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Any regular connoisseur of barbershop music knows that overtones exist and can be heard don't believe me listen for yourself there are four people singing in this clip but continents you hear in the last court there are five love this is just one example of overtones, and they don't only happen with a four-part chord they can be heard with two parts and three as well take this interval for example did you hear it this is the note you're listening for now try listening again did you hear at that time a system of two like this one is simplest so let's look at that to start let's remind ourselves how a linear system works with each frequency you put in you get the same frequency out, although the amplitudes may be different does it matter how fast the frequency is or how slow it's the same concept frequency in equals frequency out now if you put in two driving frequencies for example F 1 and F 2 when you when they come out all you get is a linear combination or in mathematical terms the superposition a nonlinear system is different however when you put in two different input frequencies like what we're dealing with what you get is the original f1 and f2, but you also get extra harmonics like QF 1 3 f1 etc now the ear is nonlinear, and it behaves as follows with one input we have an equation of MX double dot plus k1 x plus k2 x squared equals F cosine Omega Drive T which is our input our first guest will be x equals a cosine Omega drive T since the squared term gives us the harmonics we'll look at the result from that we will get a squared cosine squared Omega drive T which will give us a cosine 2 Omega drive T result our next guest is x equals a 1 cosine Omega drive T plus a 2 cosine 2 Omega drive T looking once again at the squared term we get results like cosine Omega Drive T times cosine 2 Omega drive T cosine squared Omega drive T and cosine squared 2 Omega drives T these all produce harmonics we have cosine 3 Omega drive T which is our sum term and cosine 1 Omega drive T or different term we also have cosine 2 Omega drive T and cosine 4 Omega drive T so not only do we get the simple Oh Mega Drive and two Mega Drive that we put in we also get terms like three Omega drives and 4 Omega drives and so forth so remember we're looking at a two-part harmony, so we'll have two inputs which means our equation looks as follows MX double dot plus K 1 X plus K 2 x squared equals f1 cosine Omega 1 T plus f2 cosine Omega 2 T our first guess is going to be x equals a 1 cosine Omega 1 T plus a 2 cosine Omega 2 T so since the squared term gives us our harmonics we will once again look at the results from that this gives us terms like cosine squared Omega 1 T cosine squared Omega 2t and cosine Omega 1 T times cosine Omega 2t each of these gives us harmonics this first term gives us a 2 Omega 1 the second term gives us a 2 Omega 2 and this last term gives us an Omega 1 plus Omega 2 and Omega 1 minus Omega 2 our second gas is going to be x equals a 1 cosine Omega...
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