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

Each mother-daughter arrow $x --> y$ is represented by a surjective linear map $rx,y : Ax --> Ay$ .

For all $x (- V(T)$ let $f : B --> End(A ) x x x$ be an affine map, i.e., a linear map $f0 x$ , followed by a displacement by $ft (- End(A ) x x$ . So each loop $x O$ is represented by a family of endomorphisms $(f (b) : A --> A ) x x xb (- Bx$ parameterized by the parameter space $B x$ .

For all pairs of sisters $x,y (- Dm(T )$ , the sister arrow $x --> y$ is represented by an isomorphism $ix,y : Ax --> Ay$ with $iy,x = i-x1,y$ .

Example 1

In stemma theory it is shown that a number of parameterizations of theoretical interest (Mazzola and Zahorka, 1993-1995) and of practical use in the RUBATO $®$ software (Mazzola and Garbers, 2001) can be subsumed under the method of Lie operators in the following sense.

To begin with, fix a number $n$ of musical parameters. Each vertex $x$ of the stemma tree $T0$ is associated with a closed rectangle $Rx = [lx1,ux1]×...[lxn,uxn]$ where the music events of the score are placed. For each mother $m$ and daughter $x$ , we suppose that $Rx < Rm$ , and that for each couple of sisters $x1,x2$ , $Rx1 /~\ Rx2 = Ø$ . This corresponds to a restriction of a larger portion of a musical score to a disjoint grouping of smaller portions. Here is the realization of our above system 1.-4. of quiver representations:

Consider the vector space $Fx$ of $oo C$ functions on $Rx$ . We then set $Ax = C ox R Der(Fx)$ , the space of complexified derivations, i.e., the $oo C$ vector fields on $Rx$ . In performance theory, such “performance fields” represent the deformations of the score data under a determined performance. They are the adequate generalization of tempo curves, the one-dimensional performance fields in time, (see also Mazzola, 2002). The surjective maps $rx,y$ are defined as the complexified restrictions of vector fields on the mother’s rectangle $Rx$ to the daughter’s rectangle $Ry$ . To define the representations for a sister arrow $x1-- > x2$ , consider the unique affine morphism $ax2,x1 : Rx2 --> Rx1$ on the sisters’ rectangles such that the respective vertexes are mapped onto each other. Then the sister arrow representations are the complexified isomorphisms $ix1,x2 : Ax1 --> Ax2$ induced by the transport of a vector field $Z$ on $Rx1$ to $Z .ax2,x1$ on $Rx2$ .

To define the operation of parameter family, first take a vector field $Z (- Der(Fx)$ , and a function $/\ (- Fx$ . Denote by $Aff(U, V)$ the vector space of affine morphisms from vector space $U$ to vector space $V$ . For $n n Dir (- Aff(R ,R )$ , consider the corresponding vector field $GDir(t) = (t,Dir(t))$ on $Rx$ . Then we have a new vector field $Z - LZ/\.GDir$ , where $L$ is the Lie derivative. This is a $R$ -linear operator on $Z$ , and the ‘deformation’ part $LZ/\.GDir$ is $R$ -bilinear in the function and the affine endomorphism.

We now take a finite dimensional subspace $Wx$ of $Fx$ which in performance theory represents the weight functions issued from analyses of metrical, motivic, and harmonic structures of the given score. We now set $n n Bx = C ox R Wx ox R Aff(R ,R )$ , and we obtain a $C$ -linear map $0 fx : Bx --> End(Ax)$ defined by $0 f x(/\ ox Dir)(Z) = - LZ/\.GDir$ .