MOS

A MOS (or mos, or moment of symmetry scale) is a scale where every step is either small or large (with no in-between), and the same is true with any interval formed by two adjacent steps (a "2-step"), etc. Any multiple of the period (which is usually an octave or a fraction thereof) has only one size.[1]

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 MOS form of the 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, and it can be shown that there is a unique scale (counting rotations as the same scale) that meets the MOS criteria with a given number of large and small steps. For example, 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.

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 and octave-reducing (or more generally, period-reducing).[2] 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 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...

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 2221221, so it has hardness 2/1.
• 17edo diatonic is 3331331, so it has hardness 3/1.
• 19edo diatonic is 3332332, so it has hardness 3/2.
• The equalized tuning is 7edo (1111111).
• The collapsed tuning is 5edo (1110110).

Table of MOS scales

Table of some 2/1-equivalent MOSes. Temperaments are capitalized and can be found in the List of regular temperaments. Common MOSes are highlighted.

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. Sounds like a brighter, stretched version of the diatonic scale. 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).
tetrawood	4L4s	LsLsLsLs	1\8 (150c)	1\4 (300c)	1\4 (300c)	Exists in 12edo; often called the diminished scale.
oneirotonic	5L3s	LLsLLsLs	3\8 (450c)	2\5 (480c)		Sounds like a darker, compressed version of the diatonic scale.
ekic	6L2s	LLLsLLLs	1\8 (150c)	1\6 (200c)	1\2 (600c)	The 22edo (hardness 3/2) tuning is associated with Hedgehog temperament.
pine	7L1s	LLLLLLLs	1\8 (150c)	1\7 (171.4c)		One temperament interpretation is Porcupine[8].
9-note MOSses
tcherepnin	3L6s	LssLssLss	1\9 (133.3c)	1\3 (400c)	1\3 (400c)	Exists in 12edo; associated with Augmented temperament.
gramitonic	4L5s	LsLsLsLss	2\9 (266.7c)	1\4 (300c)		So named because the generator is a grave minor third. Associated with Orwell temperament.
semiquartal	5L4s	LLsLsLsLs	2\9 (266.7c)	1\5 (240c)		So named because the generator is half a fourth.
armotonic	7L2s	LLLLsLLLs	5\9 (666.7c)	4\7 (685.7c)		Contains antidiatonic as a subset. One temperament interpretation is Mabilic[9].
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 and 46edo.
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.
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
-	4L7s	LsLssLssLss	3\11 (327.3c)	1\4 (300c)		The 11-note MOS associated with Kleismic/Cata and Orgone, both accurate temperaments.
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.
orgotonic	11L4s	LLLsLLLsLLLsLLs	4\15 (320c)	3\11 (327.3c)		The MOS unambiguously interpreted as Orgone[15].
kleistonic	15L4s	LLLLsLLLLsLLLLsLLLs	7\19 (315.8c)	4\15 (320c)		The MOS unambiguously interpreted as Kleismic/Cata[19].