Beating and Roughness

When a pair of beating sinusoids are equal in amplitude, their temporal envelope is characterized by a full-wave rectified sinusoid. This envelope has an oscillation rate at the absolute frequency difference () between the original two sinusoids. If is greater than an auditory filter bandwidth (or critical band), the combination is perceived as two distinct tones.
Temporal envelope changes produced by amplitude modulation can be perceived as fluctuation strength (or beats) or roughness, depending upon the modulation rate. Fluctuation strength has a maximum at about 4 Hz and roughness peaks at about 70 Hz (see [1]). Roughness tends to disappear when the modulation frequency is greater than 300 Hz. The continuous transition between the two perceptual modes is at about 15–20 Hz. To quantify fluctuation strength and roughness, Fastl and Zwicker proposed the respective units vacil and asper.


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[1] H. Fastl and E. Zwicker, "Psychoacoustics: Facts and Models." Springer Series in Information Sciences, 3rd ed., 22, Berlin: Springer, 2007.
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