Modulation, and what separates changing key from passing through one
The change of tonal centre, confirmed by a cadence; without one, a chord from outside is just a chord from outside, and the listener feels the step without needing to analyse it.
A song in C major has, in the middle of it, a D seventh chord. It can be two things. It can be a secondary dominant, the V7 of V, which passes and returns to C. Or it can be the start of a change: the song is now in G major and stays there. The difference between the two is not in the chord; it is in what comes after. Modulation is the change of tonal centre, and it only exists when a cadence confirms the new centre.
The test: the cadence
A chord from outside the key is, by itself, passing chromaticism. The song stays in C while the ear waits for C to come back. It comes to be in G when a cadence in G happens — a dominant of G resolving to G — and what follows behaves as though G were home.
Here D seventh resolved to G, but G went straight back to C: a secondary dominant, no modulation. Compare:
D seventh resolved to G, and then the second degree of G, A minor, the fifth of G, D, and G again. Two cadences in G, no return to C: the music modulated. The listener does not analyse, but feels that the floor has moved.
How you leave a key
There are three common ways to reach the new key, and they differ in how much they prepare the ear.
By a common chord. A chord belonging to both fields serves as a door: it is heard in the old key and, in retrospect, in the new one. A minor is the sixth degree of C and the second degree of G; pass through it, follow with the dominant of G, and the border is invisible.
By the dominant directly. With no common chord, the new key's dominant appears and resolves. It is more abrupt and clearer, and it is what most songs do when they lift the key for the last chorus.
By semitone. The most famous modulation in popular music: the whole song goes up a semitone, or a tone, and carries on the same. There is no preparation at all, and the step is felt. Technically it is a direct modulation; in practice it is an effect of energy, and everyone recognises it.
Near and far
The circle of fifths measures what a modulation costs. Neighbouring keys, a fifth above or below, share six notes out of seven, and the passage between them is smooth: from C to G, or from C to F, only one note changes.
G major F majorThe relative minor is a neighbour too, with the same key signature. Those three destinations — the dominant, the subdominant and the relative — are where tonal music modulated for two hundred years, and where popular song still goes nearly every time.
Distant keys ask for more: a common chord that does not exist, or a chain of dominants, or a modal borrowing to open the way. From C major to E major, for instance, there is no common triad, and the modulation needs a bridge.
The key signature changes, or does not
In a score, a long modulation usually brings a new key signature in the middle of the piece. A short one, lasting a phrase, is written with accidentals by hand, and the signature stays as it was. Notation follows the length of the stay, not the theory: if the music is going to remain in G, change the signature; if it is visiting G for eight bars, write the F sharps one by one.
What to take from here
Modulation is a change of centre, and the test is the cadence: without one confirming the new centre, what happened was a chord from outside passing through. You reach the new key by a common chord, by the dominant directly, or by a semitone leap, and the circle of fifths says which destinations are neighbours and which need a bridge. The key signature changes when the stay is long; for a visit, the accidentals are written by hand.