MOS

The comparison of non-MOS and MOS diatonic scales.

A MOS (or mos, or moment of symmetry scale) is a member of a family of scales which generalize properties of the diatonic and pentatonic scales within 12 equal temperament. Microtonal MOS scales are often seen as some of the easier microtonal scales to work with, because the number of distinct intervals within the scale is quite low.

As in any scale, we can define "n-step interval" to refer to one of many intervals that arise from ascending by n steps of the scale. Formally, a MOS is a scale where, for all n, there are at most two distinct sizes of n-step intervals. Any multiple of the period (which is usually an octave or a fraction thereof) has only one size.[1] MOS scales are also known as MV2 (or maximum variety 2) scales.

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

MOSses can be named according to their number of large and small steps (for example, 5L 2s for the diatonic MOS), because there is exactly one step pattern that fits the MOS criteria with any given number of small and large steps. This form of name does not specify the tuning of the MOS or which scale degree is defined as the tonic.

Examples

The most widely used MOS scale is the 12edo diatonic scale, which has five equal large steps (major seconds) and two equal small steps (minor seconds) within the octave. It can thus be notated 5L 2s. In contrast, while the melodic minor scale (LsLLLLs) has only two step sizes, it is still not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

A MOS exists for any whole number of large and small steps, for example 3L 4s (mosh), which functions as a "neutral" version of the diatonic scale, and 1L 6s (onyx), which has 1 large step and thus a very wide range of tunings.

The equave of a MOS is denoted using angle brackets: for example, 3L2s⟨3/2⟩ denotes the 3L 2s MOS pattern but using 3/2 as the interval of equivalence rather than 2/1.

Periods and generators

Every MOS scale can be generated by stacking a certain interval called the generator a number of times, then moving each note by multiples of the period (which is usually the octave) to fit within the span of one period.[2] The latter step is called period reduction. 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 produce a MOS, because there are more than 2 sizes of some interval classes.

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...

A 2/1-equivalent MOS scale aLbs always has 1\gcd(a, b) as its period. For example, 5L2s has period 1\1; 5L5s has period 1\5; 2L8s has period 1\2.

Hardness

One way to specify the tuning of a given MOS pattern (with a given equave) is hardness, which refers to the logarithmic ratio between the size of the L step versus the size of the s step. A tuning of a given MOS that has a higher hardness is harder, and one with a lower hardness is softer.

The hardness of a MOS scale is mostly associated with its melodic shape. Softer tunings (with more similar step sizes) may sound melodically smoother, softer, or more mellow. In contrast, harder tunings of the same MOS (with steps of very different sizes) may sound jagged, dramatic, or sparkly.

Hardness is usually given as L/s, and can fall anywhere between 1 and positive infinity. A hardness value of 1 is called equalized (since L = s), and positive infinity is called collapsed (since s = 0). We call hardness 2/1 the basic tuning of the MOS; the basic tuning is the smallest equal tuning that meaningfully supports the MOS scale. More generally, the basic tuning of a MOS aLbs is always (2a+b)-edo.[3]

Examples for MOS diatonic:

• 12edo diatonic is "2 2 2 1 2 2 1", so it has hardness 2/1. File:Diatonic basic hardness demo.mp3
• 17edo diatonic is "3 3 3 1 3 3 1", so it has hardness 3/1. File:Diatonic hard hardness demo.mp3
• 19edo diatonic is "3 3 3 2 3 3 2", so it has hardness 3/2. File:Diatonic soft hardness demo.mp3
• The equalized tuning is 7edo (1 1 1 1 1 1 1). File:Diatonic equalized hardness demo.mp3
• The collapsed tuning is 5edo (1 1 1 0 1 1 0). File:Diatonic collapsed hardness demo.mp3

Table of MOS scales

The following table lists common MOS scales grouped by scale size, along with their step patterns and associated temperaments. All temperaments mentioned in the table be found in the List of regular temperaments. All names in the first column do not specify anything about tuning or which scale degree is the tonic. Common MOSes are highlighted.

Table of some 2/1-equivalent MOSes.

