Finally, we're going to see a real application. A really useful application, that is. It is tuning your guitar. The abstraction of the problem is that you have reference sinusoid at some frequency omega0. You have a tunable sinusoid of frequency on omega. And we would like to make omega and omega0 as close as possible. Actually, equal, and this only by listening to it. And what we are going to do here is a beating between these two frequencies once they are close enough. And then by tuning, we can bring this beating to, essentially, frequency zero at what point omega = omega0. And we have tuned our guitar string with respect to a reference frequency. So how are we going to go about this? Well first, we bring omega close to omega0. That's sort of easy, if you have a minimum of musical ear. When these two frequencies are close, we play both sinusoids together. Then we have to remember trigonometry. And we write xn, which is a sum of cos(omega0n) + cos(omega n). In terms of a sum and a difference of these two frequencies. Which finally can be written approximately as 2 times the cos of the difference, delta of omega n, and the cosine of the base frequency, omega0. From this formula, we see there are two component. There is the error signal, the cos(delta omega n). And there is a modulation signal, the cos(omega0). When omega is close to omega0, the error signal is very low frequency. So we cannot really hear it because it's such a low frequency. So the modulation will bring up to the hearing range. And we're going to actually hear it as an oscillation of the carrier frequency. And we're going to see this picture, really, in just a moment. Let's look at the pictorial demonstration here. So we start, omega0=2pi*0.2, omega=2pi*0.22. The difference, which is actually half of the difference between the two, is 2pi*0.01. We see now, interestingly, we have the carrier frequency which is red curve modulated by the difference, by cosine of delta of omega. So we see this, the beating, in blue overlaid to the red curve, which is modulation. We can change frequency, omega, to 0.21 * 2pi. The difference now is 0.005. The beating is slower. We pick omega = 2pi*0.205. The beating is even slower. And here we take an example where omega is very close to omega0. And the beating is extremely slow. And we almost see only the modulating frequency omega0. And the very slow variation due to the beating by cosine of delta of omega. It's time see a video demonstration how to tune a guitar using this very simple principle. You probably have seen musicians do it on stage. Now you understand the math behind it. Okay, after the math, let's try to do something useful like tuning an electric bass. An electric base has an E string [SOUND], which has a frequency of 41.2 hertz. An A string [SOUND], which has a frequency of 55 hertz. To use the result we have just seen, we want to find two frequencies that we want to make equal. As long as they are not, there will be a beating that we can adjust. We are going to hear this in just a moment. So the first harmonic of the E string, [SOUND] here, is at 83.4 hertz. The third harmonic [SOUND] is at 164.8. For the A string, the first harmonic is [SOUND] at 110. And the second harmonic [SOUND] is at 165. We're going to use these two [SOUND] harmonics to do the tuning. And you can hear, when [SOUND] they're out of tune, there's a beating. [SOUND] As you get them closer and closer, the beating slows until it's actually zero beating. And then the two frequencies are similar. And then we can start playing something like [MUSIC] or something else.