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The signal you describe in your comment is very much NOT an A. look at the fourier transform of a 1kHz signal modulated by a 440Hz signal, and you won't see any

The signal you describe in your comment is very much NOT an A. look at the fourier transform of a 1kHz signal modulated by a 440Hz signal, and you won't see any frequency component at 440Hz, nor any integer multiple 440Hz! You can look at your "A440 at a constant volume" example as a 0Hz(DC) signal getting louder and softer 440 times a second, but this is the only case in which your example holds. Amplitude modulation creates sum and difference frequencies, so the A that you hear is 440Hz + 0Hz. if you change that 0Hz to 1kHz, you get a signal thats the sum of a sinewave at 560Hz and a sinewave at 1.44kHz, neither of which are an A. The distinction is that the 1Hz signal is modulating the audible signal, not adding to it, if you look at the spectrum of the sum of those frequencies there is no 1Hz component, whereas if you added a 1Hz signal you'd get something completely different. And in this case the amplitude of the signal is always changing faster than 1Hz. Edit: another way of looking at it: You wouldn't say you can "hear DC" because you can hear an A440 played at a perfectly constant volume.

thescriptkiddie 11y ago
All of that makes sense if we consider the case of amplitude modulation, which is multiplication. But if we are talking about the interference patterns caused by two overlapping audio signals, that is addition, is it not?
- hamiltonkibbe 11y ago
  The two signals which are "interfering" are added together. The amplitude of the resultant signal varies sinusoidally, as the instantaneous phase difference between the two signals goes from 0->2pi. One way of describing the signal would be that it is a separate tone (sitting in the middle of the two frequencies) being amplitude modulated by a signal at the beat frequency... which is what you hear and why you "hear" the beat frequency) I went with this way of describing the signal because you were talking about "an arbitrarily high frequency signal getting louder and softer 440 times per second" which is the definition of amplitude modulation. Counterintuitively, there is no frequency component generated at the beat frequency when you sum a 1kHz and 1.001kHz signal, its easy to test that out with matlab, octave, scipy/numpy/matplotlib, etc. Generate the two signals, add them together, and look at the Fourier transform, you'll see two components, one at 1kHz and one at 1.001kHz (assuming you take a long enough window to have that type of resolution) and no component at the beat frequency. A third sinewave doesn't just jump out of nowhere when you add two separate sinewaves together. If you take the sum of those two signals and run them through an ideal brickwall highpass filter at 999Hz so there are no frequency components below 999Hz, you'll still "hear" the beat frequency because it isn't a separate spectral component, its just the two signals slowly going out of phase, cancelling eachother out, and then going back in phase and boosting the amplitude.

thescriptkiddie 11y ago

All of that makes sense if we consider the case of amplitude modulation, which is multiplication. But if we are talking about the interference patterns caused by two overlapping audio signals, that is addition, is it not?

hamiltonkibbe 11y ago
The two signals which are "interfering" are added together. The amplitude of the resultant signal varies sinusoidally, as the instantaneous phase difference between the two signals goes from 0->2pi. One way of describing the signal would be that it is a separate tone (sitting in the middle of the two frequencies) being amplitude modulated by a signal at the beat frequency... which is what you hear and why you "hear" the beat frequency) I went with this way of describing the signal because you were talking about "an arbitrarily high frequency signal getting louder and softer 440 times per second" which is the definition of amplitude modulation. Counterintuitively, there is no frequency component generated at the beat frequency when you sum a 1kHz and 1.001kHz signal, its easy to test that out with matlab, octave, scipy/numpy/matplotlib, etc. Generate the two signals, add them together, and look at the Fourier transform, you'll see two components, one at 1kHz and one at 1.001kHz (assuming you take a long enough window to have that type of resolution) and no component at the beat frequency. A third sinewave doesn't just jump out of nowhere when you add two separate sinewaves together. If you take the sum of those two signals and run them through an ideal brickwall highpass filter at 999Hz so there are no frequency components below 999Hz, you'll still "hear" the beat frequency because it isn't a separate spectral component, its just the two signals slowly going out of phase, cancelling eachother out, and then going back in phase and boosting the amplitude.