A recent post pointed out that two pure tones that are fairly close in pitch create a rough sound. The roughness increases with the frequency difference, up to a point, then decreases.
This post will look at a roughness in a different setting, amplitude modulation. Several psychoacoustics researchers have suggested that perceived roughness increases as a power of modulation depth, up to a maximum. That is,
where the signal is
![[1 + m\cos(2\pi f_m t)] \cos(2\pi f_c t)](http://www.johndcook.com/modulation_depth.png){kind=small}
Some have suggested, based on empirical studies, that p = 2, while other have suggested thatp varies as a function of the frequency fc of the carrier wave.
Here is an audio (.wav) file where the the modulation depth varies as a function of time, m = 0.1t where t is time in seconds.
In this example the carrier frequency fc is 1000 Hz and the modulation frequency fm is 60 Hz.
Reference: Psychoacoustical Roughness: Implementation of an Optimized Model. P. Daniel and R. Weber. Acoustia 83 (1997) 113–123
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