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

Finally, the Essen Folksong Collection (see Schaffrath, 1995) provides an excellent sample for comparative studies of melodies. From this huge corpus, 55 German children songs with at most 30 tones were chosen for reasons of their relative melodic simplicity and the rather limited amount of computations necessary for the analysis.

Acknowledgements

Mathematical Music Theory, as it is understood, applied, and slightly extended in this text, ows its very existence and most of its notions and ideas to Guerino Mazzola, without whom the author had never even begun to think about music mathematically. So, he is the first I want to thank for his continuous interest in my work and the permanent encouragement. Over the years, Thomas Noll has always been a stimulating partner in a discussion that will hopefully continue for a long time still. Several years ago, Anja Fleischer talked me into starting a seminar at Humboldt University in Berlin which introduced us to the active study of music as a scientific topic. Through the work and patience of Jörg Garbers, there is now a program by means of which others may have the chance not only to reproduce the analytical results presented here but, moreover, to follow this approach in their own investigations into the structure of music. The occasion these considerations were originally written for was a talk in the Special Session on Mathematical Music Theory at the “Congreso Nacional de la Sociedad Matemática Mexicana 2000” held in Saltillo. For the invitation and the hospitality during this visit to Mexico I am indebted to Emilio Lluis-Puebla. Lastly, the author gratefully acknowledges the financial support provided by a grant of the Volkswagenstiftung within the research project “KIT-MaMuTh” at the Technical University Berlin.

2 The Mathematical Model

In this section, we describe the framework for the study of symmetry in melodies to follow. Taking the ideas of Weyl and Speiser almost literally, the (non-trivial part of the) binary relation among the subsets of a melody determined by the natural action of the paradigmatic group constitutes the symmetry. The support of this relation will be the point of departure for the different characteristics designed to express the symmetry of a melody quantitatively.

The ambient space (the Gestalt space in the sense of Mazzola), in which a melody will be assumed to be modeled, is the affine plane $2 R$ , i.e., the product of the real onset and pitch lines.

Definition 1 A melody $M$ is a finite set of integral points in the affine plane not two of which lie in a vertical line.

In other words, the first components of the points in $M$ form a strictly increasing set of integers. Using this order by increasing onsets, we will write $M = (x0,...,xn)$ where $xj = (ej,hj)$ , $j = 0,...,n$ , with $e0 < e1 <$ ... $< en$ . Note that assuming all tones of a given melody to have integer coordinates is no restriction, since this can always be arranged by suitable normalizations. Nevertheless, there are