Radical interval
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author genewardsmith and made on 2010-09-16 18:54:26 UTC.
- The original revision id was 163289187.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
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 matrices 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. If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of n-r commas c and r //eigenmonzos// e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace|eigenvector]] for the eigenvalue 1. There are various [[http://en.wikipedia.org/wiki/Matrix_norm|matrix norms]], and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the [[Monzos and interval space|Frobenius tuning]], which is the same as the unweighted rms tuning which can be found using the [[Monzos and interval space|pseudoinverse]]. If r is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(r), which is minimal; this follows from the [[http://Singular%20value%20decomposition|singular value decomposition]]. ===Algebraic considerations=== For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] (or equivalently, Z-module) of rank n 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 n) 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)^(1/26). 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 matrices 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 /> If n is the number of primes pi(p) less than or equal to p, we may define a unique nxn projection matrix by means of a list of<br /> n-r commas c and r <em>eigenmonzos</em> e. An eigenmonzo is defined as a monzo which is invariant under left multiplication by a fractional monzo projection map P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Eigenvalue,_eigenvector_and_eigenspace" rel="nofollow">eigenvector</a> for the eigenvalue 1.<br /> <br /> There are various <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Matrix_norm" rel="nofollow">matrix norms</a>, and given a matrix norm, we can define an associated tuning of a regular temperament by taking the matrix with the minimum norm subject to the condition that commas of the temperament are left eigenvectors with eigenvalue 0 (that is, elements of the null space) while vals of the temperament are right eigenvectors with eigenvalue 1. Perhaps the easiest norm to work with is the Frobenius norm, which simply treats a matrix like a vector and takes the square root of the sum of squares of the coefficients of the matrix. The associated tuning is the <a class="wiki_link" href="/Monzos%20and%20interval%20space">Frobenius tuning</a>, which is the same as the unweighted rms tuning which can be found using the <a class="wiki_link" href="/Monzos%20and%20interval%20space">pseudoinverse</a>. If r is the rank of the temperament, the Frobenius norm of the Frobenius tuning is sqrt(r), which is minimal; this follows from the <a class="wiki_link_ext" href="http://Singular%20value%20decomposition" rel="nofollow">singular value decomposition</a>.<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) we 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 n 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 n) 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>