Projection matrices: Difference between revisions

Line 23:

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===

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 [[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.

@@ Line 23: / Line 23: @@
 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===
-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 [[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.