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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Re: two sine tones simultaneously within one critical band*From*: Bob Masta <audio@xxxxxxxxxxx>*Date*: Sat, 8 Oct 2005 11:32:24 -0400*Delivery-date*: Sat Oct 8 11:45:06 2005*In-reply-to*: <43467BA4.5060004@alum.mit.edu>*Priority*: normal*Reply-to*: Bob Masta <audio@xxxxxxxxxxx>*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

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 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"! Best regards, Bob Masta audioATdaqartaDOTcom

**Follow-Ups**:**Re: two sine tones simultaneously within one critical band***From:*Reinhart Frosch

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