Name	aLbs	Brightest mode	Equalized (softest) gen.	Collapsed (hardest) gen.	Period (1\1 (1200c) unless otherwise stated)	Description
5-note MOSses
pentic	2L3s	LsLss	2\5 (480c)	1\2 (600c)		Called "pentatonic" in 12edo music theory. Five-note subset of both MOS diatonic and antidiatonic.
antipentic	3L2s	LLsLs	2\5 (480c)	1\3 (400c)		Five-note subset of both oneirotonic and checkertonic.
manual	4L1s	LLLLs	1\5 (240c)	1\4 (300c)		Five-note subset of both semiquartal and gramitonic.
6-note MOSses
machinoid	5L1s	LLLLLs	1\6 (200c)	1\5 (240c)		Some temperament interpretations: Machine[6], Gorgo[6], Slendric[6].
7-note MOSses
onyx	1L6s	Lssssss	1\7 (171.4c)	0\6 (0c)		One temperament interpretation is Porcupine[7].
antidiatonic	2L5s	LssLsss	3\7 (685.7c)	1\2 (600c)		When using a very flat fifth, reverses the interval qualities of diatonic. One temperament interpretation is Mabilic[7].
mosh	3L4s	LsLsLss	2\7 (342.9c)	1\3 (400c)		Neutral thirds generate this MOS.
smitonic	4L3s	LLsLsLs	2\7 (342.9c)	1\4 (300c)		So named because the generator is a sharp minor third. Inthar finds that it sounds like a brighter, stretched version of the diatonic scale. In fact, it can be seen as a "warped" diatonic scale, because it can be found by replacing one large step of diatonic with a small step. One temperament interpretation is Orgone[7].
(MOS) diatonic	5L2s	LLLsLLs	4\7 (685.7c)	3\5 (720c)
arch(a)eotonic	6L1s	LLLLLLs	1\7 (171.4c)	1\6 (200c)		Some temperament interpretations: Tetracot[7], Didacus[7].
8-note MOSses
checkertonic	3L5s	LsLssLss	3\8 (450c)	1\3 (400c)		Somewhat like a stretched tcherepnin scale. Associated with Squares (when relatively hard) and Sensi (hardness 3/2 or softer). In softer tunings, it is the most consonant 8-tone MOS, corresponding to the Sensi interpretation.
tetrawood	4L4s	LsLsLsLs	1\8 (150c)	1\4 (300c)	1\4 (300c)	Exists in 12edo; often called the "diminished" or "octatonic" scale in 12edo theory.[4]
oneirotonic	5L3s	LLsLLsLs	3\8 (450c)	2\5 (480c)		Sounds like a darker, compressed version of the diatonic scale according to Inthar.
ekic	6L2s	LLLsLLLs	1\8 (150c)	1\6 (200c)	1\2 (600c)	The 22edo (step ratio L:s = 3:2) tuning is associated with Hedgehog temperament. Lumi Pakkanen has described the sound of this MOS as "dreamy, yet oppressing", especially when tuned to step ratio 3:1.[5]
pine	7L1s	LLLLLLLs	1\8 (150c)	1\7 (171.4c)
9-note MOSses
tcherepnin	3L6s	LssLssLss	1\9 (133.3c)	1\3 (400c)	1\3 (400c)	Exists in 12edo; associated with Augmented temperament, which collapses the circle of just (5/4) major thirds into one augmented chord formed from 400c major thirds.
gramitonic	4L5s	LsLsLsLss	2\9 (266.7c)	1\4 (300c)		So named because the generator is a grave minor third. Associated with Orwell temperament. One of the more consonant 9-note MOSses, especially when tuned with a step ratio 4:3 < L:s < 3:2.
semiquartal	5L4s	LLsLsLsLs	2\9 (266.7c)	1\5 (240c)		So named because the generator is half a fourth. One of the more consonant 9-note MOSses.
armotonic	7L2s	LLLLsLLLs	5\9 (666.7c)	4\7 (685.7c)		Contains antidiatonic as a subset. One temperament interpretation is Mabilic[9], or it can be seen in lower complexity as Mavila[9], where three just perfect fourths stack to a just (5/2) major tenth.
subneutralic	8L1s	LLLLLLLLs	1\9 (133.3c)	1\8 (150c)		Has tunings that split the perfect fifth into 5 equal parts, e.g. in 17edo.
10-note MOSses
jaric	2L8s	LssssLssss	1\10 (120c)	1\2 (600c)	1\2 (600c)	So named because of Pajara[10] and Injera[10] interpretations. Interpreted as Diaschismic[10] in 34edo, 46edo, and 58edo.
pentawood	5L5s	LsLsLsLsLs	1\10 (120c)	0\5 (0c)	1\5 (240c)	One temperament interpretation is Blackwood[10]. Often called the Blackwood scale because composer Easley Blackwood was one of the first to use it. Contains either a major or minor triad on every note, and thus can be seen as having similar properties to MOSdiatonic in 5n-edos.
dicoid	7L3s	LLLsLLsLLs	3\10 (360c)	2\7 (342.9c)		The 10-note MOS generated by neutral thirds. So named because of the exotemperament Dichotic.
taric	8L2s	LLLLsLLLLs	1\10 (120c)	1\8 (150c)	1\2 (600c)	Generated by an oneirotonic generator (3\5 to 5\8). Named after Hindi for 18 (aṭhārah), because 18edo is the basic tuning.
sinatonic	9L1s	LLLLLLLLLs	1\10 (120c)	1\9 (133.3c)		So named because of the "sinaic" generator (named after ibn Sina), which is 1/4 of a perfect fourth.
Larger MOSses
slentonic	5L6s	sLsLsLsLsLs	2\11 (218.2c)	1\5 (240c)		The 11-note MOS of Slendric.
p-chro smitonic	4L7s	LsLssLssLss	3\11 (327.3c)	1\4 (300c)		The 11-note MOS associated with Kleismic/Cata and Orgone, both accurate temperaments. Older material may call this scale "kleistonic."
p-chromatic	5L7s	LsLsLssLsLss	7\12 (700c)	3\5 (720c)		The chromatic scale generated by sharp-of-12edo fifths. Superpyth[12] is a particularly interesting interpretation.
hexawood	6L6s	LsLsLsLsLsLs	1\12 (100c)	1\6 (200c)	1\6 (200c)	A "straddle-fifth chromatic scale", as it can be constructed by stacking alternating flat and sharp fifths as long as they stack to 1\6, in e.g. 18edo.
m-chromatic	7L5s	LLsLsLLsLsLs	7\12 (700c)	4\7 (685.7c)		The chromatic scale generated by flat-of-12edo fifths. Used in many 17th-century keyboards and still used in some church organs.
telluric	10L2s	LLLLLsLLLLLs	1\12 (100c)	1\10 (120c)	1\2 (600c)	Commonly interpreted as Pajara[12] or Diaschismic[12].
heptawood	7L7s	LsLsLsLsLsLsLs	1\14 (85.7c)	0\7 (0c)	1\7 (171.4c)	Two offset rings of 7edo fifths; the 7edo analogue of the blackwood MOS.
	11L4s	LLLsLLLsLLLsLLs	4\15 (320c)	3\11 (327.3c)		The first MOS unambiguously interpreted as Orgone.
	15L4s	LLLLsLLLLsLLLLsLLLs	7\19 (315.8c)	4\15 (320c)		The first MOS unambiguously interpreted as Kleismic.