Subject: Where's the beating?FWD from AUDITORY
From: Kevin Austin (firstname.lastname@example.org)
Date: Sun Oct 09 2005 - 12:05:12 EDT
>Topics of the day:
> 1. two sine tones simultaneously within one critical band (3)
>Date: Sat, 8 Oct 2005 11:32:24 -0400
>From: Bob Masta <audio@DAQARTA.COM>
>Subject: Re: two sine tones simultaneously within one critical band
>There is no question that the sum of two
>sines can be expressed as the product
>of two cosines of sum-and-difference frequencies,
>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,
>Those with Windows computers are welcome to
>download my free DaqGen software <www.daqarta.com>
>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.
>Beats are really, truly "all in your head"!
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