Monzos and interval space
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A [[Harmonic Limit|p-limit]] rational number q can by definition be factored into primes of size less than or equal to p, giving q = 2^e2 3^e3 ... p^ep, 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 |e2 e3 ... ep>, 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 || |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep| 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 sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2) 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]]...//
Original HTML content:
<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 q = 2^e2 3^e3 ... p^ep, 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 |e2 e3 ... ep>, 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 /> <br /> || |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p) |ep|<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> represents 2, so does |0 log3(2)>.<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> then the Tenney-Euclidean norm, or TE norm, of it is<br /> <br /> sqrt(e2^2 + (log2(3)e3)^2 ... + (log2(p)ep)^2)<br /> <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:0:<h2> --><h2 id="toc0"><a name="x-Example"></a><!-- ws:end:WikiTextHeadingRule:0 -->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>. 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.<br /> <br /> <em>see also <a class="wiki_link" href="/Fractional%20monzos">Fractional monzos</a>...</em></body></html>