I thought maybe it was time to exhume the argument and so include those in the the PF community who might have an interest and/or opinion in the subject.
Let me start by stating that modulation (a.k.a. mixing) is a nonlinear process while beating is a linear process. In physics these are distinctly different processes.
The term “mixing” has a very specific meaning in radio parlance. Mixing two signals of differing frequencies f1 and f2 results in sidebands (f1 + f2) and |f1 – f2|. These are new signals at new frequencies. Depending on the mixing circuit there can be many higher sidebands as well but let’s assume a simple multiplier as the mixer:
Mixed signal = sin(ω1t)*sin(ω2t) = ½ cos(ω1 – ω2)t – ½ cos(ω1 + ω2)t, ω= 2πf.
Note that two new frequencies are produced. (The original frequencies are lost in this case but that does not always obtain; cf. below).Now consider two signals beating against each other. Let the tuning fork be at f1 and the piano string at f2; then:
Beat signal = sin(ω1t) + sin(ω2t). This can be rewritten as
Beat signal = [cos(ω1 – ω2)t/2][cos (ω1 + ω2)t/2].
Now, this may look like the same two new signals generated by mixing. But that would be wrong. There are no new signals of frequencies (ω1+ω2) and |ω1 – ω2| generated. A look at a spectrum analyzer would quickly confirm this. (I am aware that the human ear does produce some distortion-generated higher harmonics, but these are small in a normal ear and certainly not what the piano tuner is listening to).
The beat signal is just the superposition of two signals of close-together frequencies. Assuming |f1 – f2| << f1, f2, the “carrier” frequency of the beat signal is at (f1 + f2)/2 and so approaches f1 = f2 when the piano string is perfectly strung, and the beat signal amlitude varies with a frequency of |f1 – f2|. |f1 – f2| can be extremely small before essentially disappearing altogether to the piano tuner, certainly on the order of a fraction of 1 Hz It should be obvious that no human ear could detect a sound at that low a frequency (the typical human ear lower cutoff frequency is around 15-20 Hz).
This confusion is not helped by other authors of some repute. For example, my Resnick & Halliday introductory physics textbook describes the beating process as follows:
“This phenomenon is a form of amplitude modulation which has a counterpart (side bands) in AM radio receivers”.
Most inapposite in a physics text! Beating produces no sidebands. A 550-1600 KHz AM signal is amplitude-modulated and of the form
[1 + a sin(ωmt)]sin(ωct)
which produces sidebands at (fc + fm) and (fc – fm) in addition to retention of the carrier signal at fc. Here fc is the carrier (say 1 MHz “in the middle of your dial”), ωm is the modulating signal,and a is the modulation index, |a| < 1 . Of course, in a radio signal, asin(ωmt) is really a linear superposition of sinusoids, typically music and speech, in the range 50 – 2000 Hz, .