User:Ganaram inukshuk/MOS scale: Difference between revisions
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|changes=general rewrites; definition; wrangle different ways to say "mos" | |changes=general rewrites; definition; wrangle different ways to say "mos" | ||
}}A '''moment-of-symmetry scale''' (abbreviated as '''MOS scale''', '''MOSS''', or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOSes''' or '''mosses''') is a type of [[binary]] | }}A '''moment-of-symmetry scale''' (abbreviated as '''MOS scale''', '''MOSS''', or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOSes''' or '''mosses''') is a type of [[binary]] [[scale]]. | ||
== Definition == | == Definition == | ||
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The concept of MOS scales were invented by [[Erv Wilson]] in 1975 in his paper ''Moments of Symmetry'', where, given an [[edo]] of ''n'' equal divisions, a generator of ''g'' edosteps can be stacked repeatedly to divide the octave. | The concept of MOS scales were invented by [[Erv Wilson]] in 1975 in his paper ''Moments of Symmetry'', where, given an [[edo]] of ''n'' equal divisions, a generator of ''g'' edosteps can be stacked repeatedly to divide the octave. | ||
=== | === Equivalent definitions === | ||
The definition of a moment-of-symmetry scale has several equivalences to other concepts: | |||
*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties. | |||
*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2. | |||
*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step. | |||
=== Single-period and multi-period MOS scales === | |||
== Notation == | == Notation == | ||
A moment-of-symmetry scale of ''x'' large steps and ''y'' small steps, where ''x'' and ''y'' are whole numbers, is denoted using the [[scale signature]] ''x''L ''y''s. In cases where one does not wish to distinguish between step sizes, the notation ''x''A ''y''B can be used instead, which can either refer to ''x''L ''y''s or ''y''L ''x''s. | A moment-of-symmetry scale of ''x'' large steps and ''y'' small steps, where ''x'' and ''y'' are whole numbers, is denoted using the [[scale signature]] ''x''L ''y''s. In cases where one does not wish to distinguish between step sizes, the notation ''x''A ''y''B can be used instead, which can either refer to ''x''L ''y''s or ''y''L ''x''s. | ||
By default, the [[Equave|equivalence interval]], or equave, of a MOS scale is assumed to be the [[octave]]. In discussions regarding MOS scales with [[non-octave]] equivalence intervals, the equivalence interval can be enclosed in angle brackets of either < > (less-than and greater-than symbols) or {{Angbr| }} (Unicode symbols U+27E8 and U+27E9). Whereas "5L 2s", for example, refers to an octave-equivalent pattern of 5 large and 2 small steps, 5L 2s{{Angbr|3/1}} refers to the same pattern but with 3/1 as the equivalence interval. To avoid conflicts with HTML tags, the use of Unicode symbols is advised. | By default, the [[Equave|equivalence interval]], or equave, of a MOS scale is assumed to be the [[octave]]. In discussions regarding MOS scales with [[non-octave]] equivalence intervals, the equivalence interval can be enclosed in angle brackets of either < > (less-than and greater-than symbols) or {{Angbr| }} (Unicode symbols U+27E8 and U+27E9). Whereas "5L 2s", for example, refers to an octave-equivalent pattern of 5 large and 2 small steps, 5L 2s{{Angbr|3/1}} refers to the same pattern but with 3/1 as the equivalence interval. To avoid conflicts with HTML tags, the use of Unicode symbols is advised over the former. | ||
=== Naming === | === Naming === | ||
{{Main|MOS naming}} | {{Main|MOS naming}} | ||
Although the most unambiguous way to refer to a MOS scale is by its scale signature, several naming schemes have been created that assign unique names to them. For a discussions on such names, see [[MOS naming]]. | |||
== Properties == | == Properties == | ||
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==Definition== | ==Definition== | ||
There are several equivalent definitions of MOS scales: | There are several equivalent definitions of MOS scales: | ||
# | # | ||
#Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or L − s = c) | #Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or L − s = c) | ||
#Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line ''y'' = ''p''/''q''*''x'' that stays as close to the line ''y'' = ''p''/''q''*''x'' without crossing it.) | #Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p''/''q'' is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line ''y'' = ''p''/''q''*''x'' that stays as close to the line ''y'' = ''p''/''q''*''x'' without crossing it.) | ||