Invertible Counterpoint

This is a subject so esoteric that Blogger doesn't even recognize "invertible" as a word, but underlines it in red. Trust me, it is a word and correctly spelled. I want to put up a post on counterpoint as a sequel to the one yesterday about multiculturalism. When the narrator describes the music of the Aka pygmies as "extremely complex contrapuntal polyphony" what he is really doing is attacking the worth of what Bach did, by, at the very least, implying that what Bach did was contrapuntal polyphony, sure, but since it was all rigid and didn't allow for spontaneous expression, it's not quite as good as what the Aka do, flowing as it does from their communal wonderfulness.

Now if someone can argue, explain and describe just what is going on with the Aka, then great. Go to it in the comments. From listening and doing some web research, I conclude that what the Aka are doing is an interesting kind of polyrhythmic structure similar to ones found in other places in Africa and in the gamelan music of Java and Bali. What it is not, emphatically, is counterpoint, which is a word describing pitch structures. The Aka sit on the same pitches because they are unfolding a rhythmic structure, not a tonal one. Go have a listen here, sans narrator, and you will see what I mean.

What happened in Western music history in the last thousand years is something of quite another order. True, many of the rhythmic complexities of non-Western music were ironed out, but the reason was so that the possibilities of counterpoint and harmony could be developed. One of the most interesting of these is invertible counterpoint. Here is a great article on the subject. Here is the piece they chose as example, the F minor Invention:

People often ask what is it about Bach's music that makes it so powerful, so fresh even after many listenings. Bach was a master of what is sometimes called "variety in unity". If you go look at the article on invertible counterpoint you will see that the first four measures of the invention are repeated, exactly, as measures five through eight. But they don't sound the same. The reason is that Bach has written them so that the lower voice can be placed above the upper one: this is invertible counterpoint.

Much of the history of Western music can be seen as the quest by composers of ways of structuring music so as to produce both unity and variety. This can be done in a multitude of ways as each composer strives to find a new way with each piece. One of the earliest discoveries was imitation. One voice can be imitated by another on a different pitch level:

Listen to how the eight notes of the theme of the C major invention keep returning and returning in both voices. Of course, imitation doesn't have to be quite so literal. In this canon from the Art of Fugue, the lower voice imitates exactly the upper one, but a fifth lower, in notes twice as long and upside down. Then, in the second half, the voices invert so that the whole piece is also invertible counterpoint!

These are just a couple of the basic ways composers have discovered to knit together pieces of music.