Monzos and interval space: Difference between revisions
Wikispaces>guest **Imported revision 419040128 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 419046012 - Original comment: Reverted to Mar 27, 2013 1:45 am: spam** |
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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:genewardsmith|genewardsmith]] and made on <tt>2013-03-31 11:12:36 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>419046012</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>Reverted to Mar 27, 2013 1:45 am: spam</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> | ||
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Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean metrics|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep> then the Tenney-Euclidean norm, or TE norm, of it is | Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean metrics|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep> then the Tenney-Euclidean norm, or TE norm, of it is | ||
[[math]] | [[math]] | ||
\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2} | \sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2} | ||
[[math]] | [[math]] | ||
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The TE norm is therefore | The TE norm is therefore | ||
[[math]] | [[math]] | ||
\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889. | \sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889. | ||
[[math]] | [[math]] | ||
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<!-- ws:start:WikiTextMathRule:3: | <!-- ws:start:WikiTextMathRule:3: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}&lt;br/&gt;[[math]] | \sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}&lt;br/&gt;[[math]] | ||
--><script type="math/tex"> | --><script type="math/tex">\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}</script><!-- ws:end:WikiTextMathRule:3 --><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 /> | 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 /> | ||
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<!-- ws:start:WikiTextMathRule:6: | <!-- ws:start:WikiTextMathRule:6: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.&lt;br /&gt; | \sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.&lt;br /&gt; | ||
&lt;br/&gt;[[math]] | &lt;br/&gt;[[math]] | ||
--><script type="math/tex">\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889. | --><script type="math/tex">\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889. | ||
</script><!-- ws:end:WikiTextMathRule:6 --><br /> | </script><!-- ws:end:WikiTextMathRule:6 --><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> | ||