Periodic scale
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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 numbers with values in cents. In this case, a periodic scale **s** has a nonzero quasiperiod **P** and repetition interval **O** satisfying the following conditions
[[math]]
(1) s[0] = 0
[[math]]
[[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. Neither Scala nor the above definition assumes that the scales are [[http://en.wikipedia.org/wiki/Monotonic_function|monotonically strictly increasing]], but this condition, giving a **monotone periodic scale**, is often important to add:
[[math]] (3) i < j[[math]] implies [[math]]s[i] < s[j] [[math]]
We may define an important function **class(i)** on the integers which gives the //generic intervals// of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have:
**Constant Structure**: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by [[http://en.wikipedia.org/wiki/Erv_Wilson|Erv Wilson]]) means that i<>j implies class(i) intersect class(j) = {}. In academic music theory, this is called the //partitioning property//.
**[[http://en.wikipedia.org/wiki/Rothenberg_propriety|Propriety]]** : If s is monotone, and if i <= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called //coherence//. Note that strict propriety implies constant structure.
**Epimorphic**: If the image of s consists of rational numbers, that is, if s[i] is rational for every i, and if there exists a [[Vals and Tuning Space|val]] V such that V(s[i]) = i for every integer i, then s is weakly epimorphic with val V. If s is monotone, then s is simply called epimorphic. Weakly epimorphic also implies constant structure, but not propriety.
**[[http://en.wikipedia.org/wiki/Myhill%27s_property|Myhill's property]]** : A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the **trivalence property**. If every class has less than three elements, it has the property of [[http://en.wikipedia.org/wiki/Maximal_evenness|maximal evenness]].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 numbers with values in cents. In this case, a periodic scale <strong>s</strong> has a nonzero quasiperiod <strong>P</strong> and repetition interval <strong>O</strong> satisfying the following conditions<br />
<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
(1) s[0] = 0<br/>[[math]]
--><script type="math/tex"> (1) s[0] = 0</script><!-- ws:end:WikiTextMathRule:0 --> <br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
(2) s[i + P] = s[i] + O<br/>[[math]]
--><script type="math/tex"> (2) s[i + P] = s[i] + O</script><!-- ws:end:WikiTextMathRule:1 --> <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">monotonically strictly increasing</a>, but this condition, giving a <strong>monotone periodic scale</strong>, is often important to add:<br />
<br />
<a class="wiki_link" href="/math">math</a> (3) i < j<a class="wiki_link" href="/math">math</a> implies <a class="wiki_link" href="/math">math</a>s[i] < s[j] <a class="wiki_link" href="/math">math</a><br />
<br />
We may define an important function <strong>class(i)</strong> on the integers which gives the <em>generic intervals</em> of a periodic scale. This is defined by s[j] - s[i] is in class(k) if j - i = k. Since s is quasiperiodic, class(nP) consists only of {nO}, but the rest define sets of numbers in terms of which we can define some important scale properties. In particular we have:<br />
<br />
<strong>Constant Structure</strong>: If interval classes are disjoint, then the scale is a constant structure. In other words, constant structure (a term coined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Erv_Wilson" rel="nofollow">Erv Wilson</a>) means that i<>j implies class(i) intersect class(j) = {}. In academic music theory, this is called the <em>partitioning property</em>.<br />
<br />
<strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Propriety</a></strong> : If s is monotone, and if i <= j implies every element in class(i) is less than or equal to every element in class(j), then s is (Rothenberg) proper. If i < j implies every element in class(i) is strictly less than every element in class(j), then s is strictly proper. In academic music theory circles, strict propriety is most often called <em>coherence</em>. Note that strict propriety implies constant structure. <br />
<br />
<strong>Epimorphic</strong>: If the image of s consists of rational numbers, that is, if s[i] is rational for every i, and if there exists a <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> V such that V(s[i]) = i for every integer i, then s is weakly epimorphic with val V. If s is monotone, then s is simply called epimorphic. Weakly epimorphic also implies constant structure, but not propriety.<br />
<br />
<strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a></strong> : A monotone scale in which every class but classes nP have exactly two elements has Myhill's property. If every such class has exactly three elements, it has the <strong>trivalence property</strong>. If every class has less than three elements, it has the property of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Maximal_evenness" rel="nofollow">maximal evenness</a>.</body></html>