MOS scale: Difference between revisions

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# [[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~~, 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.

# 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.

~~# 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~~.

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]].

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.

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 # [[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, 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.
+# 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.
-# 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.
-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]].
+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.