Monzo: Difference between revisions

Revision as of 12:28, 2 January 2017

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author xenwolf and made on 2017-01-02 12:28:05 UTC.

The original revision id was 602965630.

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

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

<span style="display: block; text-align: right;">[[Monzo(Esp)|Español]] - [[モンゾ|日本語]]
</span>
This page gives a pragmatic introduction to **monzos**. For the formal mathematical definition of visit the page [[Monzos and Interval Space]].

=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:
||~ Ratio ||~ Monzo ||
||= 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:
||~ Ratio ||~ Monzo ||
||= 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="/Monzo%28Esp%29">Español</a> - <a class="wiki_link" href="/%E3%83%A2%E3%83%B3%E3%82%BE">日本語</a><br />
</span><br />
This page gives a pragmatic introduction to <strong>monzos</strong>. For the formal mathematical definition of visit the page <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">Monzos and Interval Space</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><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 &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 />
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:&lt;h1&gt; --><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 | ... &gt; brackets, hence yielding | -3 1 1 &gt;.<br />
<br />
Here are some common 5-limit monzos, for your reference:<br />


<table class="wiki_table">
    <tr>
        <th>Ratio<br />
</th>
        <th>Monzo<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">3/2<br />
</td>
        <td>| -1 1 0 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5/4<br />
</td>
        <td>| -2 0 1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9/8<br />
</td>
        <td>| -3 2 0 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">81/80<br />
</td>
        <td>| -4 4 -1 &gt;<br />
</td>
    </tr>
</table>

<br />
Here are a few 7-limit monzos:<br />


<table class="wiki_table">
    <tr>
        <th>Ratio<br />
</th>
        <th>Monzo<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">7/4<br />
</td>
        <td>| -2 0 0 1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7/6<br />
</td>
        <td>| -1 -1 0 1 &gt;<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7/5<br />
</td>
        <td>| 0 0 -1 1 &gt;<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><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 &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 />
&lt; 12 19 28 | -4 4 -1 &gt;<br />
(12*-4) + (19*4) + (28*1)<span class="st"> = </span>0<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 />
<br />
<strong>In general: &lt; a b c | d e f &gt; = ad + be + cf</strong></body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2016-11-10 05:06:02 UTC</tt>.<br>
+: 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>599073962</tt>.<br>
+: The original revision id was <tt>602965630</tt>.<br>
-: The revision comment was: <tt></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>
 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>
@@ Line 11: / Line 11: @@
 =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 ... &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]].
+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]].
 Monzos can be thought of as counterparts to [[Vals|vals]].
@@ Line 18: / Line 18: @@
 =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 |.........&gt; brackets, hence yielding |-3 1 1&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 | ... &gt; brackets, hence yielding | -3 1 1 &gt;.
 Here are some common 5-limit monzos, for your reference:
-/2: |-1 1 0&gt;
+||~ Ratio ||~ Monzo ||
-/4: |-2 0 1&gt;
+||= 3/2 || | -1 1 0 &gt; ||
-/8: |-3 2 0&gt;
+||= 5/4 || | -2 0 1 &gt; ||
-/80: |-4 4 -1&gt;
+||= 9/8 || | -3 2 0 &gt; ||
+||= 81/80 || | -4 4 -1 &gt; ||
 Here are a few 7-limit monzos:
-/4: |-2 0 0 1&gt;
+||~ Ratio ||~ Monzo ||
-/6: |-1 -1 0 1&gt;
+||= 7/4 || | -2 0 0 1 &gt; ||
-/5: |0 0 -1 1&gt;
+||= 7/6 || | -1 -1 0 1 &gt; ||
+||= 7/5 || | 0 0 -1 1 &gt; ||
 =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 &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:
+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;
+&lt; 12 19 28 | -4 4 -1 &gt;
 (12*-4) + (19*4) + (28*1)&lt;span class="st"&gt; = &lt;/span&gt;0
-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 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>
+**In general: &lt; a b c | d e f &gt; = ad + be + cf**</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;monzos&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="/Monzo%28Esp%29"&gt;Español&lt;/a&gt; - &lt;a class="wiki_link" href="/%E3%83%A2%E3%83%B3%E3%82%BE"&gt;日本語&lt;/a&gt;&lt;br /&gt;
@@ Line 48: / Line 50: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h1&gt;
-  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;
+  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;
 &lt;br /&gt;
 Monzos can be thought of as counterparts to &lt;a class="wiki_link" href="/Vals"&gt;vals&lt;/a&gt;.&lt;br /&gt;
@@ Line 55: / Line 57: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Examples&lt;/h1&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;
+  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;br /&gt;
 Here are some common 5-limit monzos, for your reference:&lt;br /&gt;
-/2: |-1 1 0&amp;gt;&lt;br /&gt;
-/4: |-2 0 1&amp;gt;&lt;br /&gt;
-/8: |-3 2 0&amp;gt;&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
-/80: |-4 4 -1&amp;gt;&lt;br /&gt;
+    &lt;tr&gt;
+        &lt;th&gt;Ratio&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Monzo&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;3/2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -1 1 0 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;5/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -2 0 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;9/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -3 2 0 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;81/80&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -4 4 -1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
 &lt;br /&gt;
 Here are a few 7-limit monzos:&lt;br /&gt;
-/4: |-2 0 0 1&amp;gt;&lt;br /&gt;
-/6: |-1 -1 0 1&amp;gt;&lt;br /&gt;
-/5: |0 0 -1 1&amp;gt;&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;Ratio&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Monzo&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -2 0 0 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;7/6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| -1 -1 0 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;7/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;| 0 0 -1 1 &amp;gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Relationship with vals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Relationship with vals&lt;/h1&gt;
   &lt;em&gt;See also: &lt;a class="wiki_link" href="/Vals"&gt;Vals&lt;/a&gt;, &lt;a class="wiki_link" href="/Keenan%27s%20explanation%20of%20vals"&gt;Keenan's explanation of vals&lt;/a&gt;, &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;Vals and Tuning Space&lt;/a&gt; (more mathematical)&lt;/em&gt;&lt;br /&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;
+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;
 &lt;br /&gt;
-&amp;lt;12 19 28|-4 4 -1&amp;gt;&lt;br /&gt;
+&amp;lt; 12 19 28 | -4 4 -1 &amp;gt;&lt;br /&gt;
 (12*-4) + (19*4) + (28*1)&lt;span class="st"&gt; = &lt;/span&gt;0&lt;br /&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;
+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;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>
+&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>

Monzo: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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