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}}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]].

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

* A generator, an interval that is repeatedly stacked.
* 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, respectively.
* 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
!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.
|
|-
|2
|{{Step vis|2 5 5}}
|2 5 5
|0 2 7 12
|The next degree is at 14 edosteps. This is reduced (14 mod 12) to 2.
|
|-
|3
|{{Step vis|2 5 2 3}}
|2 5 2 3
|0 2 7 9 12
|The next degree is at 9 edosteps, but this results in three step sizes.
|
|-
|4
|{{Step vis|2 2 3 2 3}}
|2 2 3 2 3
|0 2 4 7 9 12
|The next degree is at 16 edosteps. This is reduced (16 mod 12) to 4.
|The common pentatonic scale, denoted as '''2L 3s'''.
|-
|5
|{{Step vis|2 2 3 2 2 1}}
|2 2 3 2 2 1
|0 2 4 7 9 11 12
|The next degree is at 11 edosteps, but this results in three step sizes.
|
|-
|6
|{{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 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 ===
Wilson's original definition primarily focused on the period and equivalence interval being the same.

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'''.

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.

== 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.

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 ==

=== 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==



<original stuff below here>

==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''].

Sometimes, scales are defined with respect to a period and an additional "equivalence interval," considered to be the interval at which pitch classes repeat. MOS's in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOS's'''. MOS's in which the equivalence interval is equal to the period are sometimes called '''Strict MOS's'''. MOS's in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.

With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[distributional evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt. A great deal of interesting work has been done on scales in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the pentatonics used in traditional Japanese music, where the 5 tone cycles are derived from a 7 tone MOS, which are not found in the concept of DE.

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.

== 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 ===
See:
*[[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.
**[[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.
**[[MOS scale family tree]], a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.
*[[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.
*[[MOS diagrams]], visualizations of the MOS process.
*[http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]
==Variations==
*[[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of L-s, the "chroma".
*[[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.
*[[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==
{{Main| MOS rhythm }}

M
MOS structures and thinking can be applied to the design of rhythms as well.

David Canright was the first to suggest Fibonacci Rhythms in 1/1. This led to Kraig Grady to be the first to apply MOS patterns to rhythms. Two papers on the subject can be found here:
*[http://anaphoria.com/hora.pdf A Rhythmic Application of the Horagrams] from Xenharmonikon 16
*[http://anaphoria.com/horo2.pdf More on Horogram Rhythms]
==Listen==
This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|95x95px]]


== 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]]
*[[:Category:MOS scales]], the category including all MOS-related articles on this wiki
*[[Gallery of MOS patterns]]<!-- sort order in category: this page shows above A -->

==External links==

*The Wilson Archives on moment-of-symmetry scales: https://anaphoria.com/wilsonintroMOS.html
*Erv Wilson's paper ''Moments of Symmetry'': https://anaphoria.com/mos.pdf