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.

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 (up to rotation) 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). 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...

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. Hardness is usually given as L/s, and can fall anywhere between 1 and positive infinity. A hardness value of 1 is called equalized, and positive infinity is called collapsed.

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.

Examples for 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
pentic, pentatonic	2L3s	LsLss	2\5 (480c)	1\2 (600c)
onyx	1L6s	Lssssss	1\7 (171.4c)	0\6 (0c)		One temperament interpretation is Porcupine[7].
antidiatonic	2L5s	LssLsss	3\7 (514.3c)	1\2 (600c)		When using a very flat fifth, reverses the interval qualities of diatonic. One temperament interpretation is Mavila[7].
mosh	3L4s	LsLsLss	2\7 (342.9c)	1\3 (400c)		Neutral thirds generate this MOS.
smitonic	4L3s	LLsLsLs	2\7 (342.9c)	1\3 (400c)		Sounds like a brighter, stretched version of the diatonic scale. One temperament interpretation is Orgone[7].
diatonic	5L2s	LLLsLLs	4\7 (685.7c)	3\5 (720c)
pine	7L1s	LLLLLLLs	1\8 (150c)	1\7 (171.4c)		One temperament interpretation is Porcupine[8].
oneirotonic	5L3s	LLsLLsLs	3\8 (450c)	2\5 (480c)		Sounds like a darker, compressed version of the diatonic scale.
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 Mavila[9]
pentawood, blackwood	5L5s	LsLsLsLsLs	1\10 (120c)	0\5 (0c)	1\5 (240c)	One temperament interpretation is Blackwood[10].