User:Ganaram inukshuk/MOS scale

The following is a draft for a proposed rewrite of the following page: MOS scale The primary changes are as follows: general rewrites; definition; wrangle different ways to say "mos" The original page can be compared with this page here.

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, periodic scale constructed using a generator originally invented by Erv Wilson.

Definition

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:

• A generator and an equivalence interval, called the period, which is usually the octave.
  • The generator is commonly denoted using a quantity of steps from an equal division of the octave, where both the edo and generator are coprime, meaning they do not share any common factors greater than 1.
• Two unique step sizes, called large and small, commonly denoted using the letters L and s.
  • The quantities of these steps are also coprime.

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.

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

Step visualization	Step pattern	Scale degrees	Added scale degrees	Scale produced
├──────┼────┤	7 5	0 7 12	The first scale degree is at 7 edosteps from the root.
├─┼────┼────┤	2 5 5	0 2 7 12	The next degree is at 14 edosteps. This is reduced (14 mod 12) to 2.
├─┼────┼─┼──┤	2 5 2 3	0 2 7 9 12	The next degree is at 9 edosteps, but this results in three step sizes.
├─┼─┼──┼─┼──┤	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.
├─┼─┼──┼─┼─┼┤	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.
├─┼─┼─┼┼─┼─┼┤	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.

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 xL ys. In cases where one does not wish to distinguish between step sizes, the notation xA yB can be used instead, which can either refer to xL ys or yL xs.

By default, the 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 ⟨ ⟩ (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⟨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 article: Operations on MOSes

See also: 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 frequency ratios, though the notation s:L is sometimes used to avoid division-by-zero.

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: Moments of Symmetry. There is also an introduction by Kraig Grady here: 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 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 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 aL bs are (a + b)L as (generated by generators of soft-of-basic aLbs) and aL (a + b)s (generated by generators of hard-of-basic aLbs).
• Every MOS scale (with a specified equave E), excluding aL as⟨E⟩, has a parent MOS. If a > b, the parent of aL bs is min(a, b)L|a − b|s; if a < b, the parent of aL bs is |a − 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.
• 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".
• Muddles 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 article: 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:

• A Rhythmic Application of the Horagrams from Xenharmonikon 16
• 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.

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

External links

• 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