Cartesian scale: Difference between revisions

Revision as of 13:50, 25 December 2013

IMPORTED REVISION FROM WIKISPACES

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

This revision was by author genewardsmith and made on 2013-12-25 13:50:13 UTC.

The original revision id was 479373586.

The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

A //Cartesian sca;e// is a [[periodic scale]] with an interval of equivalence ℇ (normally 2 or 1200.0 cents or an approximation to the just octave) and k generators g = [g1, g2 ... gk] with k multiplicities m = [m1,m2 ... mk], leading to a scale Descartes(ℇ, g, m) which if  ℇ and g are given multplicatively is {ℇ^n g1^i1 ... gk^ik| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. Here the multiplicities are fixed positive integers, and n ranges over al integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is {nℇ^n + k1g1 +  ... + ikgk| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. 

If the generators are odd primes and ℇ = 2, then the Cartesian scale is an [[Euler genera|Euler genus]].






[[math]]
{\Sigma^n g_1^i_1 g_2^i_2 \ldots g_k^i_k| 0 \leq i_1 \leq m_1, 0 \leq i_2 \leq m_2, \ldots, 0 \leq i_k \leq m_k}
[[math]]

Original HTML content:

<html><head><title>Cartesian scales</title></head><body>A <em>Cartesian sca;e</em> is a <a class="wiki_link" href="/periodic%20scale">periodic scale</a> with an interval of equivalence ℇ (normally 2 or 1200.0 cents or an approximation to the just octave) and k generators g = [g1, g2 ... gk] with k multiplicities m = [m1,m2 ... mk], leading to a scale Descartes(ℇ, g, m) which if  ℇ and g are given multplicatively is {ℇ^n g1^i1 ... gk^ik| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. Here the multiplicities are fixed positive integers, and n ranges over al integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is {nℇ^n + k1g1 +  ... + ikgk| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. <br />
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If the generators are odd primes and ℇ = 2, then the Cartesian scale is an <a class="wiki_link" href="/Euler%20genera">Euler genus</a>.<br />
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{\Sigma^n g_1^i_1 g_2^i_2 \ldots g_k^i_k| 0 \leq i_1 \leq m_1, 0 \leq i_2 \leq m_2, \ldots, 0 \leq i_k \leq m_k}&lt;br/&gt;[[math]]
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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:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-25 10:38:45 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-25 13:50:13 UTC</tt>.<br>
-: The original revision id was <tt>479361876</tt>.<br>
+: The original revision id was <tt>479373586</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>
 <h4>Original Wikitext content:</h4>
-<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;white-space: pre-wrap ! important" class="old-revision-html">A //Cartesian sca;e// is a [[periodic scale]] with an interval of equivalence ℇ (normally 2 or 1200.0 cents or an approximation to the just octave) and k generators g = [g1, g2 ... gk] with k multiplicities m = [m1,m2 ... mk], leading to a scale Descartes(ℇ, g, m) which if  ℇ and g are given multplicatively is
+<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;white-space: pre-wrap ! important" class="old-revision-html">A //Cartesian sca;e// is a [[periodic scale]] with an interval of equivalence ℇ (normally 2 or 1200.0 cents or an approximation to the just octave) and k generators g = [g1, g2 ... gk] with k multiplicities m = [m1,m2 ... mk], leading to a scale Descartes(ℇ, g, m) which if  ℇ and g are given multplicatively is {ℇ^n g1^i1 ... gk^ik| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. Here the multiplicities are fixed positive integers, and n ranges over al integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is {nℇ^n + k1g1 +  ... + ikgk| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}.
+If the generators are odd primes and ℇ = 2, then the Cartesian scale is an [[Euler genera|Euler genus]].
 [[math]]
 {\Sigma^n g_1^i_1 g_2^i_2 \ldots g_k^i_k| 0 \leq i_1 \leq m_1, 0 \leq i_2 \leq m_2, \ldots, 0 \leq i_k \leq m_k}
 [[math]]</pre></div>
 <h4>Original HTML content:</h4>
-<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Cartesian scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;Cartesian sca;e&lt;/em&gt; is a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; with an interval of equivalence ℇ (normally 2 or 1200.0 cents or an approximation to the just octave) and k generators g = [g1, g2 ... gk] with k multiplicities m = [m1,m2 ... mk], leading to a scale Descartes(ℇ, g, m) which if  ℇ and g are given multplicatively is &lt;br /&gt;
+<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Cartesian scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;Cartesian sca;e&lt;/em&gt; is a &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; with an interval of equivalence ℇ (normally 2 or 1200.0 cents or an approximation to the just octave) and k generators g = [g1, g2 ... gk] with k multiplicities m = [m1,m2 ... mk], leading to a scale Descartes(ℇ, g, m) which if  ℇ and g are given multplicatively is {ℇ^n g1^i1 ... gk^ik| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. Here the multiplicities are fixed positive integers, and n ranges over al integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is {nℇ^n + k1g1 +  ... + ikgk| 0 ≤ i1 ≤ m1 ... 0 ≤ ik ≤ mk}. &lt;br /&gt;
+&lt;br /&gt;
+If the generators are odd primes and ℇ = 2, then the Cartesian scale is an &lt;a class="wiki_link" href="/Euler%20genera"&gt;Euler genus&lt;/a&gt;.&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
 {\Sigma^n g_1^i_1 g_2^i_2 \ldots g_k^i_k| 0 \leq i_1 \leq m_1, 0 \leq i_2 \leq m_2, \ldots, 0 \leq i_k \leq m_k}&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;{\Sigma^n g_1^i_1 g_2^i_2 \ldots g_k^i_k| 0 \leq i_1 \leq m_1, 0 \leq i_2 \leq m_2, \ldots, 0 \leq i_k \leq m_k}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:0 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Cartesian scale: Difference between revisions

Revision as of 13:50, 25 December 2013

IMPORTED REVISION FROM WIKISPACES

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