Monzo
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<span style="display: block; text-align: right;">[[モンゾ|日本語]] </span> =Definition= A **monzo** is a way of notating a JI interval that allows us to express directly 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]]. Monzos can be thought of as counterparts to [[Vals|vals]]. For a more mathematical discussion, see also [[Monzos and Interval Space]]. =Examples= 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>. Here are some common 5-limit monzos, for your reference: 3/2: |-1 1 0> 5/4: |-2 0 1> 9/8: |-3 2 0> 81/80: |-4 4 -1> Here are a few 7-limit monzos: 7/4: |-2 0 0 1> 7/6: |-1 -1 0 1> 7/5: |0 0 -1 1> =Relationship with vals= //See also: [[Vals]], [[Keenan's explanation of vals]], [[Vals and Tuning Space]] (more mathematical)// 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)<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 general: <a b c|d e f> = ad + be + cf**
Original HTML content:
<html><head><title>monzos</title></head><body><span style="display: block; text-align: right;"><a class="wiki_link" href="/%E3%83%A2%E3%83%B3%E3%82%BE">日本語</a><br /> </span><br /> <!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> A <strong>monzo</strong> is a way of notating a JI interval that allows us to express directly 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 <a class="wiki_link" href="/harmonic%20limit">prime limit</a>.<br /> <br /> Monzos can be thought of as counterparts to <a class="wiki_link" href="/Vals">vals</a>.<br /> <br /> For a more mathematical discussion, see also <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:2 -->Examples</h1> 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>.<br /> <br /> Here are some common 5-limit monzos, for your reference:<br /> 3/2: |-1 1 0><br /> 5/4: |-2 0 1><br /> 9/8: |-3 2 0><br /> 81/80: |-4 4 -1><br /> <br /> Here are a few 7-limit monzos:<br /> 7/4: |-2 0 0 1><br /> 7/6: |-1 -1 0 1><br /> 7/5: |0 0 -1 1><br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Relationship with vals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Relationship with vals</h1> <em>See also: <a class="wiki_link" href="/Vals">Vals</a>, <a class="wiki_link" href="/Keenan%27s%20explanation%20of%20vals">Keenan's explanation of vals</a>, <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a> (more mathematical)</em><br /> <br /> 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:<br /> <br /> <12 19 28|-4 4 -1><br /> (12*-4) + (19*4) + (28*1)<span class="st"> = </span>0<br /> <br /> In this case, the val <12 19 28| is the <a class="wiki_link" href="/patent%20val">patent val</a> 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.<br /> <br /> <strong>In general: <a b c|d e f> = ad + be + cf</strong></body></html>