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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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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More Pentatonic Inversion

Taking the information about inversion that led to the last two canons, I quickly realized that since inverting at the tonic of the first scale will generate a scale a major third lower, if I wanted to maintain the same scale, i would simply need to begin the inversion a major third higher than the dux.

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Finally Caught Up Again!

I'm finally caught up again after falling behind with my posting during the grading of finals.

Today's canon is an elaboration of yesterday's 1st species-style canon in inversion.

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Tuesday, May 26, 2009

Pentatonic Canon is Inversion

If you invert a major pentatonic scale from its tonic, the result is a major pentatonic with a tonic pitch a major third lower.

To be more precise: the result is another pentatonic set a major 3rd lower but the intervals are inverted.

Major pentatonic -- M2 M2 m3 M2 m3 -- Ascending

from C -- C D E G A C --C major pentatonic.

to invert we use the same intervals descending:

from C -- C Bb Ab F Eb C

If we rearrange these pitches to start on Ab, you'll see that we have an Ab major pentatonic scale:

Ab Bb C Eb F Ab

I used this material for the canon today -- the common tone of C is used as the tonic.

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Monday, May 25, 2009

An elaboration of yesterday's canon -- quite different from either Friday's or Saturday's canons.

The instrumentation is a combination that I've always wanted to try...

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Sunday, May 24, 2009

Phrygian canon in 3 voices.

I took Friday's canon and removed the sharps from the upper two voices. This gave me a slightly odd by mostly workable canon in E phrygian. I tweaked it a bit to eliminate some voice leading problems and harsh dissonances.

Here it is (in whole and half notes):

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Saturday, May 23, 2009

An elaboration of yesterday's canon using only the tones of those three scales:

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Friday, May 22, 2009

Canon on Three Pentatonic scales.

I chose Comes I to enter a major third above the dux and Comes II to enter a major 6th above the dux (a perfect 4th above Comes I).

This gave me the following pitch material:

Dux -- C major pentatonic C D E G A
Comes I -- E major pentatonic E F# G# B C#
Comes II -- A major pentatonic A B C# E F#

But, I didn't treat them as being in these three keys. Since E is the common tone between these three scales I chose to use it as a tonal center. This makes the top two voices sound like they are in E major and the lowest voice sound like E phrygian.

Again, 1st species-style:

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Thursday, May 21, 2009

Elaboration of yesterday's 1st species-style canon.

It almost seems like cheating to write a whole-note canon one day and then elaborate it the next but I do it for numbar of reasons:

1) It facilitates composition on days that I have limited time (all too often!)

2) It explicates the process for others who may wish to try their hand at contrapuntal and/or canonic composition.

3) habit (at least by now it seems to be so...)

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Wednesday, May 20, 2009

A pentatonic canon at the tritone.

Not very traditional sounding because there are no common tones shared by these two scales.

Also, I stuck with the sharp names of the notes throughout, even though the consonances would be made more clear by occasionally spelling them enharmonically. Needless to say, this canon must be in equal temperament to work...

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Tuesday, May 19, 2009

Elaborated 3-Voice Pentatonic Canon

Yesterday's canon elaborated:

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Monday, May 18, 2009

Three-Voice Pentatonic Canon at the 4th above and below.

As I stated on Saturday, combining major pentatonic scales a fifth or fourth apart gives a set of pitches contained in the major (heptatonic) scale.

If we combine a pentatonic with one a fourth above and another a fourth below, we get the complete diatonic set.

Here the dux is C major pentatonic -- C, D, E, G, A

Comes I is F major pentatonic -- F, G, A, C, D

Comes II is G major pentatonic -- G, A, B, C, D

The complete set of pitches is C, D, E, F, G, A, B (C major)

In 1st species style:

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sunday, May 17, 2009

Elaborated Pentatonic Canon at the 4th

Yesterday's canon elaborated.

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Saturday, May 16, 2009

Pentatonic canon at the fourth.

If one takes the major pentatonic scale (scale degrees 1, 2, 3, 5 & 6) and writes a canon at the fourth (or fifth for that matter), the resulting combined scales still only use pitches contained in the major (heptatonic) scale.

That's what I explored here. The dux is in C major pentatonic C, D, E, G, A and the comes is in F major pentatonic -- F, G, A, C, D. All of the pitches are members of the C major scale.

Today, first species--elaboration tomorrow.

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Friday, May 15, 2009

4-Voice Pentatonic Canon

One interesting feature of the major pentatonic scale is that it has no half steps. A technical name for it that is sometimes used is anhemitonic pentatonic (literally "5 tone scale with no half steps").

In a practical sense this means that it is pretty difficult to produce harsh dissonance with only these tones. To demonstrate that fact, I offer this setting for string quartet of yesterday's melody. because the rhythm varies sufficiently from measure to measure, the parts will be distinct enough.

