angle brackets of either < > (less-than and greater-than symbols) or {{Angbr| }} (Unicode symbols U+27E8 and U+27E9). Whereas "5L 2s", for example, refers to an octave-equivalent pattern of 5 large and 2 small steps, 5L 2s{{Angbr|3/1}} refers to the same pattern but with 3/1 as the equivalence interval. To avoid conflicts with HTML tags, the use of Unicode symbols is advised over the former.

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=== Step ratio ===

When it comes to musical applications, the ''step ratio'', the ratio between the size of the scale's large and small step, can have a profound effect on how the overall scale sounds. The step ratio is usually denoted as L:s, to disambiguate it from [[Ratios|frequency ratios]], though the notation s:L is sometimes used to avoid division-by-zero.

===Advanced ~~discussion~~===

===Relationship between MOS scales===

=== Advanced properties ===

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*The Wilson Archives on moment-of-symmetry scales: https://anaphoria.com/wilsonintroMOS.html

*Erv Wilson's paper ''Moments of Symmetry'': ~~http~~://anaphoria.com/~~wilsonintroMOS~~.~~html~~

*Erv Wilson's paper ''Moments of Symmetry'': https://anaphoria.com/mos.pdf

@@ Line 1: / Line 1: @@
 {{User:Ganaram inukshuk/Template:Rewrite draft|MOS scale|compare=https://en.xen.wiki/w/Special:ComparePages?page1=MOS+scale&rev1=&page2=User%3AGanaram+inukshuk%2FMOS+scale&rev2=&action=&diffonly=&unhide=
-|changes=general rewrites; definition; wrangle different ways to say "mos"
+|changes=make lead section up-to-date with how mos/MOS is written; general rewrites aimed at the page being beginner page (so some stuff ''may'' need to be moved)
-}}A '''moment-of-symmetry scale''' (also called '''moment-of-symmetry''', commonly abbreviated as '''MOS scale''', '''MOSS''', or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]] originally invented by [[Erv Wilson]].
+}}A '''moment-of-symmetry scale''' (originally called '''moment of symmetry'''; commonly abbreviated as '''MOS scale''' or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''' or '''MOSS''', pronounced "moss"; plural '''moments of symmetry''', '''moment of symmetry scales''', '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]]. The concept of moment of symmetry scales were originally invented by [[Erv Wilson]].
-== Definition ==
+== An example with the diatonic scale ''(for beginner page)'' ==
+''Use sintel's example here.''
+== An example with the diatonic scale ''(for advanced page)''==
 === Erv Wilson's original definition ===
-The concept of MOS scales were invented by Erv Wilson in 1975 in his paper ''Moments of Symmetry''. A moment-of-symmetry scale consists of:
+Erv Wilson first described the concept in 1975 in ''Moments of Symmetry''. A moment-of-symmetry scale consists of:
-* A generator and an [[equivalence interval]], called the period, which is usually the [[octave]].
+* A generator, an interval that is repeatedly stacked.
-** The generator is commonly denoted using a quantity of steps from an [[EDO|equal division of the octave]], where both the edo and generator are coprime, meaning they do not share any common factors greater than 1.
+* An equivalence interval, commonly called a period, which is usually the octave.
-* Two unique step sizes, called ''large'' and ''small'', commonly denoted using the letters L and s.
+* Two unique step sizes, called ''large'' and ''small'', commonly denoted using the letters L and s, respectively.
-** The quantities of these steps are also coprime.
+* A quantity of large and small steps that is coprime, meaning they have no common factors other than 1.
+=== An example with the diatonic scale ===
 The prototypical example of a moment-of-symmetry is the common diatonic scale of [[12edo]], which can be produced using a generator of 7 edosteps.
 {| class="wikitable"
 |+Constructing a scale in 12edo using a generator of 7 edosteps
+!Generators added
 !Step visualization
 !Step pattern
@@ Line 22: / Line 26: @@
 !Scale produced
 |-
+|1
 |{{Step vis|7 5}}
 |7 5
@@ Line 28: / Line 33: @@
 |
 |-
+|2
 |{{Step vis|2 5 5}}
 |2 5 5
@@ Line 34: / Line 40: @@
 |
 |-
+|3
 |{{Step vis|2 5 2 3}}
 |2 5 2 3
@@ Line 40: / Line 47: @@
 |
 |-
+|4
 |{{Step vis|2 2 3 2 3}}
 |2 2 3 2 3
@@ Line 46: / Line 54: @@
 |The common pentatonic scale, denoted as '''2L 3s'''.
 |-
+|5
 |{{Step vis|2 2 3 2 2 1}}
 |2 2 3 2 2 1
@@ Line 52: / Line 61: @@
 |
 |-
+|6
 |{{Step vis|2 2 2 1 2 2 1}}
 |2 2 2 1 2 2 1
@@ Line 57: / Line 67: @@
 |The next degree is at 18 edosteps. This is reduced (18 mod 12) to 6.
 |The common diatonic scale, denoted as '''5L 2s'''. This is the lydian mode, equivalent to WWWHWWH.
+|-
+|7
+|{{Step vis|1 1 2 2 1 2 2 1}}
+|1 1 2 2 1 2 2 1
+|
+| rowspan="5" |The next 5 degrees are located at 1, 8, 3, 10, and 5 edosteps.
+| rowspan="5" |The common chromatic scale. At this point, the two step sizes are the same, so the scale structure is no longer valid as a MOS scale.
+|-
+|8
+|{{Step vis|1 1 2 2 1 1 1 2 1}}
