Subgroup basis matrix: Difference between revisions

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[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are ~~[[wikipedia: Linear~~ map|linear maps]] that send [[~~Monzos~~ and interval space|monzos]] to tempered monzos ~~(→ [[Tmonzos~~ and ~~tvals~~|~~tmonzos~~]]). 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.

[[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.

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 (→ [[Smonzos and svals|smonzos]]) and map them to regular monzos on the parent JI group.

And, dual to temperament mapping matrices, these subgroup matrices can also be left-multiplied by vals and thus thought of as linear maps or ~~[[wikipedia: 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 vals ~~(→ [[tvals~~]]), back to regular vals.

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.

(Note the duality here – subgroup vals are a ~~[[wikipedia: Quotient group|~~''quotient group'']] of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.)

(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 ==

As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the ~~[[wikipedia: Free abelian group~~|free abelian group]] J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\mathbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is ~~[[wikipedia: Surjective~~ function|surjective]]).

As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the {{w|free abelian group}} J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\mathbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is {{w|surjective function|surjective}}).

We can similarly look at the matrices formed by monzos, in which the column vectors are monzos, which we call a '''subgroup basis matrix'''. In mathematical terms, these represent group homomorphisms '''S''': G → J, where G is some subgroup of J, being injected back into the parent JI group J. We can view this matrix as mapping the subgroup monzos back into the parent basis, and thus translating the coordinate system from the subgroup basis to the parent basis. The integer column span of these matrices represents all the monzos within the subgroup.

Typically, for a matrix S, with column vectors as monzos, to represent a true subgroup basis matrix, it must also be of full column rank, much like a temperament matrix must be of full row rank. Another way to look at this requirement is that it is ~~[[wikipedia: Injective~~ function|injective]] into the parent group, dual to how we want mapping matrices to be surjective. However, we typically drop the restriction that this column span be [[saturated]], so that we can represent, for instance, the 2.9.5 subgroup, unlike with temperament mapping matrices, where unsaturated matrices have [[contorsion]] and are viewed as pathological.

Typically, for a matrix S, with column vectors as monzos, to represent a true subgroup basis matrix, it must also be of full column rank, much like a temperament matrix must be of full row rank. Another way to look at this requirement is that it is {{w|injective function|injective}} into the parent group, dual to how we want mapping matrices to be surjective. However, we typically drop the restriction that this column span be [[saturated]], so that we can represent, for instance, the 2.9.5 subgroup, unlike with temperament mapping matrices, where unsaturated matrices have [[contorsion]] and are viewed as pathological.

Note that, much like with temperament mapping matrices, there is not a unique basis matrix corresponding to any subgroup: for instance, the two subgroup bases "3.2.5" and "2.3.5" represent the same subgroup, but will be represented by different matrices. Similarly, these two matrices will send vals to svals on the "2.3.5" and "3.2.5" bases respectively.

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 {{Expert}}
-[[Temperament mapping matrices]] are matrices that represent [[regular temperament]]s; they are [[wikipedia: Linear map|linear maps]] that send [[Monzos and interval space|monzos]] to tempered monzos (→ [[Tmonzos and tvals|tmonzos]]). 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.
+[[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.
 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 (→ [[Smonzos and svals|smonzos]]) and map them to regular monzos on the parent JI group.
-And, dual to temperament mapping matrices, these subgroup matrices can also be left-multiplied by vals and thus thought of as linear maps or [[wikipedia: 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 vals (→ [[tvals]]), back to regular vals.
+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.
-(Note the duality here – subgroup vals are a [[wikipedia: Quotient group|''quotient group'']] of regular vals, whereas subgroup monzos are a ''subgroup'' of regular monzos.)
+(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 ==
-As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the [[wikipedia: Free abelian group|free abelian group]] J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\mathbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is [[wikipedia: Surjective function|surjective]]).
+As a preliminary, a temperament mapping matrix represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the {{w|free abelian group}} J of JI ratios to a group of "tempered intervals", which is isomorphic as a group to <math>\mathbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is {{w|surjective function|surjective}}).
 We can similarly look at the matrices formed by monzos, in which the column vectors are monzos, which we call a '''subgroup basis matrix'''. In mathematical terms, these represent group homomorphisms '''S''': G → J, where G is some subgroup of J, being injected back into the parent JI group J. We can view this matrix as mapping the subgroup monzos back into the parent basis, and thus translating the coordinate system from the subgroup basis to the parent basis. The integer column span of these matrices represents all the monzos within the subgroup.
-Typically, for a matrix S, with column vectors as monzos, to represent a true subgroup basis matrix, it must also be of full column rank, much like a temperament matrix must be of full row rank. Another way to look at this requirement is that it is [[wikipedia: Injective function|injective]] into the parent group, dual to how we want mapping matrices to be surjective. However, we typically drop the restriction that this column span be [[saturated]], so that we can represent, for instance, the 2.9.5 subgroup, unlike with temperament mapping matrices, where unsaturated matrices have [[contorsion]] and are viewed as pathological.
+Typically, for a matrix S, with column vectors as monzos, to represent a true subgroup basis matrix, it must also be of full column rank, much like a temperament matrix must be of full row rank. Another way to look at this requirement is that it is {{w|injective function|injective}} into the parent group, dual to how we want mapping matrices to be surjective. However, we typically drop the restriction that this column span be [[saturated]], so that we can represent, for instance, the 2.9.5 subgroup, unlike with temperament mapping matrices, where unsaturated matrices have [[contorsion]] and are viewed as pathological.
 Note that, much like with temperament mapping matrices, there is not a unique basis matrix corresponding to any subgroup: for instance, the two subgroup bases "3.2.5" and "2.3.5" represent the same subgroup, but will be represented by different matrices. Similarly, these two matrices will send vals to svals on the "2.3.5" and "3.2.5" bases respectively.