User:Ganaram inukshuk/MOS scale

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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 (abbreviated as MOS scale, MOSS, or MOS, pronounced "em-oh-ess"; also spelled as mos, pronounced "moss"; plural MOSes or mosses) is a type of binary, periodic scale.

Definition

Erv Wilson's original definition

The concept of MOS scales were invented by Erv Wilson in 1975 in his paper Moments of Symmetry, where, given an edo of n equal divisions, a generator of g edosteps can be stacked repeatedly to divide the octave.

Current and equivalent definitions

Notation

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 the exact sizes are not specified (that is, there is no distinction as to which step is large or small, 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.

Naming

Main article: MOS naming

Properties

Applications

Non-tuning applications

<original stuff below here>

Example: the diatonic scale

The 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:

  1. Maximum variety 2
  2. Binary and has a generator
  3. Binary and distributionally even
  4. Binary and balanced (for any k, any two k-steps u and v differ by either 0 or L − s = c)
  5. 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: 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.

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

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:

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:

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

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