Periodic scale
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- This revision was by author genewardsmith and made on 2010-05-14 02:58:52 UTC.
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Original Wikitext content:
A **periodic scale** may be defined in mathematical language as a type of [[http://en.wikipedia.org/wiki/Quasiperiodic_function|quasiperiodic function]] from the [[http://en.wikipedia.org/wiki/Integers|integers]] to musical intervals. Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real nunmbers with values in cents. In this case, a periodic scale s has a nonzero period P and repetition interval O satisfying the following conditions (1) s[0] = 0 (2) s[i + P] = s[i] + O 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. Neither Scala nor the above definition assumes that the scales are [[http://en.wikipedia.org/wiki/Monotonic_function|monotonicaally strictly increasing]], but this condition, giving a **monotone periodic scale**, is often important to add: (3) i<j implies s[i]<s[j]
Original HTML content:
<html><head><title>Periodic scale</title></head><body>A <strong>periodic scale</strong> may be defined in mathematical language as a type of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Quasiperiodic_function" rel="nofollow">quasiperiodic function</a> from the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integers" rel="nofollow">integers</a> to musical intervals. Musical intervals may be written either additively or multiplicatively, and we will assume an additive notation is used, and that intervals are given by positive or negative real nunmbers with values in cents. In this case, a periodic scale s has a nonzero period P and repetition interval O satisfying the following conditions<br /> <br /> (1) s[0] = 0<br /> <br /> (2) s[i + P] = s[i] + O<br /> <br /> Scales written in the widely used <a class="wiki_link_ext" href="http://www.huygens-fokker.org/scala/scl_format.html" rel="nofollow">Scala format</a> 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. Neither Scala nor the above definition assumes that the scales are <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Monotonic_function" rel="nofollow">monotonicaally strictly increasing</a>, but this condition, giving a <strong>monotone periodic scale</strong>, is often important to add:<br /> <br /> (3) i<j implies s[i]<s[j]</body></html>