The circle of fifths as a map

C → G changes just one note

C major contains C D E F G A B. G major contains G A B C D E F♯. The order changes and F becomes F♯; the other six notes are shared. Another step to D major also changes C to C♯. In the other direction, F major uses B♭ instead of B.

This also makes the circle a map of key signatures. C major has none, G major has one sharp (F♯), D major has two (F♯ and C♯), and F major has one flat (B♭). C major and A minor share a key signature, as do G major and E minor. The signature alone does not tell you which one is the centre.

Why do we get back to C?

Working with pitch classes modulo 12, each step is +7. Since 7 and 12 have no common divisor greater than 1, you visit all twelve positions before repeating. On the piano, F♯ and G♭ can be the same key. The spelling changes around the circle to keep key signatures readable.

A perfect fifth above B is properly spelled F♯. On an equal-tempered piano, F♯ and G♭ have the same pitch: they are enharmonically equivalent. The circle switches to flat names here to keep the keys readable. Compare the sounding pitch classes even when several letter names change at that boundary.

C → G → D → A → E → B → G♭ → D♭ → A♭ → E♭ → B♭ → F → C.

Keys and chords are related, but different

Beside C sit F and G: IV and V in C major. These are useful chords within the same key. Playing a G chord does not automatically change the key to G major. A modulation involves establishing a new tonal centre in the music.

Go the other way and hear an arrival

The roots D → G → C move down in fifths or up in fourths. In C major, Dm–G7–C makes ii–V–I. The key supplies the chord qualities while the circle shows the relationship between roots. The B → C and F → E movements in G7 → C help make the arrival clear.

A fourth spans four letter names; a perfect fourth is five semitones. Moving down a fifth or up a fourth therefore reaches the same pitch class. ii–V–I means chords on degrees two, five and one; writing G7 additionally includes the seventh of the V chord.

Try it at the piano

Compare C major and G major below and find F becoming F♯. Then play Dm–G7–C and listen to the arrival on C. In Experiments → Free exploration → Circle of fifths you can compare more neighbouring keys.