- 223 -Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory 
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Algebraic Varieties of Musical Performances


Roberto G. Ferretti
ETH Zürich, Departement Mathematik
ferretti@math.ethz.ch

Abstract

In this article, we first give an overview, without any pretension of being exhaustive, of some theoretical tool of the modern theory of musical performance. We then present a mathematical model of performance which is based on the principles of mathematical music theory. We prove a structure theorem describing the algebraic variety of performance system parameters which induce a determined performance. This algebro-geometric model has inspired a statistical model that concretely answers typical question about the way a player performs. Accordingly, a claim like ”Horowitz has a more local way than Cortot to play Schumann’s ’Träumerei’” has a strong statistical evidence.

Montez tous à la fois, océans irrités!

Astres, détachez-vous des cieux épouvantes!

Et vous, formes de l’être à jamais disparues,

Gigantesques débris que heurtaient les charrues,

Pressez-vous sous la terre et dans vos lit poudreux.

Faites-nous une place, ô frères monstrueux!

Les Fossiles, Louis Bouilhet.

1 Performance Grammars

The idea that performance should be executed along certain regular patterns comes from the fact that music is viewed as a vehicle of contents, and that these contents are, by the very nature of musical semantics, hidden, difficult and ambiguous. In other words, the way they are expressed is an essential condition for their communication. At present, only very elementary grammatical patterns are known. We distinguish rule based approaches from rule learning procedures. For an exhaustive, comprehensive overview of performance research we refer to Gabrielsson (1999).

Essentially, three directions in rule based grammars are known: the KTH school, Neil Todd’s approach and Guerino Mazzola’s work. The terminology of Johan Sundberg and his school at KTH recovers the idea of basing performance on rules in analogy with the linguistic grammar. The first attempts toward this goal were first introduced by Lussy (1874). He proposed that musical expression is not something added to a work through performers’ actions. Instead, it is inherent in the structure of music. With this work as background, the KTH school developed a system dominated by local rules which are based on low-level structural analysis of the text. Semantics of this analysis are rational and - to less degree - gestural. They are found


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