Monzos and interval space: Difference between revisions

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<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:~~xenjacob~~|~~xenjacob~~]] and made on <tt>2011-06-~~25 20~~:32:11 UTC</tt>.<br>

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:59:49 UTC</tt>.<br>

: The original revision id was <tt>~~238747703~~</tt>.<br>

: The original revision id was <tt>250543840</tt>.<br>

: The revision comment was: <tt>~~tryin out some LaTeX! hope I didn't screw it up.~~</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 [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving

<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">=Abstract=

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>.

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>

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) = 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**

=Mathematical Definition=

A [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving

[[math]]

q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}

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//see also [[Fractional monzos]], [[Vals and Tuning Space]]...//</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 and Interval Space</title></head><body>A <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> rational number q can by definition be factored into primes of size less than or equal to p, giving<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 and Interval Space</title></head><body><h1 id="toc0"><a name="Abstract"></a>Abstract</h1>

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 />

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 />

3/2: |-1 1 0&gt;<br />

5/4: |-2 0 1&gt;<br />

9/8: |-3 2 0&gt;<br />

81/80: |-4 4 -1&gt;<br />

<br />

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

7/4: |-2 0 0 1&gt;<br />

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

7/5: |0 0 -1 1&gt;<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; <h1 id="toc1"><a name="x(12*-4) + (19*4) + (28*1)"></a> (12*-4) + (19*4) + (28*1) </h1>

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><br />

<br />

<h1 id="toc2"><a name="Mathematical Definition"></a>Mathematical Definition</h1>

<br />

A <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> rational number q can by definition be factored into primes of size less than or equal to p, giving<br />

<!-- ws:start:WikiTextMathRule:0:

[[math]]&lt;br/&gt;

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and if the coordinates are the weighted interval space coordinates, then the TE norm is the <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L2-Norm.html" rel="nofollow">standard Euclidean, or L2, norm</a>.<br />

<br />

<h2 id="~~toc0~~"><a name="x-Example"></a>Example</h2>

<h2 id="toc3"><a name="Mathematical Definition-Example"></a>Example</h2>

The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&gt;, approximately |4 -1.585 -2.322&gt;. The TE norm is therefore sqrt(1^2 + log2(3)^2 + log2(5)^2) ~ sqrt(23.903) = 4.889.<br />

<br />

<em>see also <a class="wiki_link" href="/Fractional%20monzos">Fractional monzos</a>, <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">Vals and Tuning Space</a>...</em></body></html></pre></div>

@@ 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:xenjacob|xenjacob]] and made on <tt>2011-06-25 20:32:11 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-03 19:59:49 UTC</tt>.<br>
-: The original revision id was <tt>238747703</tt>.<br>
+: The original revision id was <tt>250543840</tt>.<br>
-: The revision comment was: <tt>tryin out some LaTeX!  hope I didn't screw it up.</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 [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving
+<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">=Abstract=
+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 ... &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;.
+Here are some common 5-limit monzos, for your reference:
+/2: |-1 1 0&gt;
+/4: |-2 0 1&gt;
+/8: |-3 2 0&gt;
+/80: |-4 4 -1&gt;
+Here are a few 7-limit monzos:
+/4: |-2 0 0 1&gt;
+/6: |-1 -1 0 1&gt;
+/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; = (12*-4) + (19*4) + (28*1) = 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 general: &lt;a b c|d e f&gt; = ad + be + cf**
+=Mathematical Definition=
+A [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving
 [[math]]
 q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}
@@ Line 34: / Line 60: @@
 //see also [[Fractional monzos]], [[Vals and Tuning Space]]...//</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 and Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; rational number q can by definition be factored into primes of size less than or equal to p, giving&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 and Interval Space&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Abstract"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Abstract&lt;/h1&gt;
+ 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 &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;
+&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;
+/80: |-4 4 -1&amp;gt;&lt;br /&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;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;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x(12*-4) + (19*4) + (28*1)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt; (12*-4) + (19*4) + (28*1) &lt;/h1&gt;
+&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;
+&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;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Mathematical Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Mathematical Definition&lt;/h1&gt;
+ &lt;br /&gt;
+A &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; rational number q can by definition be factored into primes of size less than or equal to p, giving&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 61: / Line 114: @@
 and if the coordinates are the weighted interval space coordinates, then the TE norm is the &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/L2-Norm.html" rel="nofollow"&gt;standard Euclidean, or L2, norm&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Example&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Mathematical Definition-Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Example&lt;/h2&gt;
   The 5-limit interval 16/15 factors as 2^4 3^(-1) 5^(-1), so it has a monzo representation of |4 -1 -1&amp;gt;. In weighted coordinates, that becomes |4 -log2(3) -log2(5)&amp;gt;, approximately |4 -1.585 -2.322&amp;gt;. The TE norm is therefore sqrt(1^2 + log2(3)^2 + log2(5)^2) ~ sqrt(23.903) = 4.889.&lt;br /&gt;
 &lt;br /&gt;
 &lt;em&gt;see also &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Fractional monzos&lt;/a&gt;, &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;Vals and Tuning Space&lt;/a&gt;...&lt;/em&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>