Roughness of amplitude modulated tones

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

Some have suggested, based on empirical studies, that p = 2, while other have suggested that p varies as a function of the frequency f_c 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 f_c is 1000 Hz and the modulation frequency f_m is 60 Hz.

Reference: Psychoacoustical Roughness: Implementation of an Optimized Model. P. Daniel and R. Weber. Acoustia 83 (1997) 113-123