Analysis and synthesis of musical harmonies based on formal grammars

Music is one of the most fundamental and universal forms of expression of humanity, being part of the human society since its origins. As any artistic manifestation, it is a cultural product and its evolution and use in all kind of scenarios show us how important it is for the humanity. The relation between music and mathematics has been a close one, investigated since the Pythagoreans, and one of the main products of this relation is tonal music, considered one of the greatest contributions of western civilization. So not only is music one of the forms of expression that most directly speak to the emotions, it is also the one whose foundations are more firmly mathematical, making music an ideal testing ground for analyzing the formalization of emotionally charged forms of expression, and to explore the limits of such formalization. Music is not subject to simple deterministic rules, but there is an obvious structure underneath it. Investigating and formalizing such structure is important to better understand music and even to develop new ways of composition using computers as a synergistic partner or as an independent composer. Thus, in this work, we consider the usefulness and the limits of applicability of formal grammars. The use of grammars can give important insights into the harmonic structure of a piece of music, and a lot of researchers have developed grammars and applied them to different styles of music. In musical theory there are harmonic concepts that are not well-defined in a mathematical sense. In this work we considered several of these concepts and see how, and whether, grammars can be modified to accommodate them. However, the ambiguous nature of music limits the possibility of formal analysis of harmonic sequences: using the grammars we could not precisely define (and, consequently, recognize) some musical concepts, such as modulation (the change of key within a musical piece). Trying to formalize modulation leads to unavoidable ambiguities in the grammar. Consequently we tried using other methods to formalize these concepts, and we successfully could determine modulations using a numerical costbased method. Then we used grammars to analyze separately the non-modulating segments in which our algorithm has divided the music. The combination of different methods and technologies developed in this work appears to be a very bright perspective for analyzing musical pieces