Temperament mapping matrix: Difference between revisions

{{Expert|Mapping}}

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 - that is, a function from J to K such that the group operation of [[stacking]] is preserved. What "quotient group" means is that the elements of this group are not "intervals", but equivalence classes of intervals separated by the comma the temperament tempers out, much like how in modular arithmetic, in a modulus of 10, "2" represents the equivalence class containing the integers 2, 12, 22, etc. 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.)