I have not altered any of the tones (except, of course, the simple transposition to a suitably resonant key for strings). Consecutive voices enter an octave apart.

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Thursday, May 14, 2009

The elaboration of yesterday's canon.

I used only the scale tones of each voice in the elaboration.

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Wednesday, May 13, 2009

Here is a 1st-species-style canon using the same two scales as yesterday but in reversed position.

I'll elaborate it tomorrow.

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Tuesday, May 12, 2009

...still playing catch-up with my posts, though I have managed to continue to write my canons daily.

Maggi and I had a concert at the Fitchburg Art Museum on Sunday that took up much of our time in preparation and rehearsal.

Here is a canon on the major pentatonic scale 1, 2, 3, 5, 6.

Because it imitates at the minor tenth, each voice is in a different scale.

Lower: C D E G A

Upper: Eb F G Bb C

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Monday, May 11, 2009

Another elaboration of Saturday's canon. This one using notes of the D minor scale. Again, recall that this symmetrical enharmonic pentatonic is a subset of the D minor scale...

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Sunday, May 10, 2009

Another very busy day.

I simply elaborated yesterday's canon using tones of the scale>

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Saturday, May 9, 2009

A three-voice canon in symmetrical enharmonic pentatonic.

Working on grading finals, this needed to be quick. It's in 1st species style.

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Friday, May 8, 2009

A second elaboration of the May 6th canon.

For yesterday's elaboration, I used only pitches of the symmetrical enharmonic scale that I used to compose the original first-species canon.

Today -- a different approach. I used the original canon as a skeletal outline and elaborated with pitches from the D minor scale. You'll notice that the sym/enh/pent is a subset of D minor...

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Thursday, May 7, 2009

The promised elaboration:

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Wednesday, May 6, 2009

This is another first-species-style canon -- this one using the same scale as my May 3 canon.

I'll elaborate it tomorrow...

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Tuesday, May 5, 2009

Elaboration of Yesterday's Canon

Simple elaboration using tones of the symmetrical enharmonic pentatonic:

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Monday, May 4, 2009

Another Symmetrical Enharmonic Scale

For this scale, I simply reversed the two tetrachords of the scale that I used for yesterday's canon.

A G# E D Bb A

A simple, first-species-style canon:

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Sunday, May 3 2009

Well, now that the semester is done, finals given and graded and semester grades submitted, I'll finally have time to catch up.

I have continued to write my daily canons but didn't have the time to enter them into Finale and post them. I will post several a day until I'm caught up.

Canon in the Symmetrical Enharmonic Pentatonic

The ancient Greek Enharmonic Pentatonic is a scale with two identical tetrachords (as were most Greek scales).

In modern terminology, a tetrachord is a group of pitches spanning the interval of a perfect fourth. The Enharmonic begins with a major third. The remaining minor second completes the tetrachord.

Example (The Greeks constructed their scales in a descending manner):

A F E (A-F is a major 3rd, F-E is a minor second)

As I said, the ancient Greek version used identical tetrachords to make a scale. These are connected by a major second so that the two tetrachords are a perfect fifth apart and will complete the octave.

A F E D Bb A

I wanted to have a scale that was symmetrical around that connective major second, so I created two versions of a symmetrical enharmonic pentatonic. Today's canon uses one of them, I'll get to the other later this week.

A F E is the first tetrachord -- M3, m2

D C# A is the second -- m2, M3

The complete scale is A F E D C# A.

As befits a symmetrical scale, here is a canon in inversion.

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Saturday, May 2, 2009

Another Canon at the Tritone.

This one is based on a scale that combines two minor chords a tritone apart.

A C E and Eb Gb Bb

The scale spelled enharmonically:

A Bb C D# E F#

See the April 30th canon.

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Lá Bealtaine 2009

Lá Bealtaine is the traditional Celtic celebration of May Day.

I have adapted phrases from a Bealtaine ritual song that Maggi and I wrote together for our book, The Measure of the Year: Singing Through the Seasons

This is an unusual canon in that the intervals of imitation change halfway through. For the first half, each following voice imitates a step lower. Beginning with the upper voice in measure six, each following voice imitates a step higher.

The title, Dlí Bealtaine is Gaeilge (Irish) for Beltane Canon.

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Thursday, April 30, 2009

A canon at the Tritone

This six-tone scale is made of two major triads a tritone apart.

C E G and F# A# C#

Written enharmonically, the scale is C Db E F# G Bb

Imitation at the tritone in this scale is a no-brainer.

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Wednesday, April 29, 2009

A canon on a scale built from the overtone series.

An equal-tempered version of the scale is:

C D E F# G A Bb C

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Tuesday, April 28, 2009

A canon at the octave, adapted from an earlier rhythmic canon.

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