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

also had no rationale for shaping the tempo hierarchy, except intuitive graphical interaction. So the RUBATO $®$ project had to deal with the question of constructing operators for shaping performance from a more analytical point of view.

The basic extension of tempo curves to higher parameter spaces is this: The performance is described by a performance mapping $p$ from the $n$ -dimensional real space $EHLD... R$ of $n$ symbolic parameters, onset $E$ , pitch $H$ , loudness $L$ , duration $D$ , etc. to the $n$ -dimensional real space $ehld... R$ of $n$ physical parameters, onset $e$ , pitch $h$ , loudness $l$ , duration $d$ , etc. Locally on the score, we suppose that $p$ is a diffeomorphism on an $n$ -dimensional cube $C$ , applied to a finite number of score events which are contained in this cube. So for a symbolic event $X$ (uppercase), $x = p(X)$ (lowercase) denotes the associated physical performance event.

The extension of the tempo concept is given by the inverse vector field $Zp$ of the constant diagonal field $D(x) = D = (1,...1)$ on the physical space, i.e., $-1 Zp(X) = (Jp(X)) (D)$ with the Jacobian $Jp(X)$ . This defines a performance field associated with the performance map $p$ . The value $x = p(X)$ can be calculated as follows (still generalizing the situation for tempo): suppose that the performance is known for a >initial set< $I < C$ of symbolic events. Suppose also that the integral curve $integral X Zp$ of $Zp$ through $X$ hits $I$ at the initial point $X0$ , and for the curve parameter time $t$ . Then we have

x = p(X) = p(X0) - tD.

(1)

So the performance map can actually be defined from a performance field $Z$ on the cube domain $C$ , together with an initial performance map $p : I --> Rehld... I$ . On the tempo level, the initial performance is the moment where the conductor lowers the baton to initialize the performance, and on the pitch level, the initial performance encodes the concert pitch!

This generalization is only possible by use of the generic mathematical concept framework of differential geometry. And it has the great advantage that it includes a very fine shaping tool for musical performance: performance vector fields! This meets the philosophy of performance as an effort of »infinite subtlety«, as it was established by Theodor W. Adorno and Walter Bejamin (W., 1976). Moreover, the shaping operators of performance can now be defined as operators which act on given performance fields and depend upon parameters which are typically available from data of harmonic, rhythmical, or motivic analysis of the underlying score. This would also meet Adorno’s principle of a performance which is based on understanding the score’s logical structure.

At present, there is no general system of performance operators. Several operators have been implemented on the RUBATO $®$ platform, and they have been tested for classical scores, such as Bach’s Kunst der Fuge (see Stange-Elbe (2000) for a very interesting performance of Contrapunctus III). The most general type of operators are linear operators in the analytical parameters as well as in the given performance field. See Mazzola and Zahorka (1993-1995); Mazzola (1995a); Ferretti and Mazzola (2001) for this subject. The formal setup of this operator type follows these lines: We are given the analytical information in form of a »weight«, i.e., a function $/\ : C --> R$ . This is what the analytical moduli of RUBATO $®$ in fact do calculate. Then, we are given an affine endomorphism $Dir$ of the symbolic