MOS scale: Difference between revisions

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{{Beginner|Mathematics of MOS}}

An '''MOS''' (sometimes '''mos'''; originally pronounced "em-oh-ess," but sometimes also pronounced "moss"; plural '''MOSes''' or '''mosses''') or '''moment of symmetry''' is a [[periodic scale]] which satisfies one of the following equivalent properties:  

# [[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.) For example, in the [[diatonic scale]], ascending by two steps can give you a major third tuned to 400c in 12edo or a minor third tuned to 300c in 12edo, but no other intervals.

# [[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.

While each characterization has a generalization to scale structures with more step sizes, the 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]].

See the [[catalog of MOS]] for a collection of MOS scales.  

The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (diatonic semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L 2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) The melodic minor scale, which is not a mode of the diatonic scale, (LsLLLLs) is not a MOS since it has three kinds of fifths: perfect, diminished, and augmented, violating the maximum variety 2 condition above.  

In general, specifying a number of large steps and small steps in a MOS scale is enough to uniquely identify the scale. (This is most evident through the "mode of a Christoffel word" definition above.)

== History and terminology ==

The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [https://anaphoria.com/mos.pdf ''Moments of Symmetry'']. There is also an introduction by [[Kraig Grady]] here: [https://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].

Sometimes, scales are defined with respect to a period and an additional "[[equivalence interval]]", considered to be 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 interesting work has been done on scales 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]]s used in traditional [[Japanese music]], where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.

As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where {{nowrap|''s'' {{=}} 0}}. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. The [[TAMNAMS#Step ratio spectrum|TAMNAMS]] system has names for both specific ratios and ranges of ratios.

Any MOS can be clearly and uniquely specified by giving its [[signature]], i.e. the number of small and large steps, which is typically notated e.g. "5L&nbsp;2s," and its equave. 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 [[5L 2s|diatonic]] or [[2L 5s|anti-diatonic]]).  

By default, the [[equave]] of a mos ''a''L&nbsp;''b''s 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>&amp;#x27E8;</code> and <code>&amp;#x27E9;</code>) is recommended; using greater-than and less-than signs ("&lt;equave&gt;") can also be done, but this can conflict with HTML and other uses of these symbols.

Several naming systems have also been proposed for MOSes, which can be seen at [[MOS naming]].

The melodic sound of a MOS is not just affected by the tuning of its intervals, but by the sizes of its steps. MOSes with ''L'' more similar to ''s'' sound smoother and more mellow. MOSes with ''L'' much larger than ''s'' sound jagged and dramatic. The ''step ratio'', the ratio between the sizes of ''L'' and ''s'', is thus important to the sound of the scale.

An in-depth analysis of this can be found at [[Step ratio]].

* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[patent val]]s) while simultaneously also being used to define the {{nowrap|''px''L (''py'' &minus; ''px'')s}} MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)

* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament'''*''', we can say that any {{w|natural number|natural}}-coefficient {{w|linear combination}} of vals {{val| ''X'' ...}} and {{val| ''Y'' ...}} (where {{nowrap|''X'' &lt; ''Y''}}) corresponds uniquely to a tuning of the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap|gcd(''a'', ''b'') {{=}} 1}}, because if {{nowrap|''k'' {{=}} gcd(''a'', ''b'') &gt; 1}} then the val {{nowrap|''a''{{val| ''X'' ...}} + ''b''{{val| ''Y'' ...}}}} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the [[Wikipedia:Rational number|rational]] ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).

: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying {{nowrap|[[step ratio]] {{=}} (''a'' + ''b'')/''b'' &ge; 1}} for [[Wikipedia:Natural number|natural]] ''a'' and ''b'', where if {{nowrap|''b'' {{=}} 0}} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is '''important to note''' that the correspondence to the {{nowrap|''X'' &amp; ''Y''}} rank 2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where {{nowrap|''X'' &amp; ''Y''}} describe a contorted temperament on the subgroup given. An example is the {{nowrap|5 &amp; 19}} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L&nbsp;14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.</ref>

* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L&nbsp;''b''s are {{nowrap|(''a'' + ''b'')L ''a''s}} (generated by generators of soft-of-basic ''a''L&nbsp;''b''s) and {{nowrap|''a''L (''a'' + ''b'')s}} (generated by generators of hard-of-basic ''a''L''&nbsp;b''s).

* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.

** [[User:Contribution/Gallery of MOS patterns (113 - 128 units)|113&ndash;128 units]]