JohnCockburn

Thank you David. You really are too kind. To be honest I don't know what came over me. Some kind of weird episode of craziness.

Don't worry. I'm sure I'll quite rightfully face some consequences.

David. I owe you a humble apology. Looking back over the way this thread developed, I can see that at one point i got hold of the wrong end of the stick about what you were saying, and ended up talking nonsense. What you were saying was right. I was wrong. And I also realise that there is a way that Tartini tones arising from beats could appear in a Fourier transform. If the difference frequency periodicity excites another resonance, such as an open string. That would do it, I think. Cheers john

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Stay safe, now.

Honestly, David. Go away and read a high school physics text book.

I edited that comment so as not to appear rude. But I see you managed to grab it. I have enormous respect for you as a violin maker.

No, a flat line has "zero amplitude". Doesn't matter where on the screen it is. That's just a DC-offset. Of course, amplitude without frequency is meaningless, because nothing is oscillating.

Right. We really are in the "post truth" world now.

David. This is just foolish. No-one was talking about a waveform without amplitude. But a waveform of constant amplitude. This really is weapons grade nonsense.

Always fancied having a go at Buddhism. Might help make me a bit less lairy.

LOL of the day.

Sound ...of.....man...banging...his.....head....against....brick .......wall...........

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

Doesn't matter. Apart from "displacement from equilibrium" which in general, isn't the definition of amplitude. Amplitude is the maximum displacement from equilibrium. So constant amplitude, ie with no modulation or fluctuation of the amplitude, for a sound wave, for example, means that the sound has constant intensity, or perceived loudness, if you will. Changing the amplitude changes the loudness. "Amplitude without modulation produces no sensation of sound, only something ranging between a weird feeling, and pain" No. Wrong.