A Mathematical Model for Tone Apperception

To deal with musical ambiguity is an emprically difficult but promising task. The paper draws a connection between very general issues of mental activity on the one hand and concrete phenomena of tonal ambiguity on the other. We think that a convincing understanding of ambiguity in the surface tonality of musical pieces (like V/I I/V substitutions or enharmonic identifications) will eventually help to gain insights into the more general principles of apperception, i.e. into the dynamics of our self-conscious mental activity. The present mathematical model of tone apperception is built upon Wilhelm Wundt’s relativistic apperception concept according to which the voluntary steering of attention is essential for the minds access to its own states. We give general heuristic as well as concrete music-theoretical arguments that the symplectic group Sp1(R) is a good candidate for a group theoretical modelling of elementary apperception processes. The rather heuristic arguments are rooted in the history of several ideas from the 19th and early 20 th century. According to Wilhelm Wundt’s eye metaphor apperception is a kind of inner vision where ideas enter the scope of attention either passively or as results of active attentional turns. In the writings of several authors (like Hugo Riemann and Ernst Kurth) musical experience is described in terms of motion metaphors. If one is looking for a mathematical bridge between these two kinds of metaphors (optical and mechanical) one has to remember William Hamilton’s successful transfer of Fermat’s principle from optics into mechanics which is now reflected in the applicatioon of symplectic geometry in both domains. In other words, there is a strong heuristic suggestion to study symplectic transformations in music theory, as far as one takes Wundt’s optical/visual and H. Riemann’s and other’s motion metaphors seriously. Also following Wundt we distinguish apperceptive acts as such from their corresponding noticability vectors, which are the infinitesimal generators of the acts themselves, i.e. elements of the Lie Algebra sp1(R). The exponential map exp : Sp1(R)→ Sp1(R) mathematically realizes Wundt’s idea to apply Weber-Fechner’s law to the very psychological domain (besides the celebrated psycho-physiological one). The apperception formula