MOS scale

This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily.
The corresponding expert page for this topic is Mathematics of MOS.

An MOS (sometimes mos; originally pronounced "em-oh-ess," but sometimes also pronounced "moss"; plural MOSes or mosses) or moment of symmetry is a periodic scale in which every interval except for the period comes in two sizes.

See the catalog of MOS.

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 (diatonic 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:

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

Naming

Any MOS can be clearly specified by giving its signature, i.e. the number of small and large steps, which is typically notated e.g. "5L 2s," and its equave. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a 2b" is used (which could refer to either diatonic or anti-diatonic).

By default, the equave of a mos aL bs is assumed to be 2/1. To specify a non-octave equave, "⟨equave⟩" is placed after the signature, e.g. 4L 5s⟨3/1⟩. Using angle brackets (⟨ and ⟩) is recommended; using greater-than and less-than signs ("<equave>") can also be done, but this can conflict with HTML and other uses of these symbols.

Several naming systems have also been proposed for MOS's, which can be seen at MOS naming.

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

For every MOS scale with an octave period (which is usually the octave), if x-edo is the collapsed tuning (where the small step vanishes) and y-edo is the equalized tuning (where the large (L) step and small (s) step are the same size), then by definition it is an xL (y − x)s MOS scale, and the basic tuning where L = 2s is thus (x + y)-edo. This is also true if the period is 1\p, that is, 1 step of p-edo, which implies that x and y are divisible by p, though note that in that case (if p > 1) you are considering a "multiperiod" MOS scale.
More generally, whenever px-edo and py-edo are used to define two vals (usually but not necessarily through taking the patent vals) while simultaneously also being used to define the pxL (py − px)s MOS scale (where p is the number of periods per octave), then the px & py temperament corresponds to that MOS scale, and adding x and/or y corresponds to tuning closer to x-edo and/or y-edo respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)
For the mathematically-inclined, we can say that whenever we consider a MOS with X/p notes per period in the collapsed tuning and Y/p notes per period in the equalized tuning and p periods per tempered octave (or more generally tempered equave), and whenever we want to associate that MOS with the X & Y rank 2 temperament*, we can say that any natural-coefficient linear combination of vals ⟨X ...] and ⟨Y ...] (where X < Y) corresponds uniquely to a tuning of the X & Y rank 2 temperament between X-ET and Y-ET (inclusive) iff gcd(a, b) = 1, because if k = gcd(a, b) > 1 then the val a⟨X ...] + b⟨Y ...] has a common factor k in all of its terms, meaning it is guaranteed to be contorted. The tuning corresponding to the rational a/b is technically only unique up to (discarding of) octave stretching (or more generally equave-tempering).

The period of this temperament is 1\gcd(X, Y), and the rational a/b is very closely related to the step ratio of the corresponding MOS scale, because 1⟨X ...] + 0⟨Y ...] is the L = 1, s = 0 tuning while 0⟨X ...] + 1⟨Y ...] is the L = 1, s = 1 tuning and 1⟨X ...] + 1⟨Y ...] is the L = 2, s = 1 tuning, so that L = a + b and s = b and therefore:

1/(step ratio) = s/L = b/(a + b) implying step ratio = (a + b)/b ≥ 1 for natural a and b, where if b = 0 then the step ratio is infinite, corresponding to the collapsed tuning.^{[note 1]}

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 aL bs) and aL (a + b)s (generated by generators of hard-of-basic aL bs).
Every MOS scale (with a specified equave Ɛ ), excluding aL as⟨Ɛ ⟩, has a parent MOS. If a > b, the parent of aL bs is bL (a − b)s; if a < b, the parent of aL bs is aL (b − a)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

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.

Notes

↑ It is important to note that the correspondence to the X & Y rank 2 temperament only works in all cases if we allow the temperament to be contorted on its subgroup; alternatively, it works if we exclude cases where X & Y describe a contorted temperament on the subgroup given. An example is the 5 & 19 temperament is contorted in the 5-limit (having a generator of a semifourth, corresponding to 5L 14s), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered ~4/3) or we exclude it because of its contortion.

[1] It is important to note that the correspondence to the X & Y rank 2 temperament only works in all cases if we allow the temperament to be contorted on its subgroup; alternatively, it works if we exclude cases where X & Y describe a contorted temperament on the subgroup given. An example is the 5 & 19 temperament is contorted in the 5-limit (having a generator of a semifourth, corresponding to 5L 14s), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered ~4/3) or we exclude it because of its contortion.

[note 1]