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)^(1/26). 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 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. ===Fractional projection maps=== A square matrix P is a [[http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29|projection]] if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvalues]] of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted [[Monzos and Interval Space|interval space]], then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v. In particular, this is true of matricies with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for 3 in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning. ===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.
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<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)^(1/26). 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 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--Fractional projection maps"></a><!-- ws:end:WikiTextHeadingRule:0 -->Fractional projection maps</h3> A square matrix P is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Projection_%28linear_algebra%29" rel="nofollow">projection</a> if P^2 = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvalues</a> of both 0 and 1 and no other eigenvalues. If the rows of P represent a tuning of a regular temperament as vectors in either weighted or unweighted <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, then a comma c of the temperament (in the appropriate coordinates) times P from the left, cP, will be the zero vector. A val of the temperament v, times P on the right, Pv, will satisfy Pv = v. <br /> <br /> In particular, this is true of matricies with rows consisting of fractional monzos. This is of interest since several of the most important tunings, in particular minimax and least squares, have tuning values which can be expressed as fractional monzos. For example, the fractional monzo we have used as an example is the tuning for 3 in the 7/26-comma Woolhouse meantone. Indeed, any meantone whose tuning is expressed as a fraction of a comma has an associated 3x3 projection matrix defining the tuning.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x--Algebraic considerations"></a><!-- ws:end:WikiTextHeadingRule:2 -->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>