2.3.  Transformations

The principle of applying transformations is standard in music. Typical examples used in tradional music are shift (e.g. repetition, transposition), shear (e.g. arpeggio), reflection (e.g. retrograde), dilatation (e.g. augmentation). In the 20th century, other transformations such as exchange of coordinates and rotation were also applied by some composers (e.g. Boulez, Kagel, Eimert, Stockhausen; also Messiaen).

The transformations that have been used by composers (in western music) throughout the centuries can be described in a unified manner by the mathematical definition of affine transformations. These are defined by f(x) = g(x) + a (a is a vector) with g linear, i.e. g(x + y) = g(x) + g(y) and g(bx) = bg(x) (b is a scalar).

2.4.  Hierarchy

Global and local structures are often shaped by starting with a global structure, then refining local structures. This is the case for the compositional process as well as for the performance (and rehearsal) of a composition. Mathematically, this can be formalized, for instance, by defining different local coverings (subsets) of a global composition (Mazzola 1990) and by hierarchical smoothing models (Beran and Mazzola 1999a,b, Mazzola and Beran 1997).

3.  Analysis: Musical analysis of historic compositions and performance

An important way of validating the relevance of mathematical models for music is to analyze existing compositions and performances. For the theory mentioned above, empirical studies are reported in Beran and Mazzola (1999a,b, 2000) and Mazzola and Beran (1997). Scores by Bach, Schumann and Webern were analyzed using the RUBATO software (Mazzola and Zahorka 1993–1995, 1996). RUBATO is based on an extension of the mathematical music theory described in Mazzola (1990). The results revealed interesting features of the compositions that were not obvious before the analysis. Moreover, for one of the scores (Träumerei by Schumann), the numeric data obtained from the analysis were compared to tempo curves from 28 historic performances by 25 pianists (Horowitz and Cortot were represented by three performances each). Using so-called HISMOOTH models (hierarchical smoothing models), common features as well stylistic differences and clusters of performers could be identified, and explained by structures in the score. In particular, the following results were obtained:

• A “Horowitz-Cluster” could be identified including Klien, Brendel and all three performances by Horowitz. In spite of a time span of several decades, the three performances by Horowitz were very similar to each other.
• A “Cortot-Cluster” could be identified including Argerich, Capova, Demus, Kubalek, Shelley and all three Cortot performances. The same remark as above applies to the three performances by Cortot.