What happens when two simple tones occur simultaneously? To find out we can start with a reference tone, f0, and add a second tone, f1 = f0 + Δf, of a slightly higher frequency. The corresponding angular frequencies are ω0 = 2πf0 and ω1 = 2πf1 = ω0 + Δω. When two cosine functions are added the result is a sum of a cosine term and a sine term which are functions of ω0 and amplitude factors A and B that are functions of Δω. These can be combined into a single wave of varying amplitude and phase, A' and φ respectively.
The second representation of the combined wave can also be split up into cosine and sine terms and the two sets of amplitudes can be used to solve for A' and φ.
To simplify the calculation of A' and φ we can write down formulas for A2 + B2 and AB.
Using the results of the derivation we can compute the result when two pitches of tones C and D of the same amplitude are combined and plot the result. The amplitude of the resulting wave is also a sinusoidal function.
The amplitude, A', is affected by the fluctuating phase but is a periodic function of the time. The period is inversely proportional to the difference of frequencies. As one decreases the interval between the two tones they will "beat" more slowly. So C and D beating together will be more easily discernable than C and E.
When the two original cosine terms are of the same amplitude the change in phase is a nearly linear function of time.
Doing a polar plot of the amplitude, A', versus phase, φ, results in a circular motion that repeats itself with the same period as the amplitude. If the amplitude of the reference wave is increased relative to that of the second the oval will move farther out from the center. The size and shape of the oval does not change with changes in the relative frequencies of the two initial waves change but it is affected by changes in the relative amplitudes. When the oval moves farther from the center the swings in phase are reduced as a result.
So we see that Harmonics has its own version of epicycles.
Supplemental (Jan 11): The difference in frequencies just "rescales" time for the amplitude function A' and phase function φ.