I've mentioned modulation quite a few times on this blog, but I haven't talked much about it. In music the term refers to the process of changing one key center for another. Here is the Wikipedia article. This Wikipedia article is not a particularly good introduction as it immediately proceeds into the technical details without giving much context.
You first have to have some idea of the notion of a "key". In music a key is a kind of basic structure that you could envision laid out like a baseball diamond:
You start at home plate, the top corner, and then proceed anti-clockwise to first base, second base and third base before returning home. Similarly, in music, the basic structure is that you start with the tonic, which is often expanded, then you go to the subdominant or related chord, then the dominant and finally return to the tonic. Here is how this looks in notation:
Of course, in actual pieces, composers depart from and vary this basic structure in many ways. But you would be surprised at how much music is written according to this structure. The movement from dominant to tonic normally creates a cadence and it is used to end most phrases and all major sections and the end of every piece. I need to qualify that by saying "a cadence ends every piece in common practice style". That is, nearly everything in the 17th and early 18th centuries (the exception being some holdovers from modal harmony) and everything in the late 18th and through most of the 19th century. I challenged a commentor to send me one example of a piece by Haydn, Mozart or Beethoven that did NOT end with an authentic cadence. I don't expect to receive any examples. This is more important than a "rule": it is a basic principle of tonal music. The key is defined by the cadence and without this, the music really isn't tonal. You define a key by running around the bases as I have illustrated above.
So, you have run around the bases a few times and start to worry that you need more variety. The solution is often to modulate, or change gears into a new key. After a section or more in the new key (or possibly more than one new key) you move back to the original key. This is called modulation. How do you do it? There are a number of ways. One of the most common is to use a chord or even just a note that is common to both keys. This is called a pivot chord or note and I suppose you could metaphorically call it a kind of "clutch" that enables you to change gears.
The most typical kind of modulation, at least in major keys, is to the key of the dominant. In C major, this would be a modulation to G major. A signal of this change would be the appearance of sharps on the Fs as G major has a key signature of one sharp, F. If you were in the key of G and modulated to the dominant, which is D, you would suddenly see C#s appearing as D has two sharps in the key signature: F and C. Let's use as an example a famous song by Franz Schubert, "Heidenröslein". Here is the composer's autograph of the song:
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That's pretty easy to read, unlike Beethoven's autographs. But we can find a clearer one:
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What you see here are two phrases. The first is in the tonic key, G major, indicated by the key signature of one sharp (F). This phrase extends for four measures and the harmonies are tonic, supertonic (with tonic pedal), dominant, tonic. The supertonic serves the same function as the subdominant. In symbols it would be I, ii 4/2, V6/5, V4/2, I6, I. The next phrase, starting in measure 5, modulates to the dominant. It starts on what was the tonic, but this is going to be the pivot chord so we should now think of it as the subdominant. G is the subdominant of D major, the new key. So the harmonies in this second phrase, which is six measures going to the end of the second line, are subdominant, dominant, tonic, supertonic, dominant, submediant, tonic, supertonic, dominant, tonic. So in this phrase, instead of just running around the bases once, he does so three times. Every time we return to the tonic it is like returning to home plate. All Schubert has to do in the final phrase is return to G major, which he does. Simple! Let's have a listen to the song. This is Peter Schreier singing in the Schönbrunn Palace in Vienna. The first verse, which encompasses the first phrase in G major, the second phrase modulating to D major and the third phrase which returns to G major, takes only 36 seconds!
So that is the basic idea of modulation. Composers have found hundreds of ways to make it more elaborate, but the basic principle is as I have explained. It is rare to find a piece of classical music that does not modulate.