Anyway, to get back to the discussion of “Are Tartini Tones just high frequency beats?"
If we want to demonstrate the phenomenon of beats, we take a signal that’s a linear superposition of 2 sinusoidal waveforms of frequencies f1 and f2.
ie: S = sin (2πf1t) + sin (2πf2t)
We arrange things so that we can vary f1 and f2. When these frequencies get within a few Hz of each other, we can clearly hear “beats” ie periodic increases and decreases in the loudness of the signal, at a frequency of f2-f1. The important thing to realize is that these beats aren’t in any sense an “additional signal”. Its just that when we add waveforms together, we will encounter situations when they will interfere constructively and destructively. Sometimes they’ll boost each other, sometimes they’ll partially cancel out. This is what gives the beats. Periodic phase cancellation. So whether we can hear beats or not, the signal is always:
S = sin (2πf1t) + sin (2πf2t)
NOT
S = sin (2πf1t) + sin (2πf2t) + woowoo
Because the woowoo is already in the addition of the sine terms.
Fourier transforming this signal is trivial. just delta functions (spikes) at the frequencies f1 and f2. So that’s all we see in the FFT. Nothing, ever, at the difference frequency f2-f1.
CTstanzio’s frequency spectrum, and ones I’ve taken myself, many times, clearly shows the Tartini tone, as well as the sum and frequency doubled signals predicted by theory. If the Tartini tone can be seen in the FFT, it can’t have its origin in beats. No way.
So, how can we generate a difference frequency signal, that will show on an FFT, from 2 waveforms of frequency f1 and f2?
We have to arrange it so that we multiply the waveforms together, rather than adding them. For waveforms on paper, we can of course, just calculate the product and display the results on our computer screen. To do it for physical waveforms we have to find a physical way to multiply or “mix” the waveforms together. This is done by passing the waves through a nonlinear system. (see the attached appendix file- I'm not going to spend ages typing out all those trig functions here )
One of the terms produced in the nonlinear mixing of our S = sin (2πf1t) + sin (2πf2t) signal is a multiplicative one. You’ll need to look at the attached file to see exactly how that arises. The upshot is, that after application of trig identities, we can write the multiplicative term as:
2Cos (2πf1t)* Cos (2πf2t) = Cos (2π(f1+f2)t) + Cos (2π(f1-f2)t)
(i've changed from sin's to cos's for consistency with the appendix. As I'm sure you know, this doesn't matter. A cos is a sin with a 90 degree phase shift)
The second term on the RHS is the tartini tone, the other the sum frequency tone.
Fourier transforming this gives you peaks at the sum and difference frequencies in the FFT spectrum.
So hopefully, we can see how nonlinearity leads to the tartini tone. This nonlinearity can be in the auditory system, the instrument , the microphone/electronice you use to record the signal, etc etc.
So there you go.
JC
JC_TARTINI_APPENDIX.pdf
