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.

The multiplicative group generated by any finite set of rational numbers is an r-rank free abelian group. Thus, an [[Abstract_regular_temperament|abstract regular temperament]], can be represented by a group homomorphism '''T''': J → K from the group J of JI rationals to a group K of tempered interval. 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. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)

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.

These are dual, in a certain sense, to [[Subgroup Basis Matrices|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~~.

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 right nullspace of M consists of the kernel of T, M is of full row rank, and the rows of M generate a subgroup of the group vals 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 group.

The column ~~module~~ of any mapping matrix is the ~~module~~ of T-tempered intervals, also known as the ~~module~~ of [[Tmonzos_and_Tvals|tmonzos]] for T. The row ~~module~~ of any mapping matrix for a temperament T is the ~~submodule~~ of all ~~vals supporting~~ T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows ~~are r linearly independent~~ vals supporting T ~~and which form a saturated submodule of the module of vals~~ will be a valid mapping for T.

The integer column span of any mapping matrix is the group of T-tempered intervals, also known as the group of '''tempered monzos''' [[Tmonzos_and_Tvals|tmonzos]] for T. The integer row span of any mapping matrix for a temperament T is the subgroup of vals that all support T. Note also that this means that if T is of rank r, then any rank-r matrix in which the rows span the subgroup of vals supporting T will be a valid mapping for T.

Note also that since all mapping matrices for T will have the same row ~~module~~, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[Normal_lists#x-Normal val lists|normal val list]], or more generally if they have the same Hermite normal form.

Note also that since all mapping matrices for T will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[Normal_lists#x-Normal val lists|normal val list]], or more generally if they have the same Hermite normal form.

=Dual Transformation=

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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.
+The multiplicative group generated by any finite set of rational numbers is an r-rank free abelian group. Thus, an [[Abstract_regular_temperament|abstract regular temperament]], can be represented by a group homomorphism '''T''': J → K from the group J of JI rationals to a group K of tempered interval. 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. (Note that many group homomorphisms can correspond to the same temperament, simply mapping to a different choice of tempered coordinates.)
-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.
+These are dual, in a certain sense, to [[Subgroup Basis Matrices|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.
+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 right nullspace of M consists of the kernel of T, M is of full row rank, and the rows of M generate a subgroup of the group vals 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 group.
-The column module of any mapping matrix is the module of T-tempered intervals, also known as the module of [[Tmonzos_and_Tvals|tmonzos]] for T. The row module of any mapping matrix for a temperament T is the submodule of all vals supporting T. Note also that this means that if T is of rank r, then a rank-r matrix in which the rows are r linearly independent vals supporting T and which form a saturated submodule of the module of vals will be a valid mapping for T.
+The integer column span of any mapping matrix is the group of T-tempered intervals, also known as the group of '''tempered monzos''' [[Tmonzos_and_Tvals|tmonzos]] for T. The integer row span of any mapping matrix for a temperament T is the subgroup of vals that all support T. Note also that this means that if T is of rank r, then any rank-r matrix in which the rows span the subgroup of vals supporting T will be a valid mapping for T.
-Note also that since all mapping matrices for T will have the same row module, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[Normal_lists#x-Normal val lists|normal val list]], or more generally if they have the same Hermite normal form.
+Note also that since all mapping matrices for T will have the same integer row span, we can easily check to see if two matrices represent the same temperament by checking to see if they generate the same [[Normal_lists#x-Normal val lists|normal val list]], or more generally if they have the same Hermite normal form.
 =Dual Transformation=