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.

A moment of symmetry (MOS or mos) scale is a periodic scale where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come in only one size.

MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.

Examples

The most widely used MOS scale is the diatonic scale. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L 2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).

Interval classes in the 5L 2s MOS scale

Interval class	Small version		Large version
	Quality	Size	Quality	Size
2nds (1 step)	minor	s	major	L
3rds (2 steps)	minor	1L + 1s	major	2L
4ths (3 steps)	perfect	2L + 1s	augmented	3L
5ths (4 steps)	diminished	2L + 2s	perfect	3L + 1s
6ths (5 steps)	minor	3L + 2s	major	4L + 1s
7ths (6 steps)	minor	4L + 2s	major	5L + 1s
8ves (7 steps)	perfect	5L + 2s	(only one version)

Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

Other MOS scales include 2L 3s, where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly 4L 4s, also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).

See the catalog of MOS for other MOS scales.

Periods and generators

Every MOS scale can be generated by stacking a certain interval called the generator and octave-reducing (or more generally, period-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as C D E F G A C does not produces a MOS, because there are more than 2 sizes of each interval class.

The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53, …. However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50, ….

Step ratio spectrum

The step ratio is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The TAMNAMS system has names for specific ratios and also ranges of ratios.

When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L 2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.

5L 2s step ratios in various edos

Example edo	Step ratio	TAMNAMS name	Likely temperament interpretations
12	2:1	basic	Meantone or Schismatic
19	3:2	soft	Meantone
22	4:1	superhard	Archy or Superpyth

Naming

Every MOS can be uniquely specified by giving its signature, i.e. the number of large and small steps, which is typically notated e.g. "5L 2s,". Every possible signature corresponds to a valid MOS scale. 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 = 2L 5s).

By default, the equave of a MOS 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 been proposed for MOSes, which can be seen at MOS naming.

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, the interval at which pitch classes repeat. MOSes 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-MOSes. For example, a MOS with a half-octave period is called a 2mos, with a 1/3-octave period a 3mos, and so on. MOSes in which the equivalence interval is equal to the period are sometimes called Strict MOSes. MOSes 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 work has been done 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 (e.g. A B C E F A), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.

Equivalent definitions and generalizations

A scale is a MOS if and only if it satisfies one of the following equivalent criteria:

1. Maximum variety 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.)
2. Binary and has a generator: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.
3. Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.

Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of balance and distributional evenness provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see Mathematics of MOS.

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 r = (a + b)/b ≥ 1 for natural a and b, where if b = 0 then the step ratio is infinite, corresponding to the collapsed tuning.

• 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

Individual pages for MOS scales

L ≤ 12, s ≤ 12

Pages for MOS scales (L ≤ 12, s ≤ 12)

1L 1s	1L 2s	1L 3s	1L 4s	1L 5s	1L 6s	1L 7s	1L 8s	1L 9s	1L 10s	1L 11s	1L 12s
2L 1s	2L 2s	2L 3s	2L 4s	2L 5s	2L 6s	2L 7s	2L 8s	2L 9s	2L 10s	2L 11s	2L 12s
3L 1s	3L 2s	3L 3s	3L 4s	3L 5s	3L 6s	3L 7s	3L 8s	3L 9s	3L 10s	3L 11s	3L 12s
4L 1s	4L 2s	4L 3s	4L 4s	4L 5s	4L 6s	4L 7s	4L 8s	4L 9s	4L 10s	4L 11s	4L 12s
5L 1s	5L 2s	5L 3s	5L 4s	5L 5s	5L 6s	5L 7s	5L 8s	5L 9s	5L 10s	5L 11s	5L 12s
6L 1s	6L 2s	6L 3s	6L 4s	6L 5s	6L 6s	6L 7s	6L 8s	6L 9s	6L 10s	6L 11s	6L 12s
7L 1s	7L 2s	7L 3s	7L 4s	7L 5s	7L 6s	7L 7s	7L 8s	7L 9s	7L 10s	7L 11s	7L 12s
8L 1s	8L 2s	8L 3s	8L 4s	8L 5s	8L 6s	8L 7s	8L 8s	8L 9s	8L 10s	8L 11s	8L 12s
9L 1s	9L 2s	9L 3s	9L 4s	9L 5s	9L 6s	9L 7s	9L 8s	9L 9s	9L 10s	9L 11s	9L 12s
10L 1s	10L 2s	10L 3s	10L 4s	10L 5s	10L 6s	10L 7s	10L 8s	10L 9s	10L 10s	10L 11s	10L 12s
11L 1s	11L 2s	11L 3s	11L 4s	11L 5s	11L 6s	11L 7s	11L 8s	11L 9s	11L 10s	11L 11s	11L 12s
12L 1s	12L 2s	12L 3s	12L 4s	12L 5s	12L 6s	12L 7s	12L 8s	12L 9s	12L 10s	12L 11s	12L 12s

Larger MOS scales

7L 34s, 9L 29s, 12L 29s, 12L 41s, 13L 14s, 14L 13s, 17L 14s, 25L 6s, 41L 12s

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

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 focused on MOS scales
• Metallic MOS, an article focusing on MOS scales based on metallic means, such as phi
• MOS rhythm
• Category:MOS scales, the category including all MOS-related articles on this wiki
• Gallery of MOS patterns