Wednesday, June 3, 2009

An Application of Double Counterpoint Technique

My Third day of working with ideas from Hugo Norden's The Technique of Canon.

Norden bases much of his technique on the principle of double counterpoint (sometimes called "invertible counterpoint"). The basic concept behind double counterpoint is that either voice of a two-voice counterpoint should be able to function as a bass to the other.

Rather than go into a discussion of how to make invertible counterpoint -- that takes at least a chapter but a thorough treatment takes a whole volume -- I'll just include a brief example. This one is double counterpoint at the 12th but any interval can be used for the inversion:

(click on image to enlarge)

Now, this example is not a canon -- just an invertible counterpoint. Norden's first two chapters are devoted to an explication of this basic technique. However, in chapter three, he begins to show how a facility with this technique becomes invaluable to a composer of canons.

When writing a perpetual canon (one that repeats, ostensibly, in perpetuity), the key problem occurs right at the juncture of the repeat itself, in the case of today's canon (as well as Monday's), the last two measures before the final repeat sign. The final note of the dux is the first note of the canon again.

The problem is that the note in the penultimate measure of the dux must act as the bass in that position but when it occurs in the other voice it must be at a proper interval above the bass note. It must function both as bass and, a measure later, as the upper voice. This is a problem of double counterpoint.

Norden gives a simple arithmetic method to determine which interval of inversion to use for the double counterpoint. You temporarily tie the previous note into each of those measures and calculate the resulting intervals -- in this case a 6th and a 9th. Add those intervals together. [strange math alert! -- when adding intervals the resulting number is one less than in simple arithmetic. This is to keep from counting a note twice.]

6 + 9 = D.C. at the 14th.

The last step is to choose the pitches informed by the inversion chart for D.C. at the 14th and complete the canon.

I'll talk about that act of informed choice a bit more later in the week.

Of course, you could avoid the learned approach and go for trial and error to complete that critical juncture. I prefer success to error though...

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