by the characteristic empirical ’analysis by synthesis’ method (see Gabrielsson, 1999).

Todd’s approach (see Todd, 1989, 1992) is backed by a systematic formalism of performance as a function of structure and specific grammatical arguments. It relates simple structural data, such as grouping boundaries, to expression by means of physically oriented transformation rules.

The method of Mazzola and collaborators is centered around the concept of weight. These are numerical functions that encode analyses and serve as input to the core of grammatical instances: trees and operators. Accordingly, a performance generated by the tree, a genealogical stemma of nodes representing local performance scores of successively refined performance quality. The generation of such node from antecedent involves performance operators. The latter are charged with weights and realize grammatical rules of different flavours. The nature of these rules is not further specified, and may include any of the system proposed by other approaches as long as they are based upon weights (in particular the KTH and Todd proposals). We will develop these aspects in the next section.

Under the term “learning grammars” we understand grammatical patterns generated by machine-based algorithmic and statistical learning from empirical performance data. A group around Giovanni de Poli has investigated various means for modeling interaction effects. They combined parameters used by KTH rules with a neural network and made it learn from performances. They further considered another alternative involving fuzzy algorithms (see R, 1993; R and Vecchio, 1994). Several other learning grammars have also been built to study music performance. An interesting approach using machine learning was taken by Gerhard Widmer (1994). He lets an artificial intelligence system infer performance rules from measured performances. We refer to Widmer (2001) for a discussion of new results in this areas. We should however note that these methods have severe limits since much of a performer’s behaviour cannot be explained at the note level (see Widmer, 2001, ), on which these methods are focused. That is why in the model that we will present in the next section, we will consider not just the “notes” by themselves, but their relations to the other parts of the piece. As we will see, even in our simple linear model, it is possible to answer coherently to important musicological questions about the comparison of several performances.

2 Analytical Stemmas

In western classical music, a composition is usually analyzed in at least three ways, namely with respect to its melodic, rhythmic, and harmonic contents. The idea of quantitative analysis is therefore to find rhythmic, melodic and harmonic structures and to quantify them in weight functions in the sense that each note event is given a weight which quantifies its “importance” with respect to the melodic, rhythmic, and harmonic structure of the composition, respectively. Clearly, methods for finding musically meaningful analytical structures must incorporate knowledge of music theory and practice. To illustrate this, we refer to the weight functions described in Mazzola et al. (Mazzola and Zahorka, 1993-1995; Mazzola, 2002).