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; the integers in this case formalize the notion of "scale degrees." 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\text{ implies }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) ∩ 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.

The set {s[i] | i∈ℤ} generates a group G, the **group of the scale**; this is a free, finitely generated subgroup of the reals ℝ. The **rank of the scale** is the rank of G.

**Epimorphic**: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by [[Yves Hellegouarch]].

**[[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 distributional evenness.

**Convexity**: The scale is [[Convex scale|convex]] if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. This entails that the equivalence classes of the notes modulo the repetition interval **O** is a [[http://en.wikipedia.org/wiki/Convex_lattice_polytope|ℤ-polytope]] in the lattice defined by a basis for G mod **O**.

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; the integers in this case formalize the notion of &quot;scale degrees.&quot; 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]]&lt;br/&gt;
(1)\ s[0] = 0&lt;br/&gt;[[math]]
 --><script type="math/tex">(1)\ s[0] = 0</script><!-- ws:end:WikiTextMathRule:0 --><br />
<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
(2)\ s[i + P] = s[i] + O&lt;br/&gt;[[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 />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
(3)\ i &lt; j\text{ implies }s[i] &lt; s[j]&lt;br/&gt;[[math]]
 --><script type="math/tex">(3)\ i < j\text{ implies }s[i] < s[j]</script><!-- ws:end:WikiTextMathRule:2 --><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) ∩ 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 &lt; 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 />
The set {s[i] | i∈ℤ} generates a group G, the <strong>group of the scale</strong>; this is a free, finitely generated subgroup of the reals ℝ. The <strong>rank of the scale</strong> is the rank of G.<br />
<br />
<strong>Epimorphic</strong>: If there exists a homomorphism h:G⟶ℤ for which s is a transversal, so that h(s[i]) = i, then s is weakly epimorphic with the homomorphism h. If s is monotone and weakly epimorphic, it is epimorphic. An important special case is where G is a JI group and h is a val. Epimorphic scales in this restricted sense were apparently first considered by <a class="wiki_link" href="/Yves%20Hellegouarch">Yves Hellegouarch</a>.<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 distributional evenness.<br />
<br />
<strong>Convexity</strong>: The scale is <a class="wiki_link" href="/Convex%20scale">convex</a> if every convex combination of notes, meaning every ℕ-linear combination of scale notes, is a scale note. This entails that the equivalence classes of the notes modulo the repetition interval <strong>O</strong> is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Convex_lattice_polytope" rel="nofollow">ℤ-polytope</a> in the lattice defined by a basis for G mod <strong>O</strong>.</body></html>