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

Harmonic Path Analysis

Thomas Noll and Jörg Garbers

Technical University of Berlin

Research Group KIT-MaMuTh for Mathematical Music Theory $*$

noll@cs.tu-berlin.de, jg@cs.tu-berlin.de

$*$ Financed by the Volkswagen-Foundation in its »Young Research Groups at the Universities« programm.

Abstract

The article proposes a conceptual framework for a special type of experiments in harmonic analysis and discusses aspects of its implementation in a software tool - called HarmoRubette. The framework comprises three basic components, namely (1) a harmonic configuration space $HARM$ equipped with harmonic tensor $ht$ quantifying the transitions between harmonic loci, (2) a Riemann Logics $RL$ , quantifying the signification of harmonic loci through chords and (3) a best path calculation method based on the assumption of a first-order-transition model.¹

¹: Mazzola (2002), chapter 27, and especially section 27.2, presents considerations on a more general framework of harmonic semantics into which the present one can be embedded as a special case.

Aside from systematic and implementatory aspects much attention is paid to the study of concrete examples: the space of Riemann functions according to the concept of the >classic< HarmoRubette, Elaine Chew’s Spiral array model, Fred Lerdahl’s hybrid chordal/regional space.

1 Paths in Harmonic Spaces

Musical harmony is a fascinating subject domain gaining a growing interest from researchers in several disciplines - music theorists, psychologists, neuro-physiologists, computer scientists, as well as mathematicians. However, this subject domain is far from being conceptually grasped. The various approaches do not just differ in their ways of interpreting or explaning commonly accepted facts, but they significantly differ in their understanding of what is the relevant data to be interpreted. The present paper investigates a special type of approach which may be labeled first order transition models. They describe analyses of chord sequences in terms of pathways through abstract harmonic spaces, where »first order« refers to the strong assumption--or restriction--, namely that pathways can be understood simply from the investigation of dyadic transitions. The interest in this study emerged from the authors’ involvement in the extension of the software $RU BAT O$ (c.f. Garbers (2004) in this volume as well as Mazzola (2002)) and was driven by the desire to make the analytical approach behind the HarmoRubette--designed by