Thursday, June 11, 2009

Invertible Canons

Chapter III of Norden's book describes how to create invertible canons. This adds a layer of complexity to the other uses of invertible counterpoint that I've been working with for the past few days.

Here is the "set-up" for today's canon:

It's a canon at the sixth above that is composed in double counterpoint at the twelfth so that, when inverted, it becomes a canon at the seventh below.

The added complexity comes at the "critical juncture" of the repeat where, in this case, I needed to write a few measures that could work in both double counterpoint at the eighth and D.C, at the sixteenth. This worked out better with half notes and passing dissonances.

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Elaborating Dissonant Counterpoint

Yesterday's basic "anti-first-species" canon used only dissonances. Continuing to "reverse the rules" of counterpoint, I elaborated that canon by using consonances approached and/or left by leap,compound leaps (2 leaps combined) of a seventh and simple leaps of a ninth. These are all expressly proscribed in species counterpoint.

We'll periodically come back to dissonant counterpoint both as a relief from the traditional stuff and sometimes to clarify traditional concepts by a negative example.

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Reversal of values

It's easy to think that these techniques only apply to certain styles -- that's the way they are traditionally taught. Usually the fledgling musician is then left on his or her own to figure out how to move beyond what is taught.

Norden says this:

"What is shown below is in accordance with the traditional rules for academic counterpoint. This is provided merely as a frame of reference. Actually, the correctness of the counterpoint as such has nothing whatever to do with the mathematical calculation of canons should a composer's artistic intentions call for the construction of contrapuntal combinations quite outside the scope of traditional academic availabilities."

Does he tell us how to achieve those other "contrapuntal combinations"?

In a word: no.

I turn for a moment to 20th century American composer, Henry Cowell. In his book, New Musical Resources, he describes another approach:

"Let us, however meet the question of what would result if we were frankly to shift the centre of musical gravity from consonance, on the edge of which it has been long poised, to seeming dissonance, on the edge of which it now rests...An examination in fact would reveal that all the rules of Bach would seem to have been reversed, not with the result of substituting chaos, but with that of substituting a new order."

So, that's what I did today. I took the rules of 1st species and reversed them. Instead of requiring all the harmonic intervals to be consonances, they are now required to be dissonances. I even went through Norden's double counterpoint technique for the critical juncture but this time chose dissonant intervals that would invert to other dissonances.

More about this tomorrow.

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A basic canon at the seventh and two elaborations

Still working with ideas from Hugo Norden's The Technique of Canon.

This time I used three measures of the dux before bringing in the comes.

This required me to use a multiple of three for the amount of measures inside the repeat signs and gave me a critical juncture of the last six measures.

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Here is my first elaboration using one measure of the basic, first species canon as one beat of the elaboration. The extension is there to make a good ending cadence:

In the second elaboration, I still use one measure of the basic canon for one beat if the new one -- this time in 9/8 time. It is transposed to suit the range and tessitura of the instruments:

Working in some suspensions.

You'll notice that in yesterday's basic (1st species) canon, I wrote a suspension in at the juncture right before the repeat signs,

Because of this and because one of Norden's techniques of embellishment (#5 --see Thursday's post) is suspension, I worked a few more into my own embellishment.

Suspensions are pitches that begin as a consonance but are tied (or otherwise heald over) to create a dissonance when the other voice changes pitch. It then resolves down by step. This is key. Essentially, I was able to include suspensions anytime my melody descended by step --as long as I could create a dissonance by hoding that pitch over, the stepwise resolution was already "built in."

I did that in measure 2 of the dux (3 of the comes). The other suspensions are in measure 5 of the dux and 6 of the comes.

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Saturday, June 6, 2009

Another perpetual canon.

This one has two notes in the dux before the repeat signs. That requires an expansion of the technique that I discussed a few days ago: simply that a double counterpoint needs to be calculated for each note that occurs at the critical juncture of the repeat.

In other words, I needed to calculate D.C. for each of those two notes. If there were three, I'd have to do it for all three etc.

I also needed to be certain that the number of measures inside the repeat signs was a multiple of the number of measure that the dux had before the repeat --this ensures that everything works out evenly on the repeat.

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More Embellishment

Another embellishment of Wednesday's basic canon with a bit of chromaticism (very little...)

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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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