Temperament mapping matrix: Difference between revisions

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=Basics=

The multiplicative group generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an [[Abstract_regular_temperament|abstract regular temperament]], which is a group homomorphism '''T''': J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a '''temperament mapping matrix'''; when context is clear enough it's also sometimes just called a '''mapping matrix''' for the temperament in question. ~~Since there is more than one type of mapping matrix which appears~~ in ~~music theory, it has also more rarely been called~~ a ~~"monzo-map" or '''M-map''' when context demands~~, ~~as opposed~~ to ~~the~~ [[~~Subgroup_Mapping_Matrices_(V-maps)|V-map~~]] which ~~is a~~ mapping ~~on vals~~.

The multiplicative group generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an [[Abstract_regular_temperament|abstract regular temperament]], which is a group homomorphism '''T''': J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a '''temperament mapping matrix'''; when context is clear enough it's also sometimes just called a '''mapping matrix''' for the temperament in question.

These are dual, in a certain sense, to [[Subgroup_Basis_Matrices]], which can be thought of as "co-tempering" vals in the same way that temperament mapping matrices "temper" monzos.

Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the null module of M consists of the kernel of T, M is of full row rank, and the rows of M generate a submodule which is [[Saturation|saturated]]. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U∙M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the tempered interval module.

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 =Basics=
-The multiplicative group generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an [[Abstract_regular_temperament|abstract regular temperament]], which is a group homomorphism '''T''': J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a '''temperament mapping matrix'''; when context is clear enough it's also sometimes just called a '''mapping matrix''' for the temperament in question. Since there is more than one type of mapping matrix which appears in music theory, it has also more rarely been called a "monzo-map" or '''M-map''' when context demands, as opposed to the [[Subgroup_Mapping_Matrices_(V-maps)|V-map]] which is a mapping on vals.
+The multiplicative group generated by any finite set of rational numbers, which is an r-rank free abelian group, also naturally has the structure of being a Z-module. Thus, an [[Abstract_regular_temperament|abstract regular temperament]], which is a group homomorphism '''T''': J → K from the group J of JI rationals to a group K of tempered intervals, also has the additional structure of being a Z-module homomorphism. This homomorphism can also be represented by a integer matrix, called a '''temperament mapping matrix'''; when context is clear enough it's also sometimes just called a '''mapping matrix''' for the temperament in question.
+These are dual, in a certain sense, to [[Subgroup_Basis_Matrices]], which can be thought of as "co-tempering" vals in the same way that temperament mapping matrices "temper" monzos.
 Assuming we use the convention that row matrices represent vals and column matrices represent monzos, then a matrix M is said to be a mapping matrix for a temperament T if and only if the null module of M consists of the kernel of T, M is of full row rank, and the rows of M generate a submodule which is [[Saturation|saturated]]. There is generally not a unique matrix M satisfying this definition for arbitrary temperament T, as for any M which is a valid mapping for T, any matrix U∙M where U is unimodular will also be a valid mapping for T. The different mapping matrices obtainable in this way still temper out the same commas, but differ in the choice of basis for the tempered interval module.