Another canon at the fifth.

Busy day today, private lessons all morning and a performance later this afternoon...

But -- the great thing about doing this kind of thing in a public forum like a blog is that it keeps you honest. I had to write a canon today...

So here it is. Simple and to the point. A canon at the fifth for two trumpets -- again in a fairly traditional style.

For a bit of interest, I broke it up into three phrases with a successively higher climax for each.

(click on image to enlarge)

Canon over a Handel Bass

When Handel was living in England, he put together a series of lessons in continuo playing and fugal improvisation for Princess Anne, one of the daughters of George II. They are remarkably well-thought out and pedagogically sound. Because the Princess was unusually gifted, Handel took great care in crafting the lessons.

I wrote today’s canon over the bass that was the first of these lessons. I held strictly to the harmonies that Handel calls for -- a series of root position triads.

My choice of canon at the fourth gave me a few challenges but nothing too difficult to overcome. The harmonic progressions are varied, even in this short exercise. In the first two measure the progressions are by fourth or fifth. From the end of measure 2 through measure 4 the progressions are by descending thirds with one interspersed fifth. Starting with the last note of measure 4, the progressions are by mostly ascending seconds with one descending third. The final cadence is, of course, V - I, the reverse of the opening progression.

This simply meant that I had to choose wisely. A little trial and error (or success) and I had it done.

(click on image to enlarge)


I’ll likely use the Handel studies as a resource several more times in this project.

Thursday, January 29, 2009 --Canon at the 6th

[I wrote this canon on Thursday but got home too late and tired to post -- here it is:]

Yesterday's canon proved to be easy because the imitation interval of a fifth facilitated circle of fifth progressions to the point of almost making them inevitable. Today I decided to explore another interval if imitation, namely the 6th and explore its relationship to another traditional element of contrapuntal/harmonic language.

My first thought was to try a chain of 7-6 suspensions -- which, of course, could be managed in any imitation interval, if the time interval is right. The 7-6 suspension chain is a contrapuntal cliché (I mean that in a good sense.) that harmonizes a descending scale.

[For the uninitiated: A suspension is a dissonance (in this case, a 7th) that is tied from the previous measure or beat, where it is consonant, and then resolves down by step to another consonance (in this case, a 6th). When the lower voice is descending stepwise through the scale, a chain of these suspensions is possible.]

The examples show the basic, unadorned 7-6 suspension in different imitation intervals and the time interval required to make it happen (for whole notes and tied half notes). As you can see, with the imitation interval of a sixth, the time interval is one half note or half a measure of 4/4 time

Also, in the examples, you'll see the underlying structure of today's canon.

(click on image to enlarge.)

I elaborated the suspensions with ornamental resolutions and fleshed out the rest with some simple passing tones to differentiate the voices.

As a further "control", I gradually moved from primarily quarter-note motion at the beginning to eighth-note motion at the end by carefully choosing my elaborations -- especially the ornamental resolutions of the suspensions.

(click on image to enlarge.)

Some of the things I am discovering about harmonic control and contrapuntal clichés in canon writing are informing my technique when improvising. This is not unexpected, but I am surprised at how soon it is manifesting.

Inventio -- a canon at the fifth

I've written a two-part invention in the form of a canon at the fifth.

Since harmonic progressions in fifths are ubiquitous in the common practice era (17th-19th centuries), a very characteristic harmonic language was easy to achieve.

I begin with I - IV - viiº - I, which helps to establish the key. Then I have a middle section that moves though the circle of fifths IV - viiº - iii - vi - ii - V, extending the V for one more measure with the rests in the right hand to set up a recap of the opening section.

I tweak the canon in measure 16 to bring about a cadence.

I should really try one of these at another interval of imitation sometime -- the choice of a fifth definitely made things easier today.

(click on image to enlarge)

16th century style canon in augmentation

In one of my classes, today, I used, as an example, a canon in augmentation by Josquin des Prez

It is in the dorian mode on D and has two clear cadences -- a weaker one at measure 5 and a stronger one at the end. "weak" and "strong" are not used here as indicators of quality but rather, referring to relative strength. If every cadence was strong a piece would sound like it was repeatedly starting and stopping. Having different kinds of cadences makes controlling the form of a piece much easier. You usually want your strongest cadence at the end...

In Josquin's canon, both voices start at the same time but the upper voice moves at half the speed of the lower. The lower part completes the canon in measure 10 and continues more freely from there to the end.

