Temperament mapping matrix: Difference between revisions
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The [[wikipedia: Multiplicative group|multiplicative group]] generated by any finite set of [[wikipedia: Rational number|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is 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.) | The [[wikipedia: Multiplicative group|multiplicative group]] generated by any finite set of [[wikipedia: Rational number|rational number]]s is an ''r''-rank [[wikipedia: Free abelian group|free abelian group]]. Thus, an [[abstract regular temperament]] can be represented by a [[wikipedia: Group homomorphism|group homomorphism]] '''T''': J → K from the group J of [[JI]] rationals to a [[wikipedia: Quotient group|quotient group]] K of tempered intervals. This homomorphism can also be represented by an [[wikipedia: Integer matrix|integer matrix]], called a '''temperament mapping matrix'''; when context is clear enough it is 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.) | ||