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I'm currently having a hard time to understand the both of these. Can anyone explain to me in a more simple yet understandable way??
FFT is fast fourier transform. It calculates the frequency domain of a wave.
Mathematically, any wave, like a sound wave, is a sum of a lot of sine waves, with different frequencies. A frequency is basically how fast the wave oscillates.
So, a low frequency is a really long wave and in sound, this would be a low bass tone. High frequencies are short waves (the peaks are close together) and they sound high.
A FFT will tell you exactly how much of each frequency is present in a wave. You give it a frequency and it will tell you the amplitude (magnitude) corresponding to that frequency. If the amplitude is 0 then that frequency is not present in the wave.
As you can see in this animation, the green wave is composed of the sum of two "pure" sine waves, one at 5 Hz (meaning it goes up and down in 1/5 = 0.2 seconds) and another at 15 Hz (one period in 1/15 = 0.066 seconds, the wave with smaller amplitude). So, if you were to plug the green wave into a FFT algorithm, it would detect these two waves and give you the blue spectrum.
Music is a lot of frequencies together. This can result in quite a complex spectrum.
In this image there is a peak at 1 KHz. So that tells you there will be a pronounced tone at that frequency. It's not a simple line as well, the tone is a bit "smeared out". This is because the tone is not a pure sine wave and because of the "window". Only a pure sine wave would give a single line in the spectrum, but for that you would need to measure infinitely long. Nobody has time for that so they measure only during a selectable amount of time. This is the window. But since stopping the measurement prematurely essentially modifies the sine wave, it distorts the spectrum a bit and you get a distribution instead of a line around the original frequency of the sine. That's advanced stuff already.
Mathematically speaking, a beat arises when you add two sine waves together:
They add up and at some places they cancel each other out and at others they amplify each other. The periodic structure that arises is called "beat".
But in music, a beat is usually referring to the unit of time in a measure. If you have a metre of 4/4 then you have four beats in one measure.
Usually such a beat (in popular music) is accompanied with percussion such as a bass drum or snare drum hit. A bass drum would show up in the lower frequencies of the spectrum as a sudden onset in amplitude. An algorithm can detect this to try and find the beats in a piece of music.
I particularly enjoyed the signal / amplitude graph.