Subgroup basis matrix: Difference between revisions

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A '''subgroup basis matrix''' is matrix consisting of columns of [[monzo]]s which is a generic representation for a basis of a [[just intonation subgroup]], as its integer column spans span the subgroup. Each column represents an entry in the basis, e.g. {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7.

A '''subgroup basis matrix''' is a matrix consisting of columns of [[monzo]]s which is a generic representation for a basis of a [[just intonation subgroup]], as its integer column spans span the subgroup. Each column represents an entry in the basis, e.g. {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7.

Subgroup basis matrices are dual to [[temperament mapping matrix|temperament mapping matrices]]. Temperament mapping matrices are matrices that represent [[regular temperament]]s; they are {{w|linear map|linear maps}} that send monzos to [[tempered monzos and vals|tempered monzos]]. The integer row span of any mapping matrix is the set of all [[vals and tuning space|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals. Subgroup basis matrices are also linear maps, but they take [[subgroup monzos and vals|subgroup monzos]] and map them to regular monzos on the parent JI group. And, dual to temperament mapping matrices, subgroup basis matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w|group homomorphism|group homomorphisms}} on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called ''restricting'' (or more rarely, ''co-tempering'') the vals. These are dual to how temperament mapping matrices send [[tempered monzos and vals|tempered vals]] back to regular vals. Note the duality here – subgroup vals are a ''{{w|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.

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 {{Expert}}
-A '''subgroup basis matrix''' is matrix consisting of columns of [[monzo]]s which is a generic representation for a basis of a [[just intonation subgroup]], as its integer column spans span the subgroup. Each column represents an entry in the basis, e.g. {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7.
+A '''subgroup basis matrix''' is a matrix consisting of columns of [[monzo]]s which is a generic representation for a basis of a [[just intonation subgroup]], as its integer column spans span the subgroup. Each column represents an entry in the basis, e.g. {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 0 1 0 }} represents the 2.3.5 subgroup of 2.3.5.7.
 Subgroup basis matrices are dual to [[temperament mapping matrix|temperament mapping matrices]]. Temperament mapping matrices are matrices that represent [[regular temperament]]s; they are {{w|linear map|linear maps}} that send monzos to [[tempered monzos and vals|tempered monzos]]. The integer row span of any mapping matrix is the set of all [[vals and tuning space|vals]] that [[support]] the temperament, which form a sublattice within the lattice of vals. Subgroup basis matrices are also linear maps, but they take [[subgroup monzos and vals|subgroup monzos]] and map them to regular monzos on the parent JI group. And, dual to temperament mapping matrices, subgroup basis matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w|group homomorphism|group homomorphisms}} on vals. They send vals to subgroup vals on the basis represented by the matrix, sometimes called ''restricting'' (or more rarely, ''co-tempering'') the vals. These are dual to how temperament mapping matrices send [[tempered monzos and vals|tempered vals]] back to regular vals. Note the duality here – subgroup vals are a ''{{w|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.