MOS scale: Difference between revisions

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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]] in which ~~every~~ [[interval]] ~~except for the~~ [[~~period~~]] comes in two sizes.

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 octave is present and taking pitches by travelling vertically and horizontally as close to the line from the origin to the octave as possible without going above it starting from the origin, then choosing any pitch within that collection of pitches as the root of the scale.

# Binary and [[distributionally even]], which is unhelpful as a definition (since distributional evenness is most conveniently defined in terms of MOS scales) but useful as a generalization.

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

While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]]. See the [[catalog of MOS]] for a collection of MOS scales.

== Example: the diatonic scale ==

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# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap|''L'' − ''s'' {{=}} ''c''}})

# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' {{sfrac|''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 {{nowrap|''y'' {{=}} {{sfrac|''p''|''q''}} ''x''}} that stays as close to the line without crossing it.)

~~While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]].~~

== History and terminology ==

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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]] in which every [[interval]] except for the [[period]] comes in two sizes.
+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 octave is present and taking pitches by travelling vertically and horizontally as close to the line from the origin to the octave as possible without going above it starting from the origin, then choosing any pitch within that collection of pitches as the root of the scale.
+# Binary and [[distributionally even]], which is unhelpful as a definition (since distributional evenness is most conveniently defined in terms of MOS scales) but useful as a generalization.
-See the [[catalog of MOS]] for a collection of MOS scales.
+While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]]. See the [[catalog of MOS]] for a collection of MOS scales.
 == Example: the diatonic scale ==
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 # Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap|''L'' &minus; ''s'' {{=}} ''c''}})
 # Mode of a Christoffel word. (A ''Christoffel word with rational slope'' {{sfrac|''p''|''q''}} is the unique path from (0,&nbsp;0) and (''p'',&nbsp;''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap|''y'' {{=}} {{sfrac|''p''|''q''}}&#x200A;''x''}} that stays as close to the line without crossing it.)
-While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]].
 == History and terminology ==