Subgroup basis matrix: Difference between revisions

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[[~~Temperament mapping matrices~~]] ~~are matrices that represent~~ [[~~regular temperament~~]]~~s; they are~~ {{w|~~linear map~~|~~linear maps~~}} ~~that send [[monzos and interval space|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~~.

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.

~~There is a "dual" set of '''subgroup~~ basis matrices~~''' (or "subgroup~~ matrices~~" for short when the context is clear), in which we look at~~ matrices ~~in which the columns~~ are ~~monzos. These~~ matrices ~~have relevance in representing~~ [[~~subgroup~~]]s~~, as their~~ integer ~~column spans~~ span ~~some subgroup~~ of [[JI]]~~. Each column represents an entry in~~ the ~~basis for~~ a ~~subgroup, 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. These~~ matrices take [[subgroup monzos and vals|subgroup monzos]] and map them to regular monzos on the parent JI group.

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.

And, dual to temperament mapping matrices, these subgroup matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w|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.

Since the kernel of any temperament is a subgroup of JI, subgroup basis matrices can thus be used to represent kernels. They can also be used to compute the subgroup restriction of a val or mapping matrix to a smaller subgroup.

~~(Note the duality here – subgroup vals are a ''{{w|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.)~~

~~Subgroup basis matrices can be used as a generic representation for a basis of any subgroups of JI.~~ Since the kernel of any temperament is a subgroup of JI, ~~they~~ can thus be used to represent kernels. They can also be used to compute the "subgroup restriction" of a val or mapping matrix to a smaller subgroup.

== Mathematical definition ==

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 {{Expert}}
-[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are {{w|linear map|linear maps}} that send [[monzos and interval space|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.
+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.
-There is a "dual" set of '''subgroup basis matrices''' (or "subgroup matrices" for short when the context is clear), in which we look at matrices in which the columns are monzos. These matrices have relevance in representing [[subgroup]]s, as their integer column spans span some subgroup of [[JI]]. Each column represents an entry in the basis for a subgroup, 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. These matrices take [[subgroup monzos and vals|subgroup monzos]] and map them to regular monzos on the parent JI group.
+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.
-And, dual to temperament mapping matrices, these subgroup matrices can also be left-multiplied by vals and thus thought of as linear maps or {{w|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.
+Since the kernel of any temperament is a subgroup of JI, subgroup basis matrices can thus be used to represent kernels. They can also be used to compute the subgroup restriction of a val or mapping matrix to a smaller subgroup.
-(Note the duality here – subgroup vals are a ''{{w|quotient group}}'' of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.)
-Subgroup basis matrices can be used as a generic representation for a basis of any subgroups of JI. Since the kernel of any temperament is a subgroup of JI, they can thus be used to represent kernels. They can also be used to compute the "subgroup restriction" of a val or mapping matrix to a smaller subgroup.
 == Mathematical definition ==