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Monzos and interval space

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Original Wikitext content:

<span style="display: block; text-align: right;">Other languages: [[xenharmonie/Intervallraum|Deutsch]]
</span>

This page gives the formal mathematical definition of a monzo. For a simpler explanation with examples, visit the [[monzos]] page.

=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]]; hence we may [[http://en.wikipedia.org/wiki/Embedding|embed]] the p-limit monzos into a normed vector I space of dimension n = π(p) via a map M:monzos ⟶ I. The monzos under this embedding now define a [[http://en.wikipedia.org/wiki/Lattice_%28group%29|lattice]], which is a discrete subgroup spanning the finite dimensional real normed vector space I. 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]].

=Alternate Definition:= 

Given a rational number q, we can rewrite it in monzo form by the following definition:
[[math]]
q = |v_2 (q) \,v_3 (q) \, v_5 (q) \dotso v_p (q)\rangle
[[math]]

The [[Tenney Height|Tenney height]] of this monzo is given by
[[math]]
\| |v_2 (q) \, v_3 (q) \dotso v_p (q) \rangle \| = |v_2 (q)| + |v_3 (q)| \log_2 3 + \dotsb + |v_p (q)| \log_2 p
[[math]]

Where vp(q) is the [[http://en.wikipedia.org/wiki/P-adic_order|p-adic valuation]] of q.

=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
[[math]]
\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.

[[math]]

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

Original HTML content:

<html><head><title>Monzos and Interval Space</title></head><body><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="https://xenharmonie.wikispaces.com/Intervallraum">Deutsch</a><br />
</span><br />
<br />
This page gives the formal mathematical definition of a monzo. For a simpler explanation with examples, visit the <a class="wiki_link" href="/monzos">monzos</a> page.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:7:&lt;h1&gt; --><h1 id="toc0"><a name="Definition:"></a><!-- ws:end:WikiTextHeadingRule:7 -->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>; hence we may <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Embedding" rel="nofollow">embed</a> the p-limit monzos into a normed vector I space of dimension n = π(p) via a map M:monzos ⟶ I. The monzos under this embedding 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 the finite dimensional real normed vector space I. 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:9:&lt;h1&gt; --><h1 id="toc1"><a name="Alternate Definition:"></a><!-- ws:end:WikiTextHeadingRule:9 -->Alternate Definition:</h1>
 <br />
Given a rational number q, we can rewrite it in monzo form by the following definition:<br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]&lt;br/&gt;
q = |v_2 (q) \,v_3 (q) \, v_5 (q) \dotso v_p (q)\rangle&lt;br/&gt;[[math]]
 --><script type="math/tex">q = |v_2 (q) \,v_3 (q) \, v_5 (q) \dotso v_p (q)\rangle</script><!-- ws:end:WikiTextMathRule:4 --><br />
<br />
The <a class="wiki_link" href="/Tenney%20Height">Tenney height</a> of this monzo is given by<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]&lt;br/&gt;
\| |v_2 (q) \, v_3 (q) \dotso v_p (q) \rangle \| = |v_2 (q)| + |v_3 (q)| \log_2 3 + \dotsb + |v_p (q)| \log_2 p&lt;br/&gt;[[math]]
 --><script type="math/tex">\| |v_2 (q) \, v_3 (q) \dotso v_p (q) \rangle \| = |v_2 (q)| + |v_3 (q)| \log_2 3 + \dotsb + |v_p (q)| \log_2 p</script><!-- ws:end:WikiTextMathRule:5 --><br />
<br />
Where vp(q) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/P-adic_order" rel="nofollow">p-adic valuation</a> of q.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc2"><a name="Example:"></a><!-- ws:end:WikiTextHeadingRule:11 -->Example:</h1>
 <br />
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;.<br />
The TE norm is therefore<br />
<!-- ws:start:WikiTextMathRule:6:
[[math]]&lt;br/&gt;
\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.&lt;br /&gt;
&lt;br/&gt;[[math]]
 --><script type="math/tex">\sqrt{(4^2 + log2(3)^2 + log2(5)^2)} ≅ \sqrt{23.903} ≅ 4.889.
</script><!-- ws:end:WikiTextMathRule:6 --><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>
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