Symmetry and asymmetry as a physical and perceptual feature of the complementary pair of bearing sinusoids. Part I. Amplitude and frequency envelope relations

Beating sinusoids are an interesting case of a simultaneous change of intensity and frequency achieved without the need of a modulator. Studies of the perception of beats provide numerous data concerning also the sound pitch perception. Hitherto, the following conclusions have been made from those studies: i) if the amplitude of one tone is much larger than the amplitude of the other one, of the two-tone complex, the pitch shifts towards the frequency of the larger amplitude tone; ii) if the amplitudes of the two tones are the same, the pitch is localized precisely at the arithmetic average of the two tone frequencies. These statements imply therefore, that a symmetry with respect to the arithmetic average frequency of the two-tone beatings is present in the pitch localization on the frequency scale. Most recent studies show, however, that this symmetry is not always maintained. In the current study, divided into Part 1 and Part 2, an attempt is made, basing on the discussion and numerical analysis of the functions which describe the beatings, to determine the cause of this asymmetry. One of the arguments may come from the fact that the narrow-band condition for beating waveforms is only partially satisfied. This implies that the consequences of the relative rate of changes of the amplitude envelope to the resultant frequency envelope should be considered in the analysis of the beatings signal. The lack of symmetry is evidenced by the functions which reflect the influence of the magnitude of the ratio of the amplitudes of two signal components on the values of the normalised parameters EWAIF (Envelope Weighted Average of Instantaneous Frequency) and IWAIF (Intensity Weighted Average of Instantaneous Frequency) correlated with the sound pitch. In Part 2, two psychoacoustic experiments are described that aimed at the examination of the pitch of beatings in view of the symmetry arguments mentioned above. Main conclusions obtained in this part of the study are used throughout together with the literature available on this subject.