Pitch class and envelope effects in the tritone paradox are mediated by differently pronounced frequency preference regions

Abstract

Shepard tones (octave complex tones) are well defined in pitch chroma but are ambiguous in pitch height. Pitch direction judgments of Shepard tones depend on the clockwise distance of the pitch classes on the pitch class circle, indicating the proximity principle in auditory perception. The tritone paradox emerges when two Shepard tones that form a tritone interval are presented successively. In this case, no proximity cue is available and judgments depend on the first tone and vary from person to person. A common explanation for the tritone paradox is the assumption of a specific pitch class comparison mechanism based on a pitch class template that is differently orientated from person to person. In contrast, psychoacoustic approaches (e.g., the Terhardt virtual pitch theory) explain it with common pitch-processing mechanisms. The present paper proposes a probabilistic threshold model, which estimates Shepard tone pitch height by a probabilistic fundamental frequency extraction. In the first processing stage, only those frequency components whose amplitudes are above specific randomly distributed threshold values are selected for further processing, and whose expected values are determined by a threshold function. The lowest of these nonfiltered components is dedicated to the pitch height. The model is designed for tone pairs and provides occurrence probabilities for descending judgments. In a pitch-matching pretest, 12 Shepard tones (generated under a cosine envelope centered at 261 Hz) were compared to pure tones, whose frequencies were adjusted by an up-down staircase method. Matched frequencies corresponded to frequency components but were ambiguous in octave position. In order to test the model, Shepard tones were generated under six cosine envelopes centered over a wide frequency range (65.41, 261, 370, 440, 523.25, 1244.51 Hz). The model predicted pitch class effects and envelope effects. Steep threshold functions caused pronounced pitch class, whereas flat threshold functions caused pronounced envelope effects. The model provides an alternative explanation to the pitch class template theory and serves as a psychoacoustic framework for the perception of Shepard tones.