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...