|
the end, they defined melodic weights designed to be used for shaping performances of the piece according to the analytical insights encoded therein by means of the software package RUBATO (see Mazzola and Zahorka, 1993-1995). In this note, a particular situation within the general framework of PMA will be discussed with the principal goal to pave the way for musicologists and others interested in the structural analysis of melodies from the point of view of symmetry. Hence, central to the following discussion will be to introduce several mathematically motivated notions that might serve to grasp, to order, and to measure symmetry in musical pieces. In addition, we try to find musical interpretations for them and begin their exploration by discussing some properties and, in particular, a series of examples. To this end, we will apply this circle of ideas to a specific range of melodies. At various places, results of the computations will serve to illustrate the approach advocated here. Moreover, we will indicate possible further applications and research directions. The musical examples are the following:
The computation for the theme from ”The Art of the Fuge” was stimulated by Buteau (2001). In that paper, the question whether the theme already ends with the eighth note or extends to the twelfth was addressed from the point of view of PMA in the contour Gestalt by means of motivic evolution trees. Of course, Buteau and Mazzola did not intend to answer this problem finally, and it cannot be resolved here, either. What may be offered as additional evidence instead, are two facts: First, the short theme is almost “rigid” in the sense that it hardly contains any inner symmetry, whereas a considerable amount of symmetry is displayed by the long one. Second, the particular constituents of the overall symmetry seem to support the derived character of the additional tones with respect to motivic relations. In Baroni and Jacoboni (1978), Baroni and Jacoboni analyzed a specific set of phrases taken from Bach chorales with respect to their rhythmic, melodic, and harmonic structure in order to formulate rules together forming a grammar grasping as much of these characteristics as possible. For reasons of comparability, the sample comprised the first two soprano phrases of 60 four-voice chorales in C major and 4/4 meter containing no modulations. To be able to test their results, Baroni and Jacoboni used a computer program to generate 40 phrases by randomly choosing tones within certain limits set by one group of the above rules, which then had to pass the test provided by the remaining ones. Apart from computing musical examples, the main purpose for considering these melodies here was to see whether there actually are significant differences between these two samples that could be detected by the methods to be described below. |