Roughness was defined by Hermann von Helmholtz in 1863 as the beating between two frequencies. He derived the Western tonal system from intervals preferred with least roughness.

Esimating roughness is performed implementing the Helmholtz idea that two frequencies with a difference of 33 Hz have a maximum roughness perception, while roughness decreases to zero when both frequencies align and it slowly turns to very small values with higher frequency differences. Helmholtz himself gave no algorithm with which he has calculated his results.

The Helmholtz roughness estimation of two complex tones is shown together with a similar algorithm of Sethares. The least roughness corresponds to the minima.

Using a cochlear model the same tones were calculated and compared to the amount of spikes at the cochlear output. The simple intervals show a sharp peak, so at right these intervals the amount of spikes is much more than with intervals having no simple ratios. Still this correponds not to perception, as roughness slowly increases and not as sharp as seen here.

 

The same tones were calculated using a Shannon entropy estimation. Again the simple intervals are preferred, still the curve is very complex and not perfectly according to perception.

So it is still open how roughness is implemented in the cochlear and the brain.