Subgroup basis matrix: Difference between revisions

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= Mathematical Definition =

As a preliminary, a [[Temperament_Mapping_Matrices|temperament mapping matrix]] represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the free abelian group J of JI ratios to a group of "tempered intervals," which is isomorphic as a group to <math>\Bbb 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 '''surjective''').

As a preliminary, a [[Temperament_Mapping_Matrices|temperament mapping matrix]] represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the free abelian group J of JI ratios to a group of "tempered intervals," which is isomorphic as a group to <math>\Bbb 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]], or [[defactored]]) and of full row rank (e.g. it is '''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 into 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 '''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 '''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]] (or are [[enfactored]]) 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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 = Mathematical Definition =
-As a preliminary, a [[Temperament_Mapping_Matrices|temperament mapping matrix]] represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the free abelian group J of JI ratios to a group of "tempered intervals," which is isomorphic as a group to <math>\Bbb 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 '''surjective''').
+As a preliminary, a [[Temperament_Mapping_Matrices|temperament mapping matrix]] represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the free abelian group J of JI ratios to a group of "tempered intervals," which is isomorphic as a group to <math>\Bbb 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]], or [[defactored]]) and of full row rank (e.g. it is '''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 into 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 '''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 '''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]] (or are [[enfactored]]) 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.

Subgroup basis matrix: Difference between revisions

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