Beating

[jh]

>Hallo Paul
>
>>You're
>>absolutely right if you say the beating will be twice as fast at
>>440/442.
>I say definitely NO! Or I do not know what is a "beatrate"? It is the  
>frequency of the chorus-like effect which is created while mixing the two VCO- 
>signals. This frequency is 440 / 442 = 0.9955 or 220 / 221 = 0.9955
>
>Gruss,
> Florian Anwander

Where did you get this from ??

This is what I read in the textbook (Bronstein, Taschenbuch der Mathematik):

sin(x) + sin(y) =

2 * sin((x+y)/2) * cos((x-y)/2)


So if x and y are close together, you get a pitch that corresponds to a frequency
*between* the original ones , (x+y)/2, and this is AM-modulated with a frequency
of half the difference, (x-y)/2. As this is a 4-quadrant-multiplication, and as your
ear will not hear the difference between an amplitude of +1 and -1, the *loudness*
will pulse with the double rate of that, i.e. (x-y). 

Therefore you will get the simple formula 

Beat Rate = Frequency 1 - Frequency 2.

If you're asking does that work for other waveforms as well, write down the waveform
of your choice as a fourier series and do the same math as an exercise (;->).

But seriously, I thought it was "so clear" until I wrote it down, and the surprise for
me was that for sine signals the *contour* of the resulting loudness modulation is 
not a cos (or sin) wave, but rather a rectified cos wave of half the beat rate
(which has the expected fundamental of (x-y) of course).
So doing the little "homework exercise" on a saw wave for instance was not an 
(entirely) ironic remak, after all. The beat *rate* is quite obvious, but 
the contour of the beating might be not.

Hope this helps,

JH.