Critical Preliminaries

The present part on harmony is a traditionally dominant and extended portion of music theory. Therefore, it is adequate to review some of the important approaches to harmony. This chapter is however far from a complete synthesis of harmony and its history. We have selected three representative approaches which are systematically elaborate and theoretically founded: H. Riemann, P. Hindemith, and H. Schenker. The following overview concentrates on the divergence between claim and realization, and it does, once again, lay bare the enormous difficulty to set up a precise discourse about music without—héelas—the power of mathematical language. Also this critique is not thought to be a preliminary to something which in the subsequent chapters of this part will be perfectly solved by mathematical music theory. The discourse simply tries to persuade music theorists that a) the commonly cultivated status quo of the subject is scientifically unacceptable, and b) that mathematically sharpened concepts, constructs, and models can show ways to more in-depth and precise understanding of harmony—without banning it to history and “atonal” negation. Generic harmony is a universal perspective of music, and it is unscientific as well as near-sighted if not antimusical to abandon harmonic paradigms instead of embedding them into a diachronically and synchronically open, unified, and universal concept framework. To be clear, the main question is not to defend or instantiate any ideology of harmony—this is the unhappy business of Pythagorean fundamentalists—but to investigate its possible semiotic functions in musical works and their communicative explication, to develop an adequate language, and to propose consistent and sound models of harmonic processes.