Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

by associating to $(am, (cmy,x)y (- Dm(T),(byj)y (- Dm(T ),j=1,...,sy)$ the sum

sy sum sum cm byjiy,x(Oyj(rm,y(am)))+ sum cm iy,x(ft(rm,y(am))). y (- Dm(T )j=1 y,x y (- Dm(T) y,x y

The equations (2) become:

ez = (3) sum mznz-1 r cynz,mznzbynzjnz ...cmy1,z1by1j1mz,y1,...,ynz,j1,...,jnz + . sum ..+ cmyznnz,zm-z1nz ...cry1,mz1mz,y1,...,ynz

where the sum is over all $y (- D (T),y (- D (T ),...,y (- D (T) 1 r 2 mz1 nz mznz-1$ and all $j = 1,...,s , k yk$ for $k = 1,...,n . z$ The leading vector summand of this linear combination equals

mz,y1,...,yn ,j1,...,jn = (4) i (O (r (.z..i z(O (r (e)))...))). ynz,mznz ynzjnz mnz-1,ynz y1,mz1 y1j1 r,y1 r

The general vector summand refers to a choice of endomorphisms $X...= Oynzjnz$ or $t X...= fynz$ etc. and is equal to

iynz,mznz(X...(rmnz-1,ynz(...iy1,mz1(X...(rr,y1(er))) ...))).

(5)

In order to describe the solution variety we can interpret (3) as a system of linear equations with parameters,

sum mnz- 1 ez = cy,z lyz. (6) y (- Dmz,nz-1(T)

Observe that $ez,lyz$ are vectors in $Az$ , so the solutions $m cyn,zz-1$ of equations (6) result from simultaneous solutions in the vectors’ coordinates if we are given a basis of each $Az$ . The coefficients $lyz$ equal linear combinations of the vector summands (4) and (5) with coefficients which are monomials in the remaining variables $c..,.$ and the $b..$ .

If we now assume that all variables $m czynz-1$ are independent, we see that for any two different $z (- Dmnz -1(T )$ , the solutions of the equations (6) do not interact with each other. This means that the solutions of the system (6) for $z (- Dmnz-1(T)$ is either empty (if the value $ez$ is not in the image of the linear map (6)) or equals

prod prod Lz (_ C#Dmnz -1(T) z (- Dmnz-1(T) z (- Dmnz-1(T)

where $Lz$ are linear subspaces of $C#Dmnz-1(T)$ . Their codimension can be read form the matrix of the coefficients. Since there is no other condition on the other variables $cmyz,xk$ we obtain altogether the following result: