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]].

}}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, 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.

* 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~~ coprime, meaning they ~~do not share any~~ common factors ~~greater~~ than 1.

* 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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|

|-

|2

|{{Step vis|2 5 5}}

|2 5 5

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|

|-

|3

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

|2 5 2 3

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|

|-

|4

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

|2 2 3 2 3

Line 46:

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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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|

|-

|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.

|}

=== Equivalent definitions ===

There are several equivalent definitions of MOS scales:

*

*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.

*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.

*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.

The term ''well-formed'', from Norman Carey and David Clampitt's paper ''Aspects of well-formed scales'', is sometimes used to equivalently describe the above definitions, and is used in academic research.

=== Single-period and multi-period MOS scales ===

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MOS scales in which the equivalence interval is a multiple of the period (or alternatively, the step pattern repeats multiple times within the equivalence interval), is commonly called a '''multi-MOS''' or '''multi-period MOS'''. This is to distinguish them from what Wilson had defined, called '''strict MOS''' or '''single-period MOS'''.

== Notation ==

An alternate definition of a multi-period MOS scale is a MOS scale in which the quantities of large and small steps are ''not'' coprime.

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.

== 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. 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.

~~=== Naming ===~~

~~{{Main|MOS naming}}~~

Although the most unambiguous way to refer to a MOS scale is by its scale signature, several naming schemes have been created that assign unique names to them. For a discussions on such names, see [[MOS naming]].

== Properties ==

~~The definition of a moment-of-symmetry scale has several equivalences to other concepts:~~

*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.

*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.

*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.

== ~~Applications~~ ==

=== 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.

===Relationship between MOS scales===

=== Advanced properties ===

== Non-tuning applications ==

==Non-tuning applications==

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~~==Example: the diatonic scale==~~

==History and terminology ==

The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L 2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) An important property of MOS scales is that all the intervals come in two sizes: major and minor seconds, major and minor thirds, perfect and augmented fourths, perfect and diminished fifths, etc. This is not true for something like the melodic minor scale (LsLLLLs), which has three kinds of fifths: perfect, diminished and augmented. Therefore, the melodic minor scale is not an MOS scale.

~~==Definition==~~

~~There are several equivalent definitions of MOS scales:~~

#

~~#Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or L − s = c)~~

#Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line ''y'' = ''p''/''q''*''x'' that stays as close to the line ''y'' = ''p''/''q''*''x'' without crossing it.)

~~While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]].~~

==History and terminology==

The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [http://anaphoria.com/mos.PDF ''Moments of Symmetry'']. There is also an introduction by Kraig Grady here: [http://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].

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As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where ''s'' = 0. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. The [[TAMNAMS #Step ratio spectrum|TAMNAMS]] system has names for both specific ratios and ranges of ratios.

~~==Step ratio spectrum==~~

The melodic sound of a MOS is not just affected by the tuning of its intervals, but by the sizes of its steps. MOSes with L more similar to s sound smoother and more mellow. MOSes with L much larger than s sound jagged and dramatic. The ''step ratio'', the ratio between the sizes of L and s, is thus important to the sound of the scale.

~~An in-depth analysis of this can be found at [[Step ratio]].~~

== Properties==

==Properties==

===Basic properties===

*Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are (''a'' + ''b'')L as (generated by generators of soft-of-basic ''a''L''b''s) and ''a''L (''a'' + ''b'')s (generated by generators of hard-of-basic ''a''L''b''s).

*Every MOS scale (with a specified [[equave]] ''E''), excluding ''a''L ''a''s⟨''E''⟩, has a ''parent MOS''. If ''a'' > ''b'', the parent of ''a''L ''b''s is min(''a'', ''b'')L|''a'' − ''b''|s; if ''a'' < ''b'', the parent of ''a''L ''b''s is |''a'' − ''b''|L min(''a'', ''b'')s.

===Advanced discussion===

===Advanced discussion ===

See:

*[[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.

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*[[MOS Cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.

