Radical interval
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A //fractional monzo// is like an ordinary [[Monzos and Interval Space|monzo]] except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/16). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents. Vectors in Tenney interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. ===Algebraic considerations=== For the mathematically inclined (other people may want to skip this paragraph) I note that monzos are elements of a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] (or equivalently, Z-module) of rank r equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a [[http://en.wikipedia.org/wiki/Divisible_group|divisible group]], meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a [[http://en.wikipedia.org/wiki/Vector_space|vector space]] (of dimension r) over the rational numbers. They are also torsion-free (equivalently, [[http://en.wikipedia.org/wiki/Flat_module|flat]]) abelian groups, and are the [[http://en.wikipedia.org/wiki/Injective_hull|injective hulls]] of the corresponding monzos.
Original HTML content:
<html><head><title>Fractional monzos</title></head><body>A <em>fractional monzo</em> is like an ordinary <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> except that coefficients have been extended to allow them to be rational numbers. If |e2 e3 ... ep> is a fractional monzo, then it represents 2^e2 3^e3 ... p^ep just as with an ordinary monzo. Hence, for instance, |14/13 -1/13 7/26> represents the interval 2^(14/13) 3^(-1/13) 5^(7/26). By taking the least common multiple of the denominators, intervals represented by a fractional monzo can always be written as an nth root of a positive rational number; for instance from our example, (20971520000000/9)^(1/16). By taking a dot product with <cents(2) cents(3) ... cents(p)| the value in cents of a monzo or fractional monzo may be obtained. For instance, in the above example (14/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 1896.1648 cents.<br /> <br /> Vectors in Tenney interval space, where the coefficients are allowed to be real numbers, do not uniquely correspond to intervals, whereas monzos do. Fractional monzos do also; for each fractional monzo there is one and only one nth root of a rational number which corresponds to it. <br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h3> --><h3 id="toc0"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:0 -->Algebraic considerations</h3> For the mathematically inclined (other people may want to skip this paragraph) I note that monzos are elements of a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> (or equivalently, Z-module) of rank r equal to the number of primes less than or equal to p for the p-limit in question. Fractional monzos do not define a free group but rather a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisible_group" rel="nofollow">divisible group</a>, meaning any element may be divided by any nonzero integer. They are Z-modules, but more than that also Q-modules, or stated equivalently, elements in a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">vector space</a> (of dimension r) over the rational numbers. They are also torsion-free (equivalently, <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Flat_module" rel="nofollow">flat</a>) abelian groups, and are the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Injective_hull" rel="nofollow">injective hulls</a> of the corresponding monzos.</body></html>