Why There Are Twelve Notes

At seven you halved a string and the note went up an octave. Stack the next simplest ratio twelve times and it lands 1.36% away from where it has to land — which is not a rounding error, it is a fact about numbers, and it cannot be fixed. So every instrument with fixed notes is tuned slightly wrong on purpose, the compromise has a name, and the frets on a guitar are where they are because of it. Arithmetic with a right answer nobody likes.