Monzo: Difference between revisions

: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2017-01-02 12:28:05 UTC</tt>.<br>

: The original revision id was <tt>602965630</tt>.<br>

: The revision comment was: <tt>tables for examples, added one space as padding as is used in most monzo expressions, for instance on the comma page</tt><br>

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 ... &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 [[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 | ... &gt; brackets, hence yielding | -3 1 1 &gt;.

||= 3/2 || | -1 1 0 &gt; ||

||= 5/4 || | -2 0 1 &gt; ||

||= 9/8 || | -3 2 0 &gt; ||

||= 81/80 || | -4 4 -1 &gt; ||

||= 7/4 || | -2 0 0 1 &gt; ||

||= 7/6 || | -1 -1 0 1 &gt; ||

||= 7/5 || | 0 0 -1 1 &gt; ||

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 &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:

&lt; 12 19 28 | -4 4 -1 &gt;

In this case, the val &lt; 12 19 28 | is the [[patent val]] 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.

**In general: &lt; a b c | d e f &gt; = ad + be + cf**</pre></div>

  A &lt;strong&gt;monzo&lt;/strong&gt; is a way of notating a JI interval that allows us to express directly how any &amp;quot;composite&amp;quot; interval is represented in terms of those simpler prime intervals. They are typically written using the notation | a b c d e f ... &amp;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 &lt;a class="wiki_link" href="/harmonic%20limit"&gt;prime limit&lt;/a&gt;.&lt;br /&gt;

  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 | ... &amp;gt; brackets, hence yielding | -3 1 1 &amp;gt;.&lt;br /&gt;

 

 

        &lt;td style="text-align: center;"&gt;3/2&lt;br /&gt;

        &lt;td&gt;| -1 1 0 &amp;gt;&lt;br /&gt;

        &lt;td style="text-align: center;"&gt;5/4&lt;br /&gt;

        &lt;td&gt;| -2 0 1 &amp;gt;&lt;br /&gt;

        &lt;td style="text-align: center;"&gt;9/8&lt;br /&gt;

        &lt;td&gt;| -3 2 0 &amp;gt;&lt;br /&gt;

        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;

        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;

 

 

 

        &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;

        &lt;td&gt;| -2 0 0 1 &amp;gt;&lt;br /&gt;

        &lt;td style="text-align: center;"&gt;7/6&lt;br /&gt;

        &lt;td&gt;| -1 -1 0 1 &amp;gt;&lt;br /&gt;

        &lt;td style="text-align: center;"&gt;7/5&lt;br /&gt;

        &lt;td&gt;| 0 0 -1 1 &amp;gt;&lt;br /&gt;

 

Monzos are important because they enable us to see how any JI interval &amp;quot;maps&amp;quot; onto a val. This mapping is expressed by writing the val and the monzo together, such as &amp;lt; 12 19 28 | -4 4 -1 &amp;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:&lt;br /&gt;

&amp;lt; 12 19 28 | -4 4 -1 &amp;gt;&lt;br /&gt;

In this case, the val &amp;lt; 12 19 28 | is the &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; for 12-equal, and | -4 4 -1 &amp;gt; is 81/80, or the syntonic comma. The fact that &amp;lt; 12 19 28 | -4 4 -1 &amp;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.&lt;br /&gt;

&lt;strong&gt;In general: &amp;lt; a b c | d e f &amp;gt; = ad + be + cf&lt;/strong&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>