Tone Apperception and Weber-Fechner’s Law

In extrapolation of WilhelmWundt’s suggestion to apply Weber-Fechner’s law to apperception and in slight modification of Davis Lewin’s definition of generalized interval systems we propose a four-dimensional space A as a model for an active tone system. This space A is also a group acting on itself. Its elements are interpreted in two ways, (1) as tone loci, i.e. as states of apperceptive processes and (2) as apperceptive acts which can be aquired by such processes as transitions between tone loci. The four dimensions are constituted by semitone steps, semitone alterations, fifth steps and fifth alterations. A fundamental ingredience to the model is the distinction between actual and virtual apperception. The former is based on the group operation of A while the latter is based on the addition of tone vectors, i.e. elements of the tangent space V of A at its identity element. Tone vectors, active tones and tone loci are connected by the apperception formula App(X, v) = X ·exp(v) which adapts Wundt’s formula to the fourdimensional situation. The interpretation of this model includes general aspects of apperception, namely a study of linear and circular apperception as well as an economy principle of reason and decision. The specific music-theoretical interpretations focus on enharmonicity. As an alternative to group factorisation as an algebraic means to describe enharmonic identification we show that the discrete quintic group G ⊂ A already yields enharmonic relations as a consequence of the apperception model. It is argued that the relations of hallucination and Escher-staircase provide suitable explanations for synquintic and enharmonic identifications. As a more sophisticated case of enharmonicity we motivate the concept of synchromatic identification. Finally, attention is paid to the geometrical study of the active tone system. A pseudo-Riemannian metrics supports the economy principle and offers relationsships to relativity. 1We mention that mathematicians use the attribute Riemannian in honor of Bernhard Riemann (1826 1866), while music theorists use it in honor of Hugo Riemann (1849 1919). The present article intends to contribute with a pseudo-Riemannian apperception model to the ongoing Neo-Riemannian discourse.