MOS scale: Difference between revisions

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 {{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.

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

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.

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 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 ''b''s) and {{nowrap| ''a''L (''a'' + ''b'')s }} (generated by generators of hard-of-basic ''a''L''&nbsp;b''s).

* 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 }}.

* [[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".