Monzo: Difference between revisions
Wikispaces>clumma **Imported revision 255365582 - Original comment: no abstract is present, nor any other headings** |
Wikispaces>mbattaglia1 **Imported revision 255424910 - 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:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-18 21:40:38 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>255424910</tt>.<br> | ||
: The revision comment was: <tt> | : 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> | ||
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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 19 28|-4 4 -1> | ||
(12*-4) + (19*4) + (28*1)<span class="st"> = </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. | ||
**In general: <a b c|d e f> = ad + be + cf** | **In general: <a b c|d e f> = ad + be + cf**</pre></div> | ||
</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</title></head><body>A monzo is the counterpart to a val. Much like vals allow us to express the way that prime intervals are mapped within an EDO, a monzo allows us to express how any &quot;composite&quot; interval is represented in terms of those simpler prime intervals. They are typically written using the notation |a b c d e f ... &gt;, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<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</title></head><body>A monzo is the counterpart to a val. Much like vals allow us to express the way that prime intervals are mapped within an EDO, a monzo allows us to express how any &quot;composite&quot; interval is represented in terms of those simpler prime intervals. They are typically written using the notation |a b c d e f ... &gt;, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<br /> | ||
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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& | &lt;12 19 28|-4 4 -1&gt;<br /> | ||
(12*-4) + (19*4) + (28*1)<span class="st"> = </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 /> | ||
<br /> | <br /> | ||
<strong>In general: &lt;a b c|d e f&gt; = ad + be + cf</strong></body></html></pre></div> | <strong>In general: &lt;a b c|d e f&gt; = ad + be + cf</strong></body></html></pre></div> | ||