The circle of fifths

Go up a perfect fifth twelve times from C and you land on C again, seven octaves higher, having passed through every major key on the way.

Twelve keys on a clock face

Put C major at 12 o'clock. Go up a perfect fifth, to G, and place G major at 1 o'clock: it has one sharp. A fifth above G is D, two sharps, at 2 o'clock. Keep going (A, E, B) and each step clockwise adds one sharp. Going the other way, a fifth down from C is F, one flat, at 11 o'clock; then B♭, E♭ and A♭ follow, each adding a flat. The result is the circle of fifths, all the key signatures in one picture.

Each new sharp is the leading tone of the new key. G major's sharp is F♯, a half step under G. D major keeps F♯ and adds C♯, a half step under D. Each new flat works the other way round: it is the fourth degree of the new key. F major adds B♭, the fourth note of F major, and B♭ major adds E♭, its own fourth note. The tricks from the last unit are this pattern seen from a different side.

Neighbours share six notes

Keys next to each other on the circle differ by exactly one note. C major and G major share C, D, E, G, A and B, and differ only in F versus F♯. Because the change is a single note, music often moves to a neighbouring key and the ear follows easily. The two neighbours of any key are its dominant (clockwise, a fifth up) and its subdominant (counterclockwise, a fifth down), the same 5 and 4 you met as scale degrees.

The farther apart two keys are on the circle, the fewer notes they share. Counting from C major: G shares six notes, D five, A four, E three, and B two. F♯ major, straight across the circle, also shares only two (B, and E♯, which sounds as F). That is why a jump from C to F♯ sounds like a sudden change of scene, while a move from C to G hardly sounds like a move at all.

The bottom of the circle is where sharps and flats meet. At 5 o'clock, B major (five sharps) sounds the same as C♭ major (seven flats). At 6 o'clock, F♯ major (six sharps) and G♭ major (six flats) are played on the same keys. At 7 o'clock, C♯ major (seven sharps) matches D♭ major (five flats). Notes or keys that sound alike but are spelled differently are called enharmonic, and that is how twelve positions hold fifteen key signatures.

Why the circle closes

A perfect fifth is seven half steps. Twelve fifths are 12 × 7 = 84 half steps, and 84 ÷ 12 = 7, exactly seven octaves, so on a piano the circle closes perfectly. Modern pianos are tuned so that every half step is the same size. If instead every fifth were tuned to the pure frequency ratio 3:2, twelve of them would overshoot seven octaves slightly: (3/2)¹² ÷ 2⁷ is about 1.0136, a gap of about 23.5 cents, where a cent is one hundredth of a half step. A keyboard with fixed pitches cannot have every fifth pure and still close the circle, so its tuning has to share that small gap out somewhere.