Spectrum of Pitch Differences Models the Perception of Octave Ambiguous Tones

In the experiment we proof the paradoxical perception of pitch. A tone sequence demonstrates this paradox: the sequence is perceived as descending, although the first and last tone are identical. The intransitivity of pitch perception generalizes (Shepard 1964). It applies also to harmonic complex tones with variate amplitude envelope and partials that are arranged in intervals different from octaves. The paradox perception in this experiment as well as in the Shepard scale is modeled by a formula (spectrum of pitch differences) that extends the virtual pitch algorithm (Terhardt 1998) to pitch intervals. Experiment: Judgment on Pitch Comparison Circular structures A relation ‘ ’ is termed circular, if there are elements with . In contrast to a linear order ‘ ’ is not transitive. The paradox of our experiments is based on this circularity and intransitivity of pitch relation. According to common notion any tone is assigned to one pitch on a linear scale. Pitch relation is thus identified with ‘ ’ resp. ‘ ’ for real numbers. But this order is linear and transitive and thereby as we will see in contradiction to our experiments. Description of the Experiment The stimuli consist of tone pairs. Six tone pairs form a logical block. To each tone pair there exists another tone pair in reverse order. In addition every tone pair exists in 9 transpositions. One block contains six distinct harmonic complex tones (cf. Figure 1). Those are combined to six tone pairs: 1 with 2; 2 with 3; ... ; 5 with 6; 6 with 1. Each of the 18 subjects has to give a judgment on pitch comparison to each tone pair: 1) ‘first tone higher’, 2) ‘second tone higher’, 3) ‘both tones are equal’, 4) ‘undecidable’. The sequence of tone pairs is presented in random order. Result A subject has perceived six tones of a block paradoxically, if there is at least one ascending tone pair and all 0 50