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Erv Wilson first described the concept in 1975 in ''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.


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.
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"
{| class="wikitable"
|+Constructing a scale in 12edo using a generator of 7 edosteps
|+Constructing a scale in 12edo using a generator of 7 edosteps
!Generators added
!Step visualization
!Step visualization
!Step pattern
!Step pattern
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!Scale produced
!Scale produced
|-
|-
|1
|{{Step vis|7 5}}
|{{Step vis|7 5}}
|7 5
|7 5
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|
|
|-
|-
|2
|{{Step vis|2 5 5}}
|{{Step vis|2 5 5}}
|2 5 5
|2 5 5
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|
|
|-
|-
|3
|{{Step vis|2 5 2 3}}
|{{Step vis|2 5 2 3}}
|2 5 2 3
|2 5 2 3
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|
|
|-
|-
|4
|{{Step vis|2 2 3 2 3}}
|{{Step vis|2 2 3 2 3}}
|2 2 3 2 3
|2 2 3 2 3
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|The common pentatonic scale, denoted as '''2L 3s'''.
|The common pentatonic scale, denoted as '''2L 3s'''.
|-
|-
|5
|{{Step vis|2 2 3 2 2 1}}
|{{Step vis|2 2 3 2 2 1}}
|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}}
|{{Step vis|2 2 2 1 2 2 1}}
|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 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.
|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). The pattern shown here can continue until every scale degree of 12edo is added, producing the common chromatic scale.
==== Splitting of large steps ====
It should be noted that the intermediate steps suggest that they are also MOS scales, 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"
!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.
|}
==== Modes ====


=== Equivalent definitions ===
=== Equivalent definitions ===
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*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
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