Monzos and interval space

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=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}
[[math]]
where the exponents are integers (positive, negative, or zero.) This is often written in [[http://mathworld.wolfram.com/Ket.html|ket vector]] ([[http://en.wikipedia.org/wiki/Bra-ket_notation|wp]]) notation as
[[math]]
|e_2 \, e_3 \, e_5 \dotso e_p\rangle
[[math]]
in which case it is called a **monzo**, where the name refers to the enthusiastic advocacy of [[Joe Monzo]].

The [[Tenney Height|Tenney height]] of this monzo is given by
[[math]]
\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p
[[math]]

which is a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]]. The monzos with this norm now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning a finite dimensional real normed vector space. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard [[http://mathworld.wolfram.com/L1-Norm.html|L1 norm]]. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0> represents 2, so does |0 log3(2)>.

Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have [[Tenney-Euclidean metrics|Tenney-Euclidean interval space]] instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep> then the Tenney-Euclidean norm, or TE norm, of it is
[[math]]
\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}
[[math]]
and if the coordinates are the weighted interval space coordinates, then the TE norm is the [[http://mathworld.wolfram.com/L2-Norm.html|standard Euclidean, or L2, norm]].

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

//see also [[Fractional monzos]], [[Vals and Tuning Space]]...//

Original HTML content:

<html><head><title>Monzos and Interval Space</title></head><body><!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc0"><a name="Abstract"></a><!-- ws:end:WikiTextHeadingRule:4 -->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;  <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc1"><a name="x(12*-4) + (19*4) + (28*1)"></a><!-- ws:end:WikiTextHeadingRule:6 --> (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 />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc2"><a name="Mathematical Definition"></a><!-- ws:end:WikiTextHeadingRule:8 -->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;
q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}&lt;br/&gt;[[math]]
 --><script type="math/tex">q = 2^{e_2} \, 3^{e_3} \, 5^{e_5} \dotso p^{e_p}</script><!-- ws:end:WikiTextMathRule:0 --><br />
where the exponents are integers (positive, negative, or zero.) This is often written in <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Ket.html" rel="nofollow">ket vector</a> (<a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Bra-ket_notation" rel="nofollow">wp</a>) notation as<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
|e_2 \, e_3 \, e_5 \dotso e_p\rangle&lt;br/&gt;[[math]]
 --><script type="math/tex">|e_2 \, e_3 \, e_5 \dotso e_p\rangle</script><!-- ws:end:WikiTextMathRule:1 --><br />
in which case it is called a <strong>monzo</strong>, where the name refers to the enthusiastic advocacy of <a class="wiki_link" href="/Joe%20Monzo">Joe Monzo</a>.<br />
<br />
The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of this monzo is given by<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p&lt;br/&gt;[[math]]
 --><script type="math/tex">\| |e_2 \, e_3 \dotso e_p \rangle \| = |e_2| + |e_3| \log_2 3 + \dotsb + |e_p| \log_2 p</script><!-- ws:end:WikiTextMathRule:2 --><br />
<br />
which is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a>. The monzos with this norm now define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Lattice_%28group%29" rel="nofollow">lattice</a>, which is a discrete subgroup spanning a finite dimensional real normed vector space. If we change coordinates by multiplying values in the coordinate belonging to the prime k by log2(k), then the norm becomes the standard <a class="wiki_link_ext" href="http://mathworld.wolfram.com/L1-Norm.html" rel="nofollow">L1 norm</a>. This vector space is Tenney interval space, and the transformed coordinates with the standard L1 norm form the standard basis for Tenney space. It should be noted that while monzos correspond uniquely to positive real numbers (always rational numbers in the case of monzos), vectors in Tenney space do not. For instance, while |1 0&gt; represents 2, so does |0 log3(2)&gt;.<br />
<br />
Because of the mathematical advantages of Euclidean norms, a Euclidean norm is often placed on the vectors in interval space instead of an L1 norm, in which case we have <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean interval space</a> instead of Tenney interval space. Explicitly, if we take the monzo |e2 e3 ... ep&gt; then the Tenney-Euclidean norm, or TE norm, of it is<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]&lt;br/&gt;
\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}&lt;br/&gt;[[math]]
 --><script type="math/tex">\sqrt{e_2^2 + (e_3\log_2 3)^2 + \dotsb + (e_p\log_2 p)^2}</script><!-- ws:end:WikiTextMathRule:3 --><br />
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 />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc3"><a name="Mathematical Definition-Example"></a><!-- ws:end:WikiTextHeadingRule:10 -->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>