Subgroup basis matrix: Difference between revisions

[[Temperament_Mapping_Matrices|Temperament mapping matrices]] are matrices in that represent regular temperaments; they are linear maps that send monzos to "tempered monzos" or "tmonzos." The integer row span of any mapping matrix is the set of all [[Vals|vals]] that support the temperament, which form a sublattice within the lattice of vals.

Subgroup basis matrices can be used as a generic representation for a basis of any subgroups of JI. Since the kernel of any temperament is a subgroup of JI, they can thus be used to represent kernels. They can also be used to compute the "subgroup restriction" of a val or mapping matrix to a smaller subgroup.

As a preliminary, a [[Temperament_Mapping_Matrices|temperament mapping matrix]] represents some particular basis of a temperament. In mathematical terms, it represents a group homomorphism '''T''': J → K from the free abelian group J of JI ratios to a group of "tempered intervals," which is isomorphic as a group to <math>\Bbb Z^n</math>. Using the usual convention, we have that column vectors are monzos and row vectors are vals, so that the rows of these matrices are vals, and typically we will have more rows than columns. The integer row span of these matrices represent all the vals which "support" the temperament; typically we require the matrix to not be [[contorted]] (meaning the subgroup of supporting vals is [[saturated]]) and of full row rank (e.g. it is '''surjective''').

The integer column span of any subgroup basis matrix is said to '''generate''' the subgroup G. The integer row span of any subgroup basis matrix generates the "dual subgroup" of </span>[[Smonzos_and_Svals|svals]] in which the coefficients represent, in order, the mappings for the intervals specified by the columns of S.

We can also multiply a subgroup basis matrix with another matrix on the left, one in which the rows are vals. This gives the "dual transformation" of that subgroup basis. Since multiplication from the right represents a linear transformation '''S:''' G → J, mapping from subgroup monzos to monzos, the associated dual transformation is '''S*:''' J* → G*. A bit of analysis will reveal that these homomorphisms are maps which restrict vals to [[Smonzos_and_Svals|svals]] on a certain subgroup, and that the subgroup L which the elements of G* act on are [[Smonzos_and_Svals|smonzos]]. Put another way, svals are thus quotients of vals, similarly to how tmonzos are quotients of monzos; we call this '''restricting''' (or sometimes "co-tempering") the vals.

S can also represent an arbitrary subgroup of JI, such as ones with monzos we'd like to play (rather than just representing the kernel for some temperament). In this situation, it is useful to view S as a map from vals to svals on S's subgroup basis. With this interpretation, S still has a left kernel of vals, which is the set of vals that are '''restricted away''' (or "co-tempered out"), as their subgroup restriction under S is the zero sval. The vals in the left kernel have the property that, for any v and any other val k in the left kernel, we have (v+k)∙S = v∙S + k∙S = v∙S + 0= v∙S. ''In other words, any two vals differing by an element in the left kernel will restrict to the same sval.''

Say that our JI parent group J is in the 7-limit, and we want to look at temperaments on the 2.9/7.5/3 subgroup. We can create the subgroup mapping matrix by forming a matrix in which the columns are the monzo representation of these intervals:

This matrix will be called '''S''' in the examples below.

'''S''' can be viewed as a mapping from smonzos to monzos. As an example, we'll consider the matrix of smonzos [|0 1 0&gt;, |0 -2 1&gt;|] on the 2.9/7.5/3 subgroup, which represent 9/7 and 245/243.

These monzos are the 7-limit representation of 9/7 and 245/243, respectively, in 2.3.5.7 coordinates.

To restrict a val to the subgroup defined by the subgroup basis matrix, we'll left-multiply '''S''' by a val '''V'''. In this case, our val '''V''' will be the 7-limit patent val for [[12edo|12-EDO]]:

[[Category:Math]]

[[Category:Subgroup]]

[[Category:Mapping]]