Monzos and interval space: 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:~~hythloday~~|~~hythloday~~]] and made on <tt>2013-03-~~26 22~~:19:33 UTC</tt>.<br>

: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2013-03-27 04:45:21 UTC</tt>.<br>

: The original revision id was <tt>~~418151424~~</tt>.<br>

: The original revision id was <tt>418200132</tt>.<br>

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

: The revision comment was: <tt>math formatting - the sqare root argument is now easier to see</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>

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=Example:=

The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1>. In weighted coordinates, that becomes |4 -log2(3) -log2(5)>, approximately |4 -1.585 -2.322>. The TE norm is therefore √(4^2 + log2(3)^2 + log2(5)^2) ≅ ~~√23~~.903 ≅ 4.889.

The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1>. In weighted coordinates, that becomes |4 -log2(3) -log2(5)>, approximately |4 -1.585 -2.322>.

The TE norm is therefore

[[math]]

\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.

[[math]]

//see also [[Fractional monzos]], [[Vals and Tuning Space]]...//</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>Monzos and Interval Space</title></head><body><h1 id="toc0"><a name="Definition:"></a>Definition:</h1>

<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>Monzos and Interval Space</title></head><body><h1 id="toc0"><a name="Definition:"></a>Definition:</h1>

<br />

A <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> rational number q can by definition be factored into primes of size less than or equal to p, giving<br />

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and if the coordinates are the weighted interval space coordinates, then the TE norm is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L2-Norm.html" rel="nofollow">standard Euclidean, or L2, norm</a>.<br />

<br />

<h1 id="toc1"><a name="Alternate Definition:"></a>Alternate Definition:</h1>

<h1 id="toc1"><a name="Alternate Definition:"></a>Alternate Definition:</h1>

<br />

Given a rational number q, we can rewrite it in monzo form by the following definition:<br />

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Where vp(q) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/P-adic_order" rel="nofollow">p-adic valuation</a> of q.<br />

<br />

<h1 id="toc2"><a name="Example:"></a>Example:</h1>

<h1 id="toc2"><a name="Example:"></a>Example:</h1>

<br />

The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&gt;, approximately |4 -1.585 -2.322&gt;. The TE norm is therefore √(4^2 + log2(3)^2 + log2(5)^2) ≅ ~~√23~~.903 ≅ 4.889.<br />

The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&gt;, approximately |4 -1.585 -2.322&gt;.<br />

The TE norm is therefore<br />

<!-- ws:start:WikiTextMathRule:6:

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

\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.&lt;br /&gt;

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

--><script type="math/tex">\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.

</script><br />

<br />

<em>see also <a class="wiki_link" href="/Fractional%20monzos">Fractional monzos</a>, <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a>...</em></body></html></pre></div>

@@ 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:hythloday|hythloday]] and made on <tt>2013-03-26 22:19:33 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2013-03-27 04:45:21 UTC</tt>.<br>
-: The original revision id was <tt>418151424</tt>.<br>
+: The original revision id was <tt>418200132</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>math formatting - the sqare root argument is now easier to see</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>
@@ Line 47: / Line 47: @@
 =Example:=
-The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&gt;, approximately |4 -1.585 -2.322&gt;. The TE norm is therefore √(4^2 + log2(3)^2 + log2(5)^2) ≅ √23.903 ≅ 4.889.
+The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&gt;, approximately |4 -1.585 -2.322&gt;.
+The TE norm is therefore
+[[math]]
+\sqrt{(4^2 + log2(3)^2 + log2(5)^2)}  ≅ \sqrt{23.903}  ≅ 4.889.
+[[math]]
 //see also [[Fractional monzos]], [[Vals and Tuning Space]]...//</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;Monzos and Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Definition:&lt;/h1&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;Monzos and Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:7:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:7 --&gt;Definition:&lt;/h1&gt;
   &lt;br /&gt;
 A &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; rational number q can by definition be factored into primes of size less than or equal to p, giving&lt;br /&gt;
@@ Line 80: / Line 85: @@
 and if the coordinates are the weighted interval space coordinates, then the TE norm is the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L2-Norm.html" rel="nofollow"&gt;standard Euclidean, or L2, norm&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Alternate Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Alternate Definition:&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:9:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Alternate Definition:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:9 --&gt;Alternate Definition:&lt;/h1&gt;
   &lt;br /&gt;
 Given a rational number q, we can rewrite it in monzo form by the following definition:&lt;br /&gt;
@@ Line 96: / Line 101: @@
 Where vp(q) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/P-adic_order" rel="nofollow"&gt;p-adic valuation&lt;/a&gt; of q.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Example:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Example:&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:11:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Example:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:11 --&gt;Example:&lt;/h1&gt;
   &lt;br /&gt;
-The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&amp;gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&amp;gt;, approximately |4 -1.585 -2.322&amp;gt;. The TE norm is therefore √(4^2 + log2(3)^2 + log2(5)^2) ≅ √23.903 ≅ 4.889.&lt;br /&gt;
+The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&amp;gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&amp;gt;, approximately |4 -1.585 -2.322&amp;gt;.&lt;br /&gt;
+The TE norm is therefore&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:6:
+[[math]]&amp;lt;br/&amp;gt;
+\sqrt{(4^2 + log2(3)^2 + log2(5)^2)}  ≅ \sqrt{23.903}  ≅ 4.889.&amp;lt;br /&amp;gt;
+&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\sqrt{(4^2 + log2(3)^2 + log2(5)^2)}  ≅ \sqrt{23.903}  ≅ 4.889.
+&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:6 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;em&gt;see also &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Fractional monzos&lt;/a&gt;, &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;Vals and Tuning Space&lt;/a&gt;...&lt;/em&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>

Monzos and interval space: Difference between revisions

Wikispaces>xenwolf