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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