Embellishment according to Norden

In Chapter XIV of The Technique of Canon, Norden lists the following ways to embellish simple canons:

1) Rhythmic

2) Neighbor tones, passing tones and appoggiaturas

3) "other notes of the harmony"

4) shift of notes of the basic canon up or down an octave

5) ties and suspensions

Use of any of these techniques by them selves and in combination leads to a wide variety of solutions.
For today's canon, I've used only the first three of these techniques.

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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:

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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...

Perpetual Canon Elaborated

Here is yesterday's perpetual canon elaborated.

I'll explain these techniques tomorrow.

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Monday, June 1, 2009

Getting Systematic Again

What is it about turning a calendar page that makes one want to begin anew?

Well, here it is June 1st and I am feeling the need to take a more systematic approach. It could have something to do with the fact that I simplified my process greatly when I had very busy weeks. Well -- for whatever reason...

Here's what I've decided to do. I'm going to work through the book The Technique of Canon by Hugo Norden (Boston: Brandon Press, [no date]).

Norden is an interesting character to me. He taught for many years in Boston (including at the Boston Conservatory where he taught some of the courses that I teach there now!)and there are still many of his students in my circle of colleagues. The first book that I learned counterpoint from was his Fundamental Counterpoint.

The Technique of Canon (like many of his other books) takes a fairly mathematical approach to things, Don't rush to decry that as unmusical, it is anything but. Counterpoint is to music as perspective is to the visual arts. There are techniques and the ARE mathematical, no matter how one might choose to approach them.

Anyway, the math is really more like arithmetic to be sure.

I'll talk about that over the next few days. For now, here is a 1st species style perpetual canon written according to principles in the Norden book.

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Canon in mensuration and inversion

I seem to be drawn towards making some of these short canons into fanfares -- not sure why...

Anyway, The 1st Trumpet and the trombone are inversions of each other. So are the 2nd trumpet and the bass trombone but these two are an augmentation os the former two.

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Saturday, May 30, 2009

Pentatonic Canon at the Octave

Busy day (how often do I say that?) -- Maggi and I had a concert followed by a dinner with our audience (a group of Japanese tourists visiting Salem).

So, I took the easy way out and wrote a simple pentatonic canon at the octave for harpsichord.

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Hidden Elaboration

Elaboration of yesterday's canon.

The basic canon in inversion stays the same but is hidden by the different types of elaboration in the two parts.

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