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~~]], or [[defactored~~]]) 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]]) 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.

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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]], or [[defactored]]) 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]]) 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.