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~~, then choosing any pitch within that collection of pitches as the root of the scale~~.

# 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.

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

~~== Example: the diatonic scale ==~~

The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (diatonic semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L 2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) The melodic minor scale, which is not a mode of the diatonic scale, (LsLLLLs) is not a MOS since it has three kinds of fifths: perfect, diminished, and augmented, violating the maximum variety 2 condition above.

The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (diatonic semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L 2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) The melodic minor scale (LsLLLLs) is not a MOS since it has three kinds of fifths: perfect, diminished, and augmented, violating the maximum variety 2 condition above.

In general, specifying a number of large steps and small steps in a MOS scale is enough to uniquely identify the scale. (This is most evident through the "mode of a Christoffel word" definition above.)

== History and terminology ==

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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, then choosing any pitch within that collection of pitches as the root of the scale.
+# 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.
 # 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]]. See the [[catalog of MOS]] for a collection of MOS scales.
-== Example: the diatonic scale ==
+The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (diatonic semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L&nbsp;2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) The melodic minor scale, which is not a mode of the diatonic scale, (LsLLLLs) is not a MOS since it has three kinds of fifths: perfect, diminished, and augmented, violating the maximum variety 2 condition above.
-The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (diatonic semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L&nbsp;2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) The melodic minor scale (LsLLLLs) is not a MOS since it has three kinds of fifths: perfect, diminished, and augmented, violating the maximum variety 2 condition above.
+In general, specifying a number of large steps and small steps in a MOS scale is enough to uniquely identify the scale. (This is most evident through the "mode of a Christoffel word" definition above.)
 == History and terminology ==