View source for User:Nick Vuci/Moments of Symmetry

 WORK-IN-PROGRESS AS OF 25 OCT 2025
'''Moments of Symmetry (MOS)''' are scales created by a simple procedure that generates the common [[2L 3s|pentatonic]] and [[5L 2s|diatonic]] scales, but also a wide range of novel xenharmonic scales that share similar melodic coherence and structural balance. First described by [[Erv Wilson]] in the 1970's, the concept shares fundamental similarities and is often thought of as synonymous with the concept of Well-Formed scales, as well as the more generalized concept of [[Maximum variety|MV2 scales]]. Over time, MOS have become a fundamental concept in xenharmonic theory, inspiring a wide range of musical uses, analytical approaches, and derivative concepts such as [[MODMOS]], multi-MOS, and [[MOS rhythm|MOS-based rhythm]]. 

MOS are commonly notated with either a [[Step pattern|scale signature or a step pattern]]. For example, the common major scale of 12-EDO would be notated: 

As a concrete step pattern: 2 2 1 2 2 2 1 

As an abstract step pattern: L L s L L L s 

As a scale signature: 5L 2s 

== Construction ==
<gallery mode="nolines" widths="200" heights="200">
File:1-1 MOS Construction.png|An interval is chosen to be the period (the '''1200 cent octave''' in this example). The period is represented with a circle; intervals are represented with purple lines and points radiating out from the middle which move clockwise.
File:1-2 MOS Construction.png|An interval is chosen to be a generator (the '''700 cent perfect fifth''' in this example). This is the '''[[1L 1s]]''' MOS pattern.
File:1-3 MOS Construction.png|Stacking the generator upon itself yields a scale with <u>2 step sizes</u>. This is the '''[[2L 1s]]''' MOS pattern.
File:1-4 MOS Construction.png|Stacking the generator 2 times yields a scale with 3 step sizes ('''not MOS''').
File:1-5 MOS Construction.png|Stacking the generator 3 times yields a scale with <u>2 step sizes</u>. This is the '''[[2L 3s]]''' MOS pattern.
File:1-6 MOS Construction.png|Stacking the generator five times yields a scale with 3 step sizes ('''not MOS''').
File:1-7 MOS Construction.png|Stacking the generator 6 times yields a scale with <u>2 step sizes</u>. This is the '''[[5L 2s]]''' MOS pattern.
File:700centMOS.gif|The generator sequence animated with the moments of symmetry labelled.
</gallery>

== Step Ratios ==
The [[step ratio]]—also referred to as the ''hardness''—of MOS denote the relative sizes of the large and small steps and is a key factor in classifying MOS patterns. A step ratio of 2:1, meaning the large steps are twice the size of the small steps, is considered the ''basic'' form of the MOS. When the difference between the large and small steps increases (i.e., the large step becomes larger and the small step smaller), the MOS is considered ''harder'', as the contrast in step sizes becomes more pronounced. Conversely, when the size difference decreases (i.e., the large step becomes smaller and the small step larger), the MOS is considered ''softer'', due to the more subtle contrast.

To find the equal tuning that supports a given MOS pattern at a particular hardness, simply multiply the number of each step type by its relative size and sum the results. For example, for the 5L 2s pattern at a hardness of 2:1, calculate 5×2+2×1=12, showing that 12-EDO supports this pattern. 


For a more thorough discussion on the spectrum of step ratios, please see [[TAMNAMS#Step ratio spectrum|TAMNAMS]].<gallery mode="nolines" widths="200" heights="200">
File:5L 2s hard.png|When the L:s ratio is '''3:1''', the MOS is '''hard'''. The '''hard 5L 2s''' is supported by '''17-EDO'''.
File:Basic 5L 2s.png|When the L:s ratio is '''2:1''', the MOS is '''basic'''. The '''basic 5L 2s''' is supported by '''12-EDO'''.
File:5L 2s soft.png|When the L:s ratio is '''3:2''', the MOS is '''soft'''. The '''soft 5L 2s''' is supported by '''19-EDO.'''.
</gallery>

== Spectrum of MOS ==
As a generator moves through all possible positions within a fixed period, MOS emerge in specified ranges bounded by equal divisions. This spectrum is symmetric and mirrored at the midpoint of the period, with generators larger than half the period produce the same MOS as their complements.<gallery mode="slideshow" widths="200" heights="200">
File:MOS Spectrum.gif|The entire MOS spectrum at 6-levels
File:MOS Spectrum mirror.gif|Showing the mirror-like symmetry of the half period. The same patterns repeat from 0-600 cents and 1200-600 cents. 
File:MOS Spectrum Level2.gif
File:MOS Spectrum Level3.gif
</gallery>

== Formal definitions and conditions of MOS ==
.

.

.

.

.

== See also ==

* [[Interval variety]]
* [[Step ratio]]
* [[TAMNAMS]]
* [[MODMOS]]
* [[MOS rhythm]]

== External links ==

* [https://www.anaphoria.com/wilsonintroMOS.html "Introduction to Moments of Symmetry"] in the ''[https://www.anaphoria.com/wilson.html Wilson Archives]'', hosted on [[Kraig Grady|Kraig Grady's]] website [[Anaphoria]]
* [https://www.wilsonic.co/ Wilsonic MTS-ESP and iOS apps] by [[Marcus Hobbs]]
* [https://wilson-tunings.netlify.app/moments-of-symmetry-v2 Moments of symmetry calculator (V2)] by [[Diego Villaseñor]] 
* [https://www.anaphoria.com/MOSedo-app.html Moments of Symmetry in equal divisions of the octave calculator] by [[Billy Stiltner]]
* [https://untwelve.org/static/javascript_demos/MOSring_white.html Moment of Symmetry generator demo] by [[Aaron Krister Johnson]]

== References ==