MOS scale: Difference between revisions

| de = MOS-Skala

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]] where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come 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).

{| class="wikitable"

|+ style="font-size: 105%;" | Interval classes in the 5L&nbsp;2s MOS scale

! rowspan="2" | Interval class

! colspan="2" | Small version

! colspan="2" | Large version

|-

! 2nds (1 step)

|-

! 3rds (2 steps)

| {{nowrap|1L + 1s}}

! 4ths (3 steps)

! 5ths (4 steps)

| {{nowrap|3L + 1s}}

|-

! 6ths (5 steps)

| {{nowrap|3L + 2s}}

| {{nowrap|4L + 1s}}

! 7ths (6 steps)

| {{nowrap|4L + 2s}}

| {{nowrap|5L + 1s}}

! 8ves (7 steps)

| {{nowrap|5L + 2s}}

| colspan="2" | (only one version)

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.

Other MOS scales include [[2L&nbsp;3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L&nbsp;4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).

See the [[catalog of MOS]] for other MOS scales.

== 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 {{nowrap| 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.

|+ style="font-size: 105%;" | 5L&nbsp;2s step ratios in various edos

|-

|-

| 2:1

|-

| 3:2

|-

| 4:1

== 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 {{nowrap| [[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]].

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]], 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'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. 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]]s used in traditional [[Japanese music]] (e.g. {{nowrap| 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]].

* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (''L'') step and small (''s'') step are the same size), then by definition it is an {{nowrap| ''x''L (''y'' − ''x'')s }} MOS scale, and the [[basic]] tuning where {{nowrap| ''L'' {{=}} 2''s'' }} is thus {{nowrap|(''x'' + ''y'')}}-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if {{nowrap| ''p'' > 1 }}) you are considering a "multiperiod" MOS scale.

* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[Patent val|patent vals]]) while simultaneously also being used to define the {{nowrap|''px''L (''py'' − ''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'' & ''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'' < ''Y'' }}) corresponds uniquely to a tuning of the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap| gcd(''a'', ''b'') {{=}} 1 }}, because if {{nowrap| ''k'' {{=}} gcd(''a'', ''b'') > 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 {{w|Rational number|rational}} ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).

: The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap| 1{{val| ''X'' … }} + 0{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap| 0{{val| ''X'' … }} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap| 1{{val| ''X'' … ;}} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:

: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying [[step ratio]] {{nowrap| ''r'' {{=}} (''a'' + ''b'')/''b'' ≥ 1 }} for {{w|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'' & ''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'' & ''Y'' }} describe a contorted temperament on the subgroup given. An example is the {{nowrap| 5 & 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 (with a specified [[equave]] ''Ɛ''), excluding {{nowrap|''a''L ''a''s{{angbr|''Ɛ''}} }}, has a ''parent MOS''. If {{nowrap| ''a'' > ''b'' }}, the parent of ''a''L&nbsp;''b''s is {{nowrap| ''b''L (''a'' − ''b'')s }}; if {{nowrap| ''a'' < ''b'' }}, the parent of ''a''L&nbsp;''b''s is {{nowrap| ''a''L (''b'' − ''a'')s }}.

* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.

* [[MOS diagrams]], visualizations of the MOS process.

* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]

| [[7L&nbsp;2s]]

| [[7L&nbsp;3s]]

| [[7L&nbsp;5s]]

| [[7L&nbsp;10s]]

| [[7L&nbsp;11s]]

| [[7L&nbsp;12s]]

| [[10L&nbsp;10s]]

| [[10L&nbsp;11s]]

| [[10L&nbsp;12s]]

| [[11L&nbsp;7s]]

| [[12L&nbsp;1s]]

| [[12L&nbsp;2s]]

| [[12L&nbsp;3s]]

| [[3L&nbsp;19s]]

| [[3L&nbsp;20s]]

| [[3L&nbsp;21s]]

| [[3L&nbsp;22s]]

| [[3L&nbsp;23s]]

| [[3L&nbsp;24s]]

| [[4L&nbsp;16s]]

| [[4L&nbsp;20s]]

| [[4L&nbsp;21s]]

| [[4L&nbsp;22s]]

| [[4L&nbsp;23s]]

| [[4L&nbsp;24s]]

| [[6L&nbsp;15s]]

| [[6L&nbsp;23s]]

| [[6L&nbsp;24s]]

| [[8L&nbsp;24s]]

| [[9L&nbsp;21s]]

| [[9L&nbsp;22s]]

| [[10L&nbsp;23s]]

| [[11L&nbsp;15s]]

| [[11L&nbsp;16s]]

| [[11L&nbsp;17s]]

| [[11L&nbsp;18s]]

| [[11L&nbsp;19s]]

| [[12L&nbsp;13s]]

| [[13L&nbsp;1s]]

| [[23L&nbsp;4s]]

| [[23L&nbsp;5s]]

| [[24L&nbsp;12s]]

[[7L&nbsp;34s]], [[9L&nbsp;29s]], [[12L&nbsp;29s]], [[12L&nbsp;41s]], [[13L&nbsp;14s]], [[14L&nbsp;13s]], [[17L&nbsp;14s]], [[25L&nbsp;6s]], [[41L&nbsp;12s]]

* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap| L − s }}, the "chroma".

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

* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.

* [[Operations on MOSes]]

[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}}

* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales

* [[:Category:MOS scales|Category:MOS scales]], the category including all MOS-related articles on this wiki

* [[Gallery of MOS patterns]]

[[Category:Math]]

[[Category:MOS scale| ]] <!-- Sort order in category: this page shows above A -->

[[Category:Scale]]

[[Category:Erv Wilson]]