Temperament mapping matrix: Difference between revisions

This page gives a formal mathematical approach to [[RTT]] mapping. For a page with a simpler introduction, see [[mapping]].

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.)

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.

Any mapping matrix can be said to represent a linear map '''M:''' J → K, where J is a group of JI intervals and K is a quotient group of tempered intervals. There is thus an associated dual transformation '''M*:''' K* → J*, where J* and K* are the dual groups to J and K, respectively. J* is the dual group of vals on J, and K* is the group of '''tempered vals''' or [[Tmonzos_and_Tvals|tvals]] on K. As tempered vals can naturally be viewed as a subgroup of all vals, so '''M'''* represents a linear transformation mapping from tvals to vals. This mapping will generally be injective but not surjective; its image is the subgroup of vals supporting the associated temperament, and no two tvals map to the same val.

These two transformations correspond to different types of matrix multiplication: the ordinary transformation '''M''' corresponds to multiplication with the mapping on the left and a matrix whose columns are monzos on the right, and the dual transformation '''M'''* corresponds to multiplication with the mapping on the right and a matrix whose rows are tvals on the left.

11-limit porcupine tempers out 55/54, 64/63, and 100/99, and is supported by the patent vals for [[15-EDO|15-EDO]] and [[22-EDO|22-EDO]]. Since these two vals form a saturated subgroup of the dual group of 11-limit vals, then the matrix formed by setting the rows to these vals is a valid mapping matrix for porcupine: