Monzos and interval space: Difference between revisions
Wikispaces>genewardsmith **Imported revision 357345852 - Original comment: ** |
Wikispaces>guest **Imported revision 362413524 - 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: | : This revision was by author [[User:guest|guest]] and made on <tt>2012-09-05 22:47:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>362413524</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> | ||
<h4>Original Wikitext content:</h4> | <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 [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving | <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">=Definition:= | ||
A [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving | |||
[[math]] | [[math]] | ||
q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p} | q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p} | ||
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and if the coordinates are the weighted interval space coordinates, then the TE norm is the [[http://mathworld.wolfram.com/L2-Norm.html|standard Euclidean, or L2, norm]]. | and if the coordinates are the weighted interval space coordinates, then the TE norm is the [[http://mathworld.wolfram.com/L2-Norm.html|standard Euclidean, or L2, norm]]. | ||
==Example= | =Alternate Definition:= | ||
Given a rational number q, we can rewrite it in monzo form by the following definition: | |||
[[math]] | |||
q = |v_2 q \,v_3 q \, v_5 q \dotso v_p q\rangle | |||
[[math]] | |||
The [[Tenney Height|Tenney height]] of this monzo is given by | |||
[[math]] | |||
\| |v_2 q \, v_3 q \dotso v_p q \rangle \| = |v_2 q| + |v_3 q| \log_2 3 + \dotsb + |v_p q| \log_2 p | |||
[[math]] | |||
Where vp(q) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of q. | |||
=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 √(1^2 + 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 √(1^2 + log2(3)^2 + log2(5)^2) ≅ √23.903 ≅ 4.889. | ||
//see also [[Fractional monzos]], [[Vals and Tuning Space]]...//</pre></div> | //see also [[Fractional monzos]], [[Vals and Tuning Space]]...//</pre></div> | ||
<h4>Original HTML content:</h4> | <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>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 /> | <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><!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:6 -->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 /> | |||
<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
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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 /> | 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 /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt; | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc1"><a name="Alternate Definition:"></a><!-- ws:end:WikiTextHeadingRule:8 -->Alternate Definition:</h1> | ||
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 √(1^2 + log2(3)^2 + log2(5)^2) ≅ √23.903 ≅ 4.889.<br /> | <br /> | ||
Given a rational number q, we can rewrite it in monzo form by the following definition:<br /> | |||
<!-- ws:start:WikiTextMathRule:4: | |||
[[math]]&lt;br/&gt; | |||
q = |v_2 q \,v_3 q \, v_5 q \dotso v_p q\rangle&lt;br/&gt;[[math]] | |||
--><script type="math/tex">q = |v_2 q \,v_3 q \, v_5 q \dotso v_p q\rangle</script><!-- ws:end:WikiTextMathRule:4 --><br /> | |||
<br /> | |||
The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of this monzo is given by<br /> | |||
<!-- ws:start:WikiTextMathRule:5: | |||
[[math]]&lt;br/&gt; | |||
\| |v_2 q \, v_3 q \dotso v_p q \rangle \| = |v_2 q| + |v_3 q| \log_2 3 + \dotsb + |v_p q| \log_2 p&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\| |v_2 q \, v_3 q \dotso v_p q \rangle \| = |v_2 q| + |v_3 q| \log_2 3 + \dotsb + |v_p q| \log_2 p</script><!-- ws:end:WikiTextMathRule:5 --><br /> | |||
<br /> | |||
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 /> | |||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc2"><a name="Example:"></a><!-- ws:end:WikiTextHeadingRule:10 -->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 √(1^2 + log2(3)^2 + log2(5)^2) ≅ √23.903 ≅ 4.889.<br /> | |||
<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> | <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> | ||