# Re: two sine tones simultaneously within one critical band

```Dear list:

There is no question that the sum of two
sines can be expressed as the product
of two cosines of sum-and-difference frequencies,
as in:

sin(A) + sin(B) = 2*cos(.5(A-B))*sin(.5(A+B))

But this most definitely does NOT demonstrate the
production of real acoustic beat frequencies.
It just says that you can think of the two original
sines as the product of two particular cosines.
This is essentially amplitude modulation, with
the sides of the equation reversed.  Yes, we
know that if you *multiply* two frequencies, the
product will consist of two other frequencies.

That's not what we are talking about with beats.
When you add two frequencies linearly, the sum
contains *only* those two frequencies. This is
the definition of linearity.  If you get anything else you
have intermodulation distortion from some system
nonlinearity. In fact, the standard test for IM is to drive a
system at two frequencies and look for difference tones.

The fact that we can hear beats tells us about
the auditory system, not about external reality.
They are not present as real acoustical sound
components, assuming you have taken reasonable
care in the production of the two tones.  (Not too
loud from a single speaker or other nonlinear system,
for example.)

Those with Windows computers are welcome to
and visualize this for yourself.  You can generate two sine
waves and look at the the waveform and spectrum of the
total before it goes out to your sound card, so it does not
include speaker distortions, etc.  You will see only
two spectral peaks.  (Assuming you set each primary
"Stream" level at 50% or less, so the sum doesn't clip.)
You will want to use higher frequencies than 99 and 101, to
get better spectral resolution.  Try 4000 and 5000. You will
see a 1 msec periodicity in the waveform, but you will not
see any component at 1000 Hz in the spectrum.