+|1 1 2 2 1 1 1 2 1
+|
+|-
+|9
+|{{Step vis|1 1 1 1 2 1 1 1 2 1}}
+|1 1 1 1 2 1 1 1 2 1
+|
+|-
+|10
+|{{Step vis|1 1 1 1 2 1 1 1 1 1 1}}
+|1 1 1 1 2 1 1 1 1 1 1
+|
+|-
+|11
+|{{Step vis|1 1 1 1 1 1 1 1 1 1 1 1}}
+|1 1 1 1 1 1 1 1 1 1 1 1
+|0 1 2 ... 11 12
+|}
+With the above example, valid MOS scales are produced at 2L 3s (the common pentatonic scale) and 5L 2s (the common diatonic scale).
+A familiar property with the diatonic scale is that every interval – seconds, thirds, etc – has two sizes of major and minor. With the perfect 4th, these sizes are perfect and augmented, and with the perfect 5th, these sizes are perfect and diminsihed. These different sizes are accessed through the scale's different modes: lydian, ionian, mixolydian, dorian, aeolian, phrygian, and locrian. This property holds for all MOS scales, ''regardless of how many large and small steps there are''.
+It should be noted that the intermediate steps (adding generators 7 through 10) suggest that they are also MOS scales, as there are two unique step sizes of 2 and 1, but this is not the case. Looking at 2L 3s and 5L 2s, a pattern can be observed in which the large step of the preceding scale splits into both a large and small step of the next scale. This observation allows for this construction to be simplified further, and disallows the intermediate scales (7 to 10 generators added) from being counted as MOS scales.
+{| class="wikitable"
+|+Constructing a scale in 12edo using a generator of 7 edosteps, simplified
+!Generators added
+!Step visualization
+!Step pattern
+!Scale degrees
+!Added scale degrees
+!Scale produced
+|-
+|1
+|{{Step vis|7 5}}
+|7 5
+|0 7 12
+|The first scale degree is at 7 edosteps from the root.
+|'''1L 1s'''. Included for completeness.
+|-
+|2
+|{{Step vis|2 5 5}}
+|2 5 5
+|0 2 7 12
+|The next MOS scale is reached by adding one scale degree at 2 edosteps.
+|'''2L 1s'''. Included for completeness.
+|-
+|5
+|{{Step vis|2 2 3 2 3}}
+|2 2 3 2 3
+|0 2 4 7 9 12
+|The next MOS scale is reached by adding two scale degrees at 4 and 9 edosteps.
+|The common pentatonic scale, denoted as '''2L 3s'''.
+|-
+|7
+|{{Step vis|2 2 2 1 2 2 1}}
+|2 2 2 1 2 2 1
+|0 2 4 6 7 9 11 12
+|The next MOS scale is reached by adding two scale degrees at 6 and 11 edosteps.
+|The common diatonic scale, denoted as '''5L 2s'''.
+|-
+|12
+|{{Step vis|1 1 1 1 1 1 1 1 1 1 1 1}}
+|1 1 1 1 1 1 1 1 1 1 1 1
+|0 1 2 ... 11 12
+|Adding the remaining scale degrees.
+|The common chromatic scale.
 |}
@@ Line 76: / Line 162: @@
 == Notation and naming==
 {{See also|MOS naming}}
-A moment-of-symmetry scale of ''x'' large steps and ''y'' small steps, where ''x'' and ''y'' are whole numbers, is denoted using the [[scale signature]] ''x''L ''y''s. In cases where one does not wish to distinguish between step sizes, the notation ''x''A ''y''B can be used instead, which can either refer to ''x''L ''y''s or ''y''L ''x''s.
+A moment-of-symmetry scale of ''x'' large steps and ''y'' small steps, where ''x'' and ''y'' are whole numbers, is denoted using the [[scale signature]] ''x''L ''y''s. In cases where one does not wish to distinguish between step sizes, the notation ''x''A ''y''B can be used instead, which can either refer to ''x''L ''y''s or ''y''L ''x''s. Other notations may use different symbols for ''x''L ''y''s, such as ''a''L ''b''s, but these notations are identical.
 By default, the [[Equave|equivalence interval]], or equave, of a MOS scale is assumed to be the [[octave]]. In discussions regarding MOS scales with [[non-octave]] equivalence intervals, the equivalence interval can be enclosed in angle brackets of either < > (less-than and greater-than symbols) or {{Angbr|&nbsp;}} (Unicode symbols U+27E8 and U+27E9). Whereas "5L 2s", for example, refers to an octave-equivalent pattern of 5 large and 2 small steps, 5L 2s{{Angbr|3/1}} refers to the same pattern but with 3/1 as the equivalence interval. To avoid conflicts with HTML tags, the use of Unicode symbols is advised over the former.
@@ Line 85: / Line 171: @@
 === Step ratio ===
-{{Main|Operations on MOSes}}{{See also|Step ratio}}{{See also|TAMNAMS#Step ratio spectrum}}
+{{Main|Step ratio}}{{See also|TAMNAMS#Step ratio spectrum}}
 When it comes to musical applications, the ''step ratio'', the ratio between the size of the scale's large and small step, can have a profound effect on how the overall scale sounds. The step ratio is usually denoted as L:s, to disambiguate it from [[Ratios|frequency ratios]], though the notation s:L is sometimes used to avoid division-by-zero.
-===Advanced discussion===
+===Relationship between MOS scales===
+{{Main|Operations on MOSes}}{{See also|Recursive structure of MOS scales}}{{See also|MOS scale family tree}}
+=== Advanced properties ===
@@ Line 145: / Line 234: @@
 *The Wilson Archives on moment-of-symmetry scales: https://anaphoria.com/wilsonintroMOS.html
-*Erv Wilson's paper ''Moments of Symmetry'': http://anaphoria.com/wilsonintroMOS.html
+*Erv Wilson's paper ''Moments of Symmetry'': https://anaphoria.com/mos.pdf