Notes that are close together on the piano keyboard are in fact harmonically far apart. Try playing C and C# together - it will sound dissonant and unpleasant. These two notes are far apart on the circle of fifths, and they'll rarely be heard together in music.
But play C and G together and it will sound sweet. G is a fifth higher than C, it is the closest note to C harmonically, and they are neighbours on the circle of fifths.
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The reasons for this can be analysed mathematically, in fact G is 1.5 times the frequency of C. So three cycles of G will fit into the same time as 2 cycles of C. But when playing music there's not much time to think this way, we just need a feeling for what sounds good.
Look at the circle of fifths and you'll see that G is one place clockwise from C. Now, go one place anticlockwise and you'll find F. In other words, C is the fifth note of the scale of F.
Say we are playing a tune in the scale of C. The two notes most closely related to the home key are G and F. And you'll find that the chords C, G and F are the ones that will pop up most often when trying to harmonise an accompaniment with a tune in C. The most commonly needed chords are the two nearest neighbours on the circle of fifths.
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