Monzo: Difference between revisions

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:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2012-01-~~21 12~~:40:27 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 04:24:02 UTC</tt>.<br>

: The original revision id was <tt>~~294107264~~</tt>.<br>

: The original revision id was <tt>325924106</tt>.<br>

: 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>

<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">A **monzo** is ~~the counterpart to~~ a ~~[[Vals|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]].

<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">=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]].

~~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>~~.

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:

Line 22:

Line 26:

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:

Line 32:

Line 39:

**In general: <a b c|d e f> = 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"><html><head><title>monzos</title></head><body>A <~~strong~~>~~monzo~~<~~/strong~~> ~~is the counterpart to a~~ <a ~~class~~="~~wiki_link" href=~~"/~~Vals"~~>~~val~~</a>~~. 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><h1 id="toc0"><a name="Definition"></a>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 />

~~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;. <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 />

<h1 id="toc1"><a name="Examples"></a>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 />

Line 48:

Line 59:

7/6: |-1 -1 0 1&gt;<br />

7/5: |0 0 -1 1&gt;<br />

<br />

<h1 id="toc2"><a name="Relationship with vals"></a>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 />

@@ 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:genewardsmith|genewardsmith]] and made on <tt>2012-01-21 12:40:27 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2012-04-27 04:24:02 UTC</tt>.<br>
-: The original revision id was <tt>294107264</tt>.<br>
+: The original revision id was <tt>325924106</tt>.<br>
 : 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>
 <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">A **monzo** is the counterpart to a [[Vals|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 ... &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]].
+<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">=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]].
-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;.
+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 |.........&gt; brackets, hence yielding |-3 1 1&gt;.
 Here are some common 5-limit monzos, for your reference:
@@ Line 22: / Line 26: @@
 /6: |-1 -1 0 1&gt;
 /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:
@@ Line 32: / Line 39: @@
 **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;A &lt;strong&gt;monzo&lt;/strong&gt; is the counterpart to a &lt;a class="wiki_link" href="/Vals"&gt;val&lt;/a&gt;. 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 &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;
+<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;!-- 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;
 &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;
+Monzos can be thought of as counterparts to &lt;a class="wiki_link" href="/Vals"&gt;vals&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 For a more mathematical discussion, see also &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;Monzos and Interval Space&lt;/a&gt;.&lt;br /&gt;
+&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;
 &lt;br /&gt;
 Here are some common 5-limit monzos, for your reference:&lt;br /&gt;
@@ Line 48: / Line 59: @@
 /6: |-1 -1 0 1&amp;gt;&lt;br /&gt;
 /5: |0 0 -1 1&amp;gt;&lt;br /&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;