Radical interval: Difference between revisions
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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. | ||
Starting with the next section, we discuss projection matrices in the formal mathematical language. For a beginner-level introduction, see [[Projection]]. | |||
== Fractional projection matrices == | == Fractional projection matrices == | ||
A square matrix P is a [[Wikipedia: Projection (linear algebra)|projection]] if P<sup>2</sup> = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[Wikipedia: 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 [[Wikipedia: Projection (linear algebra)|projection]] if P<sup>2</sup> = P. A nontrivial projection, meaning one which is neither the zero matrix nor the identity matrix, has [[Wikipedia: 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. | ||
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== Tunings in terms of fractional monzos == | == Tunings in terms of fractional monzos == | ||
If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' 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 matrix P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1. | If ''n'' is the number of primes π (''p'') less than or equal to ''p'', we may define a unique ''n''×''n'' 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 matrix P, so that uP = u where u is the eigenmonzo. The name refers to the fact that u is a left [[Wikipedia: Eigenvalue, eigenvector and eigenspace|eigenvector]] for the eigenvalue 1. | ||
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== Algebraic considerations == | == Algebraic considerations == | ||
For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a [[Wikipedia: 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 [[Wikipedia: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 [[Wikipedia:Vector space|vector space]] (of dimension ''n'') over the rational numbers. They are also torsion-free (equivalently, [[Wikipedia:Flat module|flat]]) abelian groups, and are the [[Wikipedia: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 [[Wikipedia: 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 [[Wikipedia: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 [[Wikipedia:Vector space|vector space]] (of dimension ''n'') over the rational numbers. They are also torsion-free (equivalently, [[Wikipedia:Flat module|flat]]) abelian groups, and are the [[Wikipedia:Injective hull|injective hulls]] of the corresponding monzos. | ||
[[Category:Regular temperament theory]] | |||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Tuning]] | [[Category:Tuning]] | ||