Radical interval: Difference between revisions
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<span style="display: block; text-align: right;">[[de:Nichtganzzahlige_Intervallvektoren]]</span> | <span style="display: block; text-align: right;">[[de:Nichtganzzahlige_Intervallvektoren]]</span> | ||
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, |1/13 -1/13 7/26> represents the interval 2^(1/13) 3^(-1/13) 5^(7/26). By taking the [[Least_common_multiple|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, (312500/9)^(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 (1/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 696.1648 cents. | 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, |1/13 -1/13 7/26> represents the interval 2^(1/13) 3^(-1/13) 5^(7/26). By taking the [[Least_common_multiple|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, (312500/9)^(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 (1/13)*1200.0 - (1/13)*cents(3) + (7/26)*cents(5) = 696.1648 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 positive rational number which corresponds to it. | 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 positive 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. | 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 a fifth 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. | 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 a fifth 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. | ||
== Tunings in terms of fractional monzos == | |||
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. | 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. | ||
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If the target set is a q limit diamond, eigenmonzos in the range 1<x<sqrt(2) may be found by simply finding all of the sets of r-1 q-limit diamond intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error. | If the target set is a q limit diamond, eigenmonzos in the range 1<x<sqrt(2) may be found by simply finding all of the sets of r-1 q-limit diamond intervals in that range which together with 2 define an independent set of intervals, and computing the corresponding projection matrix. The matrix leading to the least maximum error on elements of the diamond will be the minimax tuning. If there is a tie or ties, it may be broken by choosing the tuning with the smallest sum of squares of the error. | ||
== 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. | 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. | ||
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