Periodic scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-13 11:44:27 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-13 12:55:51 UTC</tt>.

: The original revision id was <tt>~~508611636~~</tt>.

: The original revision id was <tt>508629270</tt>.

: The revision comment was: <tt></tt>

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==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**.

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==

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<h2 id="toc1"><a name="Definition-Rotations"></a>Rotations</h2>

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] &lt; O, which entails 0 ≤ i &lt; P.

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] &lt; O, which entails 0 ≤ i &lt; P.

<h2 id="toc2"><a name="Definition-Classes"></a>Classes</h2>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2014-05-13 11:44:27 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-13 12:55:51 UTC</tt>.<br>
-: The original revision id was <tt>508611636</tt>.<br>
+: The original revision id was <tt>508629270</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>
@@ Line 26: / Line 26: @@
 ==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] &lt; **O**, which entails 0 ≤  i &lt; **P**.
+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] &lt; **O**, which entails 0 ≤  i &lt; **P**.
 ==Classes==
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:5:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Definition-Rotations"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:5 --&gt;Rotations&lt;/h2&gt;
-By a &lt;em&gt;rotation&lt;/em&gt; or mode of a periodic scale s is meant a scale r such that r[i] = s[i + &lt;strong&gt;N&lt;/strong&gt;] - s[n], where &lt;strong&gt;N&lt;/strong&gt; is a fixed integer. Since s[i + &lt;strong&gt;P&lt;/strong&gt;] - s[&lt;strong&gt;P&lt;/strong&gt;] = 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] &amp;lt; &lt;strong&gt;O&lt;/strong&gt;, which entails 0 ≤  i &amp;lt; &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
+By a &lt;em&gt;rotation&lt;/em&gt; or mode of a periodic scale s is meant a scale r such that r[i] = s[i + &lt;strong&gt;N&lt;/strong&gt;] - s[&lt;strong&gt;N&lt;/strong&gt;], where &lt;strong&gt;N&lt;/strong&gt; is a fixed integer. Since s[i + &lt;strong&gt;P&lt;/strong&gt;] - s[&lt;strong&gt;P&lt;/strong&gt;] = 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] &amp;lt; &lt;strong&gt;O&lt;/strong&gt;, which entails 0 ≤  i &amp;lt; &lt;strong&gt;P&lt;/strong&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Definition-Classes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Classes&lt;/h2&gt;