Hearing a trigonometric identity

If you force a harmonic oscillator with natural frequency w_0 with a sinusoidal force of frequency w, the resulting steady-state is a linear combination of those two frequencies. If its something in the audible range, you hear either the two seperate frequencies or one frequency with a beat if they are close together. In other words, you hear either the right or left hand side of the identity shown below.



Sage source code available here. Ironically, because of my use of html I can't seem to show the source directly here.

Small frequency difference.

Smallish frequency difference (close to the Ma Bell tone).

Intermediate frequency difference.

Large frequency difference - you definitely hear it as a 2-note chord.