MOS scale: Difference between revisions

A '''moment of symmetry''' ('''MOS''' or '''mos'''<ref group="note">The acronym "MOS" is generally pronounced ''em-oh-ess'', while the {{w|anacronym}} "mos", more common in informal and experimental settings, is generally pronounced  ''moss''. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.</ref>) '''scale''' is a [[periodic scale]] with two properties. 1) It has only two step sizes, where size means the size in cents. 2) not just the steps (i.e. all the 2nds) but also all the 3rds, all the 4ths, and so on (i.e. every [[interval class]]) occur in only two sizes. However, the octave occurs in only one size.

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

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

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.

 

 

== 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&nbsp;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...  

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#Step ratio spectrum|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&nbsp;2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.

|[[Meantone]] or [[Schismatic]]

|[[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&nbsp;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&nbsp;2b" is used (which could refer to either diatonic or [[2L 5s|anti-diatonic]] = 2L 5s).

By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x26;#x27E8;</code> and <code>&#x26;#x27E9;</code>) is recommended; using greater-than and less-than signs ("&#x3C;equave&#x3E;") 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]].

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'''. 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 [[Distributional evenness|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<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. 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 [[Pentatonic|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:

# [[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.)

# [[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.

# 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 [[Balanced word|balance]] and [[distributional evenness]] provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see [[Mathematics of MOS]].