Projection matrices: Difference between revisions
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A '''projection matrix''', or '''projection''' for short, is an object in [[regular temperament theory]] (RTT) that uniquely identifies a specific tuning of a specific regular temperament. | A '''projection matrix''', or '''projection''' for short, is an object in [[regular temperament theory]] (RTT) that uniquely identifies a specific tuning of a specific regular temperament in terms of eigenmonzos or fractional-commas - that is, while meantone can be identified by a mapping, quarter-comma meantone is identified by a projection. | ||
Starting with the next section, we discuss projection matrices in the formal mathematical language. For a beginner-level introduction, see [[Projection]]. | Starting with the next section, we discuss projection matrices in the formal mathematical language. For a beginner-level introduction, see [[Projection]]. | ||
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; as an interval, an [[eigenmonzo]] can be called an "unchanged interval", as the interval is left unchanged by the tuning of the temperament. The use of fractional monzos appears to correspond with the reference to fractional commas. | ||
There are various [[Wikipedia: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 [[Tenney-Euclidean tuning|Frobenius tuning]], which is the same as the unweighted RMS tuning which can be found using the [[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 [[Wikipedia:Singular value decomposition|singular value decomposition]]. | There are various [[Wikipedia: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 [[Tenney-Euclidean tuning|Frobenius tuning]], which is the same as the unweighted RMS tuning which can be found using the [[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 [[Wikipedia:Singular value decomposition|singular value decomposition]]. | ||
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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 tuning|minimax]] and [[Least squares tuning|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 3×3 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 tuning|minimax]] and [[Least squares tuning|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 3×3 projection matrix defining the tuning. | ||
===Algebraic considerations=== | ===Algebraic considerations=== | ||
For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a [[ | For the mathematically inclined (other people may want to skip this paragraph) we note that monzos are elements of a [[Stacking|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. | ||
== See also == | |||
* [[Projection]]: an introduction to the topic | |||