The circle of fifths is a beautiful thing, fundamental to music theory.
Sound is vibrations in air. Start with some note on the piano. Then play another note that vibrates 3/2 times as fast. Do this 12 times. Since
(3/2)¹² ≈ 128 = 2⁷
when you're done your note vibrates about 2⁷ times as fast as when you started! We say it's 7 octaves higher.
Notes have letter names, and the notes you played form this 12-pointed star: the circle of fifths!
It's great! But....
(1/n)
I said
(3/2)¹² ≈ 128
but this is just approximate! In reality
(3/2)¹² = 129.746....
so the circle of fifths does not precisely close - see below!
This is called the Pythagorean comma, and you can hear the problem here:
https://en.wikipedia.org/wiki/Pythagorean_comma
As a result, the equal-tempered 12-tone scale now used on most pianos doesn't have 'perfect fifths' - frequency ratios of 3/2.
People have dealt with this in many, many ways. No solution makes everyone happy.
(2/2)
@johncarlosbaez A great number of songs use this circle. Off the top of my head, Autumn Leaves and Hey Joe
