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~~; the~~ integers ~~in this case~~ formalize the notion of "scale degrees"~~. Here, the~~ intervals ~~are~~ given in cents ~~from an arbitrary "root" note~~. In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' and repetition interval ''O'' ~~satisfying~~ the ~~following conditions~~

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. 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

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Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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|~~monotonically~~ strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add:

Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period in scale steps equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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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 == 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; the integers in this case formalize the notion of "scale degrees". Here, the intervals are given in cents from an arbitrary "root" note. In this case, a periodic scale ''s'' has a nonzero quasiperiod ''P'' and repetition interval ''O'' satisfying the following conditions
+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. 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
 <math>(1)\ s[0] = 0</math>
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 <math>(2)\ s[i + P] = s[i] + O</math>
-Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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|monotonically strictly increasing]], but this condition, giving a '''monotone periodic scale''', is often important to add:
+Scales written in the widely used [http://www.huygens-fokker.org/scala/scl_format.html Scala format] are implicitly assumed to be periodic, with the repetition interval equal to the last scale entry, and the period in scale steps equal to the number of notes (on the second line) of the scale. Informally, a periodic scale could be defined as the kind of scale a Scala .scl file is intended to denote. Of course, 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:
 <math>(3)\ i < j\text{ implies }s[i] < s[j]</math>