My question is regarding the interference in sound waves
of slightly different frequencies which indeed produces beats. If
three tuning forks of frequencies, say, 401Hz, 402Hz and 403Hz are
vibrated simultaneously then what will be the beat frequency heard?
Since the first two forks differ by one Hz The second two do also. Each pair
(401 and 402 plus 402 and 403), will generate a 1 hz beat. The difference in
frequency between first and third (401 and 403), is 2 Hz., They will generate a
2 Hz. beat. But since the 1 beat per second fits neatly in the 2 beat per
second , you will just hear 2 beats per second.
Beat frequencies in general are the sum and difference of the two frequencies
being mixed by a nonlinear process.
For 401Hz and 402Hz, the heterodyne frequencies are 1Hz and 803 Hz.
The nonlinearity in your hearing is:
the mind's estimation of the sound intensity envelope.
That's a non-linear function, like rectifying a sine-wave
or using a square-law detector to get the RMS (root-mean-square) amplitude.
But it is slow, <30Hz, so the 803Hz will probably be lost / not created / not perceived.
The 1Hz will be perceived as an amplitude-modulation of an approximately-400Hz tone,
getting louder and softer with 1/second periodicity and a simple sine-wave envelope shave.
Jumbling three frequencies together definitely makes it less simple.
Then you have
401Hz&402Hz -> 1Hz,
402Hz&403Hz -> 1Hz,
401Hz&403Hz -> 2Hz,
The 2Hz will be quite noticeable as a jauntiness or
non-sinusoidal character within the 1-Hz louder-softer cycle.
It is even possible, depending on the relative intensities and phases
of the 401/402/403Hz signals,
to have only 2Hz beat frequency, that also being a simple sinusoidal envelope.
That would happen when the 402Hz was very weak or absent,
or maybe if the 1Hz beats can be of proper phase to cancel each other.
If the two 1-Hz frequency-separations are slightly different
(i.e. 1.0 Hz and 0.9 Hz),
as would often happen with a string instrument rather than digital electronics,
then the sound would slowly drift through a range of perceptually differing
combinations of the 1Hz and 2Hz beat-frequencies.
At one moment it would seem to be lopsided 1Hz modulation,
then about 5 seconds later it might sound more like 2Hz ripples on top of a larger constant tone ,
then back to varyingly lopsided 1Hz beats.
The transitions between all those would be gradual.
If a second stage of nonlinear signal-processing is somehow added,
such as strong electronic distortion (i.e., an electric-guitar's fuzz-box),
then it is quite possible for the 1Hz and 2Hz to beat together
making some 3Hz, and even higher harmonics of 1Hz running to above 10Hz.
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Update: June 2012