Monzos and interval space: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 250543856 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 250543914 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 20: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 20:01:03 UTC</tt>.<br> | ||
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Monzos are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as <12 19 28|-4 4 -1>. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example: | Monzos are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as <12 19 28|-4 4 -1>. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example: | ||
<12 19 28|-4 4 -1> = (12*-4) + (19*4) + (28*1) = 0 | <12 19 28|-4 4 -1> = (12*-4) + (19*4) + (28*1)<span class="st"> &#61; </span>0 | ||
In this case, the val <12 19 28| is the [[patent val]] for 12-equal, and |-4 4 -1> is 81/80, or the syntonic comma. The fact that <12 19 28|-4 4 -1> tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation. | In this case, the val <12 19 28| is the [[patent val]] for 12-equal, and |-4 4 -1> is 81/80, or the syntonic comma. The fact that <12 19 28|-4 4 -1> tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation. | ||
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Monzos are important because they enable us to see how any JI interval &quot;maps&quot; onto a val. This mapping is expressed by writing the val and the monzo together, such as &lt;12 19 28|-4 4 -1&gt;. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:<br /> | Monzos are important because they enable us to see how any JI interval &quot;maps&quot; onto a val. This mapping is expressed by writing the val and the monzo together, such as &lt;12 19 28|-4 4 -1&gt;. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example:<br /> | ||
<br /> | <br /> | ||
&lt;12 19 28|-4 4 -1&gt; | &lt;12 19 28|-4 4 -1&gt; = (12*-4) + (19*4) + (28*1)<span class="st"> &#61; </span>0<br /> | ||
<br /> | <br /> | ||
In this case, the val &lt;12 19 28| is the <a class="wiki_link" href="/patent%20val">patent val</a> for 12-equal, and |-4 4 -1&gt; is 81/80, or the syntonic comma. The fact that &lt;12 19 28|-4 4 -1&gt; tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation.<br /> | In this case, the val &lt;12 19 28| is the <a class="wiki_link" href="/patent%20val">patent val</a> for 12-equal, and |-4 4 -1&gt; is 81/80, or the syntonic comma. The fact that &lt;12 19 28|-4 4 -1&gt; tells us that 81/80 is mapped to 0 steps in 12-equal - aka it's tempered out - which tells us that 12-equal is a meantone temperament. It is noteworthy that almost the entirety of western music, particularly western music composed for 12-equal or 12-tone well temperaments, is made possible by the above equation.<br /> | ||
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<strong>In general: &lt;a b c|d e f&gt; = ad + be + cf</strong><br /> | <strong>In general: &lt;a b c|d e f&gt; = ad + be + cf</strong><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc1"><a name="Mathematical Definition"></a><!-- ws:end:WikiTextHeadingRule:6 -->Mathematical Definition</h1> | ||
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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 /> | 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 /> | 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 /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc2"><a name="Mathematical Definition-Example"></a><!-- ws:end:WikiTextHeadingRule:8 -->Example</h2> | ||
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 sqrt(1^2 + log2(3)^2 + log2(5)^2) ~ sqrt(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;. The TE norm is therefore sqrt(1^2 + log2(3)^2 + log2(5)^2) ~ sqrt(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> | ||