Periodic scale: Difference between revisions

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== Mathematical definition ==

A periodic scale may be defined in mathematical language as a type of [[Wikipedia: Quasiperiodic function|quasiperiodic function]] from the [[Wikipedia: Integer|integers]] to musical intervals, or in layman's terms, a "table" that maps integers (which formalize the notion of "scale degrees") to intervals given in cents (hence, an additive notation will be used, with the stacking of intervals notated by addition). In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' (the period in scale steps) and repetition interval ''O'' (the period in cents) where by adding P to the scale degree, O is always added to the resulting interval.

A periodic scale may be defined in mathematical language as a type of [[Wikipedia: Quasiperiodic function|quasiperiodic function]] from the [[Wikipedia: Integer|integers]] to musical intervals, or in layman's terms, a "table" that maps integers (which formalize the notion of "scale degrees") to intervals given in cents (hence, an additive notation will be used, with the stacking of intervals notated by addition). In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' (the period in scale steps) and repetition interval ''O'', also notated s[P] (the period in cents) where by adding P to the scale degree, O is always added to the resulting interval.

Since arbitrarily high and low pitches go beyond the [[human hearing range|range of human hearing]], this definition is a mathematical idealization, but it is much simpler to adopt the idealization than to worry about that. Neither Scala nor the above definition assumes that the scales are [[Wikipedia: Monotonic function|strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add.

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=== Modes ===

A mode, or "rotation", of a periodic scale is a scale r such that r[''i''] = s[''i'' + ''N''] - s[''N''], where ''N'' is a fixed integer; in other words, it is the same scale pattern, but starting on a different scale degree. Since s[''i'' + ''P''] - s[''P''] = s[''i''], there are only a finite number of rotations, equal to the number of notes of the scale reduced to the range of the interval of equivalence, 0 ≤ s[''i''] < ''s[P]'', which entails 0 ≤ ''i'' < ''P''.

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=== Classes ===

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 == Mathematical definition ==
-A periodic scale may be defined in mathematical language as a type of [[Wikipedia: Quasiperiodic function|quasiperiodic function]] from the [[Wikipedia: Integer|integers]] to musical intervals, or in layman's terms, a "table" that maps integers (which formalize the notion of "scale degrees") to intervals given in cents (hence, an additive notation will be used, with the stacking of intervals notated by addition). In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' (the period in scale steps) and repetition interval ''O'' (the period in cents) where by adding P to the scale degree, O is always added to the resulting interval.
+A periodic scale may be defined in mathematical language as a type of [[Wikipedia: Quasiperiodic function|quasiperiodic function]] from the [[Wikipedia: Integer|integers]] to musical intervals, or in layman's terms, a "table" that maps integers (which formalize the notion of "scale degrees") to intervals given in cents (hence, an additive notation will be used, with the stacking of intervals notated by addition). In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' (the period in scale steps) and repetition interval ''O'', also notated s[P] (the period in cents) where by adding P to the scale degree, O is always added to the resulting interval.
 Since arbitrarily high and low pitches go beyond the [[human hearing range|range of human hearing]], this definition is a mathematical idealization, but it is much simpler to adopt the idealization than to worry about that. Neither Scala nor the above definition assumes that the scales are [[Wikipedia: Monotonic function|strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add.
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 === Modes ===
 A mode, or "rotation",  of a periodic scale is a scale r such that r[''i''] = s[''i'' + ''N''] - s[''N''], where ''N'' is a fixed integer; in other words, it is the same scale pattern, but starting on a different scale degree. Since s[''i'' + ''P''] - s[''P''] = s[''i''], there are only a finite number of rotations, equal to the number of notes of the scale reduced to the range of the interval of equivalence, 0 ≤ s[''i''] &lt; ''s[P]'', which entails 0 ≤ ''i'' &lt; ''P''.
+In termso f mo
 === Classes ===