Cartesian 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:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-30 16:05:34 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-30 16:08:19 UTC</tt>.<br>

: The original revision id was <tt>~~479904012~~</tt>.<br>

: The original revision id was <tt>479904292</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 scale// 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 scale// is a [[periodic scale|monotone 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

[[math]]

\{{ℇ^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]]

Here the multiplicities are fixed positive integers, and n ranges over all integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is

Here the multiplicities are fixed positive integers, and n ranges over all integers, with the scale sorted in ascending size with al duplicates removed. If intervals are written additively as cents, then Descartes(ℇ, g, m) is

[[math]]

\{{nℇ + i_1g_1 + \ldots + i_kg_k| 0 \leq i_1 \leq m_1 \ldots 0 \leq i_k \leq m_k}\}.

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</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"><html><head><title>Cartesian scales</title></head><body>A <em>Cartesian scale</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<br />

<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>Cartesian scales</title></head><body>A <em>Cartesian scale</em> is a <a class="wiki_link" href="/periodic%20scale">monotone 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<br />

<!-- ws:start:WikiTextMathRule:0:

[[math]]&lt;br/&gt;

\{{ℇ^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]]

--><script type="math/tex">\{{ℇ^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}\}.</script><br />

Here the multiplicities are fixed positive integers, and n ranges over all integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is <br />

Here the multiplicities are fixed positive integers, and n ranges over all integers, with the scale sorted in ascending size with al duplicates removed. If intervals are written additively as cents, then Descartes(ℇ, g, m) is <br />

<!-- ws:start:WikiTextMathRule:1:

[[math]]&lt;br/&gt;

@@ 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>2013-12-30 16:05:34 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-12-30 16:08:19 UTC</tt>.<br>
-: The original revision id was <tt>479904012</tt>.<br>
+: The original revision id was <tt>479904292</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 scale// 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 scale// is a [[periodic scale|monotone 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
 [[math]]
 \{{ℇ^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]]
-Here the multiplicities are fixed positive integers, and n ranges over all integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is
+Here the multiplicities are fixed positive integers, and n ranges over all integers, with the scale sorted in ascending size with al duplicates removed. If intervals are written additively as cents, then Descartes(ℇ, g, m) is
 [[math]]
 \{{nℇ + i_1g_1 + \ldots + i_kg_k| 0 \leq i_1 \leq m_1 \ldots 0 \leq i_k \leq m_k}\}.
@@ Line 23: / Line 23: @@
 </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 scale&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 scale&lt;/em&gt; is a &lt;a class="wiki_link" href="/periodic%20scale"&gt;monotone 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;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
 \{{ℇ^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;\{{ℇ^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;br /&gt;
-Here the multiplicities are fixed positive integers, and n ranges over all integers. If intervals are written additively as cents, then Descartes(ℇ, g, m) is &lt;br /&gt;
+Here the multiplicities are fixed positive integers, and n ranges over all integers, with the scale sorted in ascending size with al duplicates removed. If intervals are written additively as cents, then Descartes(ℇ, g, m) is &lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:1:
 [[math]]&amp;lt;br/&amp;gt;