*[[Operations on MOSes]]

==As applied to rhythms==

== As applied to rhythms==

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==See also==

== See also==

*[[Diamond-mos notation]], a microtonal notation system focussed on MOS scales

*[[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]

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*[[Gallery of MOS patterns]]

== External links ==

==External links==

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

*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]].
+}}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, 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.
+* 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 coprime, meaning they do not share any common factors greater than 1.
+* 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.
 |}
+=== Equivalent definitions ===
+There are several equivalent definitions of MOS scales:
+*
+*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.
+*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.
+*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.
+The term ''well-formed'', from Norman Carey and David Clampitt's paper ''Aspects of well-formed scales'', is sometimes used to equivalently describe the above definitions, and is used in academic research.
 === Single-period and multi-period MOS scales ===
@@ Line 64: / Line 158: @@
 MOS scales in which the equivalence interval is a multiple of the period (or alternatively, the step pattern repeats multiple times within the equivalence interval), is commonly called a '''multi-MOS''' or '''multi-period MOS'''. This is to distinguish them from what Wilson had defined, called '''strict MOS''' or '''single-period MOS'''.
-== Notation ==
+An alternate definition of a multi-period MOS scale is a MOS scale in which the quantities of large and small steps are ''not'' coprime.
-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.
+== 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. 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.
-=== Naming ===
-{{Main|MOS naming}}
 Although the most unambiguous way to refer to a MOS scale is by its scale signature, several naming schemes have been created that assign unique names to them. For a discussions on such names, see [[MOS naming]].
 == Properties ==
-The definition of a moment-of-symmetry scale has several equivalences to other concepts:
-*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.
-*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.
-*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.
-== Applications ==
+=== Step ratio ===
+{{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.
+===Relationship between MOS scales===
+{{Main|Operations on MOSes}}{{See also|Recursive structure of MOS scales}}{{See also|MOS scale family tree}}
+=== Advanced properties ===
-== Non-tuning applications ==
+==Non-tuning applications==
@@ Line 87: / Line 186: @@
 <original stuff below here>
-==Example: the diatonic scale==
+==History and terminology ==
-The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L 2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) An important property of MOS scales is that all the intervals come in two sizes: major and minor seconds, major and minor thirds, perfect and augmented fourths, perfect and diminished fifths, etc. This is not true for something like the melodic minor scale (LsLLLLs), which has three kinds of fifths: perfect, diminished and augmented. Therefore, the melodic minor scale is not an MOS scale.
-==Definition==
-There are several equivalent definitions of MOS scales:
-#
-#Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or L &minus; s = c)
-#Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line ''y'' = ''p''/''q''*''x'' that stays as close to the line ''y'' = ''p''/''q''*''x'' without crossing it.)
-While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]].
-==History and terminology==
 The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [http://anaphoria.com/mos.PDF ''Moments of Symmetry'']. There is also an introduction by Kraig Grady here: [http://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].
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 As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where ''s'' = 0. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. The [[TAMNAMS #Step ratio spectrum|TAMNAMS]] system has names for both specific ratios and ranges of ratios.
-==Step ratio spectrum==
-The melodic sound of a MOS is not just affected by the tuning of its intervals, but by the sizes of its steps. MOSes with L more similar to s sound smoother and more mellow. MOSes with L much larger than s sound jagged and dramatic. The ''step ratio'', the ratio between the sizes of L and s, is thus important to the sound of the scale.
-An in-depth analysis of this can be found at [[Step ratio]].
+== Properties==
-==Properties==
 ===Basic properties===
 *Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are (''a'' + ''b'')L as (generated by generators of soft-of-basic ''a''L''b''s) and ''a''L (''a'' + ''b'')s (generated by generators of hard-of-basic ''a''L''b''s).
 *Every MOS scale (with a specified [[equave]] ''E''), excluding ''a''L ''a''s⟨''E''⟩, has a ''parent MOS''. If ''a'' > ''b'', the parent of ''a''L ''b''s is min(''a'', ''b'')L|''a'' &minus; ''b''|s; if ''a'' < ''b'', the parent of ''a''L ''b''s is |''a'' &minus; ''b''|L min(''a'', ''b'')s.
-===Advanced discussion===
+===Advanced discussion ===
 See:
 *[[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.
@@ Line 124: / Line 212: @@
 *[[MOS Cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.
 *[[Operations on MOSes]]
-==As applied to rhythms==
+== As applied to rhythms==
 {{Main| MOS rhythm }}
@@ Line 137: / Line 225: @@
-==See also==
+== See also==
 *[[Diamond-mos notation]], a microtonal notation system focussed on MOS scales
 *[[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]
@@ Line 143: / Line 231: @@
 *[[Gallery of MOS patterns]]<!-- sort order in category: this page shows above A -->
-== External links ==
+==External links==
-* The Wilson Archives on moment-of-symmetry scales: https://anaphoria.com/wilsonintroMOS.html
+*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