Periodic scale

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[[toc]]

=Definition= 
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. 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 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 [[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]]

==Rotations== 
By a //rotation// or mode of a periodic scale s is meant a scale r such that r[i] = s[i + **N**] - s[**N**], where **N** is a fixed integer. 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] < **O**, which entails 0 ≤ i < **P**.

==Classes== 
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.

=Scale properties= 
==[[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//.

==[[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.

==Epimorphism== 
If there exists a homomorphism h: G → ℤ 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]]. The name comes from the fact that h is an epimorphism onto ℤ.

==[[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**. Myhill's property is synonymous with **strict [[MOS]]**, though some authors prefer to identify MOS itself with Myhill's property.

==Distributional evenness== 
A monotone scale in which every class comes in exactly n elements is n-distributionally even, or **n-DE**. If n=2, then we can simply say that it is distributionally even. <span style="line-height: 1.5;">Distributional evenness is also synonymous with **[[MOS]]**, though some authors prefer a stricter definition of MOS identifying it with Myhill's property.

==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. If the quasiperiod **P** is normalized so as to be positive and minimal, this is equivalent to the condition 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**.

==[[Maximal evenness]]== 
Maximally even scales of n notes in m edo are any mode of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the "floor" function rounds down to the nearest integer.

==Numerical properties== 
[[Scale diversity]]
[[Lumma stability]]

=See also=
[[Scale properties simplified]]

Original HTML content:

<html><head><title>Periodic scale</title></head><body><!-- ws:start:WikiTextTocRule:29:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><div style="margin-left: 1em;"><a href="#Definition">Definition</a></div>
<!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><div style="margin-left: 2em;"><a href="#Definition-Rotations">Rotations</a></div>
<!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><div style="margin-left: 2em;"><a href="#Definition-Classes">Classes</a></div>
<!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --><div style="margin-left: 1em;"><a href="#Scale properties">Scale properties</a></div>
<!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><div style="margin-left: 2em;"><a href="#Scale properties-Constant Structure">Constant Structure</a></div>
<!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><div style="margin-left: 2em;"><a href="#Scale properties-Propriety">Propriety</a></div>
<!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><div style="margin-left: 2em;"><a href="#Scale properties-Epimorphism">Epimorphism</a></div>
<!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><div style="margin-left: 2em;"><a href="#Scale properties-Myhill's property">Myhill's property</a></div>
<!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><div style="margin-left: 2em;"><a href="#Scale properties-Distributional evenness">Distributional evenness</a></div>
<!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 2em;"><a href="#Scale properties-Convexity">Convexity</a></div>
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 2em;"><a href="#Scale properties-Maximal evenness">Maximal evenness</a></div>
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Scale properties-Numerical properties">Numerical properties</a></div>
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 1em;"><a href="#See also">See also</a></div>
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --></div>
<!-- ws:end:WikiTextTocRule:43 --><br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:3 -->Definition</h1>
 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. 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 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 <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 />
<!-- ws:start:WikiTextHeadingRule:5:&lt;h2&gt; --><h2 id="toc1"><a name="Definition-Rotations"></a><!-- ws:end:WikiTextHeadingRule:5 -->Rotations</h2>
 By a <em>rotation</em> or mode of a periodic scale s is meant a scale r such that r[i] = s[i + <strong>N</strong>] - s[<strong>N</strong>], where <strong>N</strong> is a fixed integer. Since s[i + <strong>P</strong>] - s[<strong>P</strong>] = 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; <strong>O</strong>, which entails 0 ≤ i &lt; <strong>P</strong>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h2&gt; --><h2 id="toc2"><a name="Definition-Classes"></a><!-- ws:end:WikiTextHeadingRule:7 -->Classes</h2>
 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc3"><a name="Scale properties"></a><!-- ws:end:WikiTextHeadingRule:9 -->Scale properties</h1>
 <!-- ws:start:WikiTextHeadingRule:11:&lt;h2&gt; --><h2 id="toc4"><a name="Scale properties-Constant Structure"></a><!-- ws:end:WikiTextHeadingRule:11 --><a class="wiki_link" href="/Constant%20Structure">Constant Structure</a></h2>
 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 />
<!-- ws:start:WikiTextHeadingRule:13:&lt;h2&gt; --><h2 id="toc5"><a name="Scale properties-Propriety"></a><!-- ws:end:WikiTextHeadingRule:13 --><a class="wiki_link" href="/Rothenberg%20propriety">Propriety</a></h2>
 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 />
<!-- ws:start:WikiTextHeadingRule:15:&lt;h2&gt; --><h2 id="toc6"><a name="Scale properties-Epimorphism"></a><!-- ws:end:WikiTextHeadingRule:15 -->Epimorphism</h2>
 If there exists a homomorphism h: G → ℤ 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>. The name comes from the fact that h is an epimorphism onto ℤ.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:17:&lt;h2&gt; --><h2 id="toc7"><a name="Scale properties-Myhill's property"></a><!-- ws:end:WikiTextHeadingRule:17 --><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Myhill%27s_property" rel="nofollow">Myhill's property</a></h2>
 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>. Myhill's property is synonymous with <strong>strict <a class="wiki_link" href="/MOS">MOS</a></strong>, though some authors prefer to identify MOS itself with Myhill's property.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:19:&lt;h2&gt; --><h2 id="toc8"><a name="Scale properties-Distributional evenness"></a><!-- ws:end:WikiTextHeadingRule:19 -->Distributional evenness</h2>
 A monotone scale in which every class comes in exactly n elements is n-distributionally even, or <strong>n-DE</strong>. If n=2, then we can simply say that it is distributionally even. <span style="line-height: 1.5;">Distributional evenness is also synonymous with <strong><a class="wiki_link" href="/MOS">MOS</a></strong>, though some authors prefer a stricter definition of MOS identifying it with Myhill's property.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:21:&lt;h2&gt; --><h2 id="toc9"><a name="Scale properties-Convexity"></a><!-- ws:end:WikiTextHeadingRule:21 -->Convexity</h2>
 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. If the quasiperiod <strong>P</strong> is normalized so as to be positive and minimal, this is equivalent to the condition 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>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:23:&lt;h2&gt; --><h2 id="toc10"><a name="Scale properties-Maximal evenness"></a><!-- ws:end:WikiTextHeadingRule:23 --><a class="wiki_link" href="/Maximal%20evenness">Maximal evenness</a></h2>
 Maximally even scales of n notes in m edo are any mode of the sequence ME(n, m) = [floor(i*m/n) | i=1..n], where the &quot;floor&quot; function rounds down to the nearest integer.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:25:&lt;h2&gt; --><h2 id="toc11"><a name="Scale properties-Numerical properties"></a><!-- ws:end:WikiTextHeadingRule:25 -->Numerical properties</h2>
 <a class="wiki_link" href="/Scale%20diversity">Scale diversity</a><br />
<a class="wiki_link" href="/Lumma%20stability">Lumma stability</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:27:&lt;h1&gt; --><h1 id="toc12"><a name="See also"></a><!-- ws:end:WikiTextHeadingRule:27 -->See also</h1>
<a class="wiki_link" href="/Scale%20properties%20simplified">Scale properties simplified</a></body></html>