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Re: two sine tones simultaneously within one critical band
I'm not quite sure if everyone including myself means the same when
talking about (monaural) beats.
May I express my thoughts through the following three examples:
First, a tone "rides" on a LF pressure wave, i.e. a 1 kHz on a 6 Hz -
represented by the sum of the two sine waves.
This is a typical scenario for environmental sounds which reach our ears
through air exposed to atmospheric turbulence.
Sound pressure analysis shows both of these two components in the f
domain. The wave envelope visible in the t domain has either the looks
the 6 Hz sine wave, drawn with a thick pencil or the looks of an
or descending 1 kHz wave, depending on the frequency resolution applied.
We only hear one steady 1 kHz tone though, no 6 Hz beats.
Second, two similar tones at similar amplitude i.e. 1 kHz and 994 Hz are
emitted from the same or different sources:
Again, sound pressure analysis shows both components in the f domain.
In the time domain, the resulting sound pressure shows maxima and minima
in the wave envelope.
The peaks and zero crossings of the envelope appear regularly every 1/12
s, hence we hear correctly a tone which has beats at 1/6 s intervals.
Presumably, the pitch of the perceived tone would be 997 Hz, and as such
is an illusion, while we hear the correct beat.
The focus of the auditory system shifts from correct spectral resolution
to correct temporal resolution, as both at the same time is impossible
Third, a loudspeaker membrane is overdriven with a strong 6 Hz and a
Hz signal. The resulting IM components contain the multiplication
of the two sines as per previous discussion: sin(A) × sin(B) = 1/2 ×
[cos(A-B) - cos(A+B)]
In the f domain, this will be visible as a 994 Hz and as a 1006 Hz tone.
The wave envelope visible in the t domain will again show regular peaks
and zero crossings, but this time around at 1/24 s intervals. Hence
hear the original 1000 Hz tone beating every 1/12 s, which is also
I think, the confusion which arose in this discussion lies solely in the
different interpretation of "beats".
Just because the "beat interval" does not show up as "beat frequency" in
fourier transform of real world sound pressure waves per se, it doesn't
mean that our ears are playing tricks on us.
The auditory system merely shifts its focus of attention from spectral
temporal resolution under certain conditions i.e. phase inversion at