Periodic scale: Difference between revisions
Wikispaces>genewardsmith **Imported revision 477282996 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 481921260 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-01-10 13:37:47 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>481921260</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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=Convexity= | =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**.</pre></div> | 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]] | |||
</pre></div> | |||
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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Periodic scale</title></head><body><!-- ws:start:WikiTextTocRule: | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Periodic scale</title></head><body><!-- ws:start:WikiTextTocRule:19:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --><a href="#Constant Structure">Constant Structure</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Propriety">Propriety</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Epimorphic">Epimorphic</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Myhill's property">Myhill's property</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#Distributional evenness">Distributional evenness</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: --> | <a href="#Convexity">Convexity</a><!-- ws:end:WikiTextTocRule:25 --><!-- ws:start:WikiTextTocRule:26: --> | <a href="#Maximal evenness">Maximal evenness</a><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --> | <a href="#Numerical properties">Numerical properties</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --> | ||
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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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc5"><a name="Convexity"></a><!-- ws:end:WikiTextHeadingRule:13 -->Convexity</h1> | <!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc5"><a name="Convexity"></a><!-- ws:end:WikiTextHeadingRule:13 -->Convexity</h1> | ||
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>.</body></html></pre></div> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc6"><a name="Maximal evenness"></a><!-- ws:end:WikiTextHeadingRule:15 --><a class="wiki_link" href="/Maximal%20evenness">Maximal evenness</a></h1> | |||
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:17:&lt;h1&gt; --><h1 id="toc7"><a name="Numerical properties"></a><!-- ws:end:WikiTextHeadingRule:17 -->Numerical properties</h1> | |||
<a class="wiki_link" href="/Scale%20diversity">Scale diversity</a><br /> | |||
<a class="wiki_link" href="/Lumma%20stability">Lumma stability</a></body></html></pre></div> | |||