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

can not be too general. Indeed, one can select subschemes that are appropriate for the study of a particular problem. For example, if one is interested in comparing several performers under the point of view of the local/global way of playing, one could try to restrict the research just at the level of the daughters of the root, and to use some more of the structures of the vector space $Ax$ (if any). An example of this procedure will be presented in the next section. Another question that could be asked is that of the final/causal way of playing. This requires to impose mutual dependence conditions upon the variables $m cy,zkx$ for varying $z (- Dmzk$ (see Mazzola and Zahorka, 1993-1995).

3 Consequence for the choice of Statistical Models

An increasing number of numerical data is becoming available so that a statistical approach to answering these questions is feasible, and also necessary in order to cope with the large amount of data. Two particular classes of statistical methods, which are particularly suitable for performance analysis, have been introduced in the papers Beran and Mazzola (1999b), Beran and Mazzola (1999a), and Beran and Mazzola (2001). Let us describe them briefly:

Approach 1: Decompose the weight functions $w$ into components $wi$ that correspond to different levels $(i = 1,2,3,...)$ of refinement. Fit a model of the form $y(si) = g(si,w1,w2,...) +ei,$ where $ei$ is a suitable zero mean process (see Beran and Mazzola, 1999b,a).
Approach 2: Decompose not only $w$ but also the tempo series $y$ into subseries $y1,y2,...$ that corresponds to different levels of “zooming-in”. For each level, fit a model of the form $yj(si) = g(si,wj)+ eij.$ The final model is then $sum yj(si) = jg(si,wj)+ eij$ (see Beran and Mazzola, 2001).

The first approach has been encoded in the so-called HISMOOTH models (Beran and Mazzola, 1999b,a) and formalizes the idea that a musical performance combines local and global structures in a final as well causal way. This is done in a way that may be interpreted as complementary to the rooted tree introduced in section 2. Instead of looking for deep and finer structures of the explanatory time series $x$ , one decomposes it ‘horizontally’ as a sum of simple pieces. This has however the advantage to provide a method allowing the classification of performances. In fact, the estimated bandwidth values $brj$ can be used for clustering (Beran and Mazzola, 1999b, $§5$ ). Whence the claims like the fact that the $m-$ curve emphasizes features that may correspond to personal style, or the fact that Horowitz plays more ‘locally’ than Cortot, (at least for what concerns Schumann’s “Träumerei”) are now supported by strong statistical evidences (Beran and Mazzola, 1999b, $§6.3$ ).

The second approach can be realized studying an alternative to HISMOOTH models, called HIWAVE models (see Beran and Mazzola, 2001), obtained using wavelet smoothing. The idea is now that one can use the information of hierarchical structure defined in 2, and assume that $y$ is related to several “layers” of the weight function that are constructed by classifying the coefficients in the wavelet