Monzo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 250597108 - Original comment: ** |
Wikispaces>clumma **Imported revision 255365582 - Original comment: no abstract is present, nor any other headings** |
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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:clumma|clumma]] and made on <tt>2011-09-18 18:08:57 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>255365582</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt>no abstract is present, nor any other headings</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"> | <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 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 "composite" interval is represented in terms of those simpler prime intervals. They are typically written using the notation |a b c d e f ... >, 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 [[harmonic limit|prime limit]]. | ||
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 "composite" interval is represented in terms of those simpler prime intervals. They are typically written using the notation |a b c d e f ... >, 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 [[harmonic limit|prime limit]]. | |||
For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the |.........> brackets, hence yielding |-3 1 1>. | For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the |.........> brackets, hence yielding |-3 1 1>. | ||
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<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> | <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 /> | ||
<br /> | <br /> | ||
For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the |.........&gt; brackets, hence yielding |-3 1 1&gt;.<br /> | For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the |.........&gt; brackets, hence yielding |-3 1 1&gt;.<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&gt; <!-- ws:start:WikiTextHeadingRule: | &lt;12 19 28|-4 4 -1&gt; <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x(12*-4) + (19*4) + (28*1)"></a><!-- ws:end:WikiTextHeadingRule:0 --> (12*-4) + (19*4) + (28*1) </h1> | ||
0(28*1)<span class="st"> &#61; </span>0<br /> | 0(28*1)<span class="st"> &#61; </span>0<br /> | ||
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