What this is, is a close examination of the music to see what chords are being used and where the cadences are. It is a way of looking behind the surface activity of the music, all those melodies and figurations, to the harmonic background that also provides the underlying structure.
Before we can do a harmonic analysis, we have to know a bit about chords. The basic chord in music is a triad, three notes consisting of a root, third and fifth. This is constructed by simply choosing a note and adding two notes above it from the scale, skipping in between notes. Here is the C major scale:
Then, starting with C, we take every other note:
Which gives us a C major triad (because we started on C):
You can build a triad on every note of the scale:
But some are more important than others:
These three are important because the C major triad is the tonic, the "home base" that the music will end on (and usually begin on). The triad built on G is the dominant, a fifth above the tonic, and every piece (and most phrases) will end with a cadence from the dominant to the tonic. The other triad, the one built on F, is called the subdominant because it is a fifth below the tonic (UPDATED). The other triads have names as well, but we can skip them for now. These three chords are important because they serve three different functions.
Musical structure, at least in tonal music, is based on a couple of basic ideas. One is that of time regularity: most music has a steady pulse or beat (that can be distorted for expressive purposes) and these beats are grouped in regular patterns and all this falls into equal-sized larger groups called phrases. The other basic idea is that of harmonic function. Harmonies create and resolve tension. This tension is normally resolved at the end of the phrase with a cadence: a perfect authentic cadence to be precise.
The three functions served by the three chords above are of resolution, tension and the building of tension. How a piece normally goes is by starting with the tonic chord, the C major in the above examples. Then the music moves through the subdominant or some version thereof, to the dominant, the area of maximum tension. Finally this tension is resolved by moving from the dominant back to the tonic. End of the journey! Sounds simple and indeed, on the most basic level, it is. A great deal of the organized power of music consists in following this simple journey in an interesting way. The vast, vast majority of pieces written between 1600 and 1900 follow this simple structure both on the small scale of the phrase and the large scale of the whole piece.
One reason that we can focus on just these three chords is that the other triads share a lot of notes with these and along with the notes, some of the function. For example, the triad on A shares two notes with the C triad. This is used in what is known as a deceptive cadence. You prepare for a standard cadence, but at the last moment you go to A minor instead of C major. Another example is the triad on B. This shares two notes with the dominant so it can function as a dominant. Added to this is that the third note of the B triad is an F, which is the seventh of a dominant seventh on G. The triad on D shares two notes with the F triad, so it can function as a subdominant.
Add it all together and there are only three basic harmonic functions: the tonic, or point of arrival, the dominant or point of maximum tension and the preparation for the dominant, everything that leads up to the dominant.
OK, now let's look at a piece. This little minuet in G major from the Anna Magdalena Notebook was for a long time thought to be written by her husband, J. S. Bach. Recent scholarship reveals it to be by the rather obscure composer Christian Pezold.
Since this is in G major, we should adjust our scale and triads to G:
Let's listen to the piece first. It takes up the first 1:30 of this clip:
Starting with the first line:
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The first two measures are the tonic chord. As you can see, the G doesn't always have to be in the bass. In the second measure the B is the bass note, but the harmony is still G major. The third measure moves to the subdominant, a C triad. Then we move back to the tonic. The last measure in this line is a triad on A, which is the supertonic. As this triad shares two notes with the subdominant, it functions the same way. Here is the second line of the minuet:
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The first measure of this line returns again to the tonic. The first beat of the next measure briefly touches the dominant, but the rest of the measure is tonic G major. Then the third measure of this line moves solidly to the dominant. The fourth and fifth measures return to the tonic G major. This is not a perfect authentic cadence, notice, because neither the dominant nor the tonic is in root position. Here is the third and last line of this first half of the menuet:
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The first measure of this line returns once more to the subdominant, C major. The second measure is the tonic again. The third measure of this line outlines the dominant (with a seventh), but it is not in root position. Then we have the tonic yet again. The final two measures finally give us our necessary perfect authentic cadence with the dominant in root position followed by the tonic in root position. And that's the first half. I won't do the second half because it adds some other elements and because just the first half illustrates the basic principle. This is a complete sixteen measure period in G major.
There are thousands of pieces that follow this same kind of structure. The miracle is that, because of the way the chords are articulated, the rhythms and most of all the melody, they all sound different!