Temperament mapping matrix: Difference between revisions

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=Basics=

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 quotient group K of tempered intervals. 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.)

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 quotient group K of tempered intervals. 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''' or even just a '''mapping''' 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|subgroup basis matrices]], which can be thought of as "co-tempering" vals in the same way that temperament mapping matrices "temper" monzos.

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 =Basics=
-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 quotient group K of tempered intervals. 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.)
+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 quotient group K of tempered intervals. 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''' or even just a '''mapping''' 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|subgroup basis matrices]], which can be thought of as "co-tempering" vals in the same way that temperament mapping matrices "temper" monzos.