The circle of fifths, and the map it draws
The twelve keys laid out by fifths, in a circle that explains the order of the accidentals, measures how closely keys are related, and shows where chord sequences prefer to walk.
Go up a perfect fifth from C and you reach G. Up another and you reach D. Carry on and, twelve fifths later, you are back at C. That closed path is the circle of fifths, and it is not a decorative diagram: it is the map of how keys relate to one another, and of how chords move inside one.
One step, one accidental
Put C major at the top of a clock face. Each step clockwise rises a fifth and adds one sharp to the signature: G has one, D has two, A has three. Each step anticlockwise falls a fifth and adds one flat: F has one, B flat has two, E flat has three.
G major F majorThis is no coincidence. Going up a fifth from any major key preserves six of the scale's seven notes and raises the seventh by a semitone. From C to G, the only new note is F sharp. From G to D, C sharp. One step, one altered note, one more accidental.
The order of the accidentals is the circle
Sharps appear in the signature in a fixed order: F, C, G, D, A, E, B. Read those names as notes and each is a fifth above the one before. The flats — B, E, A, D, G, C, F — are the same list backwards: each a fifth below the one before.
Whoever memorised the order of the accidentals memorised the circle without knowing it. Whoever understood the circle has nothing to memorise.
Where the circle closes
After six steps in either direction, you arrive at the same sound with two names: F sharp major, with six sharps, and G flat major, with six flats.
F♯ major G♭ majorThey are the same key and the same scale to the ear; they are different spellings on paper, and the choice between them usually follows what comes before and after in the music. That is the point where the two sides of the clock meet, and it is why it is a circle and not a line.
Neighbours sound alike
Neighbouring keys on the circle share six notes out of seven. C major and G major differ by one note; C major and F sharp major, on opposite sides of the circle, differ by six. Distance on the circle measures kinship: the closer they are, the easier it is to move from one to the other without the ear flinching.
The chords of neighbouring keys fit almost entirely inside the starting key. That is why a song in C visits G or F effortlessly, and needs preparation to visit E or A flat.
The circle inside one key
So far the circle has organised keys. It also organises chords: the commonest sequence of roots in Western music walks by descending fifths, which is the anticlockwise direction of the clock.
E, A, D, G, C: each root a fifth below the one before. The ear recognises that movement as the most natural there is, because each chord stands as the dominant of the next. Half the progressions you know are a stretch of that path.
What to take from here
The circle of fifths is one thing seen from three sides. As a list of key signatures, it says that each step adds an accidental and explains the order they appear in. As a measure of distance, it says which keys are neighbours and why a modulation sounds smooth or abrupt. As a path of chords, it says where progressions prefer to walk. The harmonic functions and cadences of the next readings are that path seen close up.