MOS scale: Difference between revisions

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== 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.) An important property of MOS scales is that all the intervals come in two sizes: major and minor seconds, major and minor thirds, perfect and augmented fourths, perfect and diminished fifths, etc. This is not true for something like the melodic minor scale (LsLLLLs)~~, which~~ has three kinds of fifths: perfect, diminished, and augmented~~. Therefore~~, the ~~melodic minor scale is not an MOS 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 (LsLLLLs) is not a MOS since it has three kinds of fifths: perfect, diminished, and augmented, violating the maximum variety 2 condition above.

~~== Definition ==~~

~~There are several equivalent definitions of MOS scales:~~

~~# [[Maximum~~ variety~~]] 2~~

~~# [[Binary]] and has a generator~~

~~# Binary and [[distributionally even]]~~

~~# 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~~.)

== History and terminology ==

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 == 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.) An important property of MOS scales is that all the intervals come in two sizes: major and minor seconds, major and minor thirds, perfect and augmented fourths, perfect and diminished fifths, etc. This is not true for something like the melodic minor scale (LsLLLLs), which has three kinds of fifths: perfect, diminished, and augmented. Therefore, the melodic minor scale is not an MOS 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 (LsLLLLs) is not a MOS since it has three kinds of fifths: perfect, diminished, and augmented, violating the maximum variety 2 condition above.
-== Definition ==
-There are several equivalent definitions of MOS scales:
-# [[Maximum variety]] 2
-# [[Binary]] and has a generator
-# Binary and [[distributionally even]]
-# 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.)
 == History and terminology ==