Mathematics of MOS: Difference between revisions

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An MOS scale consists of:

1. A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)

# A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)

# A generator "g" (of any size, for example 700 cents in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period

# No more than two sizes of scale steps (Large and small, often written "L" and "s")

# Where ''each'' number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period except in such cases as an ET.

# The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal.

2. ~~A generator "g" (~~of ~~any size, for example 700 cents in 12edo) which~~ is ~~added repeatedly to make~~ a ~~chain~~ of scale steps~~, starting from~~ the ~~unison~~ or ~~0 cents scale step,~~ and ~~then reducing to within~~ the period

Condition 4 is [[Wikipedia:Myhill's property|Myhill's property]] where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[Wikipedia:Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period.

3. No more than two sizes of ~~scale~~ steps (~~Large~~ and ~~small~~, ~~often written "L"~~ and ~~"s"~~)

These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.

=== Characterizations ===

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 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.)

~~4. Where ''~~each~~'' number of scale steps, or generic interval, within the~~ scale ~~occurs in no~~ more ~~than two different~~ sizes, ~~and in exactly two if the interval is not a multiple of~~ the ~~period except in such cases as an ET.~~

While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:

# Maximum variety n

~~5. The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS~~ are ~~legal.~~

# [[Generator-offset property]]

# [[Distributional evenness]]

~~Condition Four is~~ [[~~Wikipedia:Myhill's property|Myhill's~~ property]] ~~where, as a~~ [[~~periodic scale~~]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[~~Wikipedia:Continued_fraction|convergent or semiconvergent~~]] ~~of the ratio g/P of the generator and the period.~~

# [[Balance]] 1

# [[Billiard scale]]s

~~These conditions entail that the generated~~ scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.

== Properties ==

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 An MOS scale consists of:
-. A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)
+# A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)
+# A generator "g" (of any size, for example 700 cents in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period
+# No more than two sizes of scale steps (Large and small, often written "L" and "s")
+# Where ''each'' number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period except in such cases as an ET.
+# The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal.
-. A generator "g" (of any size, for example 700 cents in 12edo) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period
+Condition 4 is [[Wikipedia:Myhill's property|Myhill's property]] where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[Wikipedia:Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period.
-. No more than two sizes of scale steps (Large and small, often written "L" and "s")
+These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.
+=== Characterizations ===
+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 L &minus; 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.)
-. Where ''each'' number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period except in such cases as an ET.
+While each characterization has a generalization to scale structures with more step sizes, the generalizations are no longer equivalent:
+# Maximum variety n
-. The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal.
+# [[Generator-offset property]]
+# [[Distributional evenness]]
-Condition Four is [[Wikipedia:Myhill's property|Myhill's property]] where, as a [[periodic scale]], the scale has every generic interval aside from the initial unison interval and intervals some number of periods from it having exactly two specific intervals. Another characterization of when a generated scale is a MOS is that the number of scale steps is the denominator of a [[Wikipedia:Continued_fraction|convergent or semiconvergent]] of the ratio g/P of the generator and the period.
+# [[Balance]] 1
+# [[Billiard scale]]s
-These conditions entail that the generated scale has exactly two sizes of steps when sorted into ascending order of size, and usually that latter condition suffices to define a MOS. However, when the generator is a rational fraction of the period and the number of steps is more than half of the total possible, a generated scale can have only two sizes of steps and the pseudo-Myhill property, meaning that not all non-unison classes have only two specific intervals.
 == Properties ==