What is the circle of fifths, and why do musicians use it as a map?

The circle of fifths is a way of arranging the twelve notes of the chromatic scale in a cycle of perfect fifths: C, G, D, A, E, B, F sharp, C sharp, A flat, E flat, B flat, F, and back to C. Drawn as a clock face, it puts the most closely related keys next to one another. Starting from C major at the top, which has no sharps or flats, each step clockwise adds a sharp (G has one, D has two) and each step counterclockwise adds a flat (F has one, B flat has two).

That is why composers use it as a map. Tonal music often modulates to a key whose key signature differs from the original by only one flat or sharp, and those closely related keys are neighbours on the circle. Chord progressions likewise often move between chords whose roots are a fifth apart, so the circle shows the harmonic distance between chords.

There is a catch hidden in the drawing. Twelve pure 3:2 fifths overshoot seven octaves by about 23.46 cents, roughly a quarter of a semitone, the Pythagorean comma, so with pure fifths the circle would not quite close. It closes on a modern keyboard because twelve-tone equal temperament makes each fifth about two cents narrower than a pure one, so twelve of them land exactly seven octaves up.