(click on image to enlarge)



In my canon (also in D dorian), I chose to start the augmentation (in the lower part) one measure later. I used a typical 16th century style cadential figure and a leading tone (C#) to make the first cadence (measure 5) but by having it occur over F (making the interval of a 6th) instead of D (which would have given an octave), this cadence is weaker than the final. After the first cadence, I begin the next section with some imitation to further unify the piece,

I crossed the voices at measure 8 because I wanted to hear the leading tone-to-tonic motion on top.

(click on image to enlarge)

a quicky...

I worked this one out on the train this morning -- first making a note against note version and then elaborating it and adding a bass to clarify the harmonies.

Oddly, it reminds me of some of the string parts to the Beatles recordings -- probably the plagal cadence with a suspension.

Pretty simple to understand: The violin part is an inversion of the viola part.

A quick little thing for an overloaded day...

(click on image to enlarge)

another one on Perle's 12 tone row

Using the same 12 tone row as yesterday, I've written a very different canon.

This is a canon at the 5th in 2 parts with the upper part being a free voice. By "free" I mean non-canonic. It does seem a bit odd to refer to a melody written on a row as free. I guess it is all relative.

For the two canonic voices, I did not restrict myself in any way -- other than the canon, that is...

The structure of the piece is ABA. I tried to make the B section (measure 7 to 10) a bit more dissonant than the A section -- but not too much so. In a short piece like this, a little contrast is sufficient.

(click on image to enlarge)

George Perle (1915-2009)

American composer and Pulitzer Prize winner, George Perle died on Friday.
Here is the NY Times obit. of Perle by Allan Kozinn.

I didn't know him personally but his books were important in the development of my approach to composition and analysis. Don't get me wrong -- my music sounds nothing like his. It's just something about the way that he thought that has crept into my methodology.

His books are staples of graduate work in music: Serial Composition and Atonality, Twelve-Tone Tonality and The Operas of Alban Berg are the most well-known. My personal favorite is The Listening Composer.

My little homage to George Perle is today's canon (which I'll probably expand into a larger work at some point.)

I based it loosely on a 12-rone row that was the basis of one of his string quartets. It is a canon in retrograde with the row in the first half of the right hand part and its retrograde in the second half of the left hand part.

I the ornamentation is free (i.e. non-canonic).

(click on image to enlarge)


I find Perle to be one of the most expressive of the "12 tone" composers (whatever that name really means). The best tribute to a composer is to listen to his music. I'll spend some time with Perle's this evening.

Alex Ross has posted a movement of Perle's Serenade #3 on his blog.

canon in retrograde

Today's entry is a canon on retrograde. The bassoon part is the same as the oboe part but backwards. As you may have gathered by now, I have a great interest in symmetries, mirrors, retrogrades, inversions, palindromes etc.

(click on image to enlarge)

The technique for writing these is surprisingly simple:

Write the first half in invertible counterpoint. I'll explain how to do this another time but it simply means that either part can function as the bass line. Then copy the upper part backwards onto the end of the lower part and vice versa.

The trick is to make it sound musical. I'll talk about this another day. I have an important meeting to prepare for today.

another puzzle canon and solution

This is another little canon based on the idea of Bach's Trias Harmonica (see yesterday's post). This time I'm also using the idea of exact inversion, used by Bartok among others.

I've chosen a symmetrical chord and derived a passing chord that is also symmetrical.

(Click on image to enlarge) The puzzle:

Because of this symmetry, I am able to use strict inversion

Again, this is very condensed -- more an intellectual exercise than an artistic statement, to be sure.


(Click on image to enlarge) The solution:

condensare

There is an odd little canon by J.S. Bach called Trias Harmonica (The Harmonic Triad) which was transmitted to us by F.W. Marpurg in his Abhandlung von der Fuge (1753).

In its written form, it has eight pitches -- a C major scale that ascends stepwise from C to G and back. The dotted rhythm makes it obvious that the Ds and Fs are simply passing tones. We are told that this is an "eight-part canon in contrary motion." Typical of a puzzle canon, this is all we get.

(Click on image to enlarge)

Now to the nuts and bolts: A very little thought reveals the solution. There are eight beats to the theme and the chord tones are articulated on every other beat. If we add three entrances of simple canon at the time interval of a half note and, beginning on G, add the four contrary-motion voices on the even numbered beats, we get the full canon.

The result is a C major chord that is constantly in motion with passing tones occurring in unaccented positions between the beats.

(Click on image to enlarge)

A work of genius? Well...no...but it is a clear demonstration of some basic principles of counterpoint and Bach's ability to condense his musical gestures.
------------------------------------------------------------------------------------

I've chosen a different type of chord for my canon -- a four note chord of stacked perfect fourths. This is not an 18th century chord. Its most frequent use came in the twentieth century. You'll find this sonority in works by composers as diverse as Paul Hindemith, Charles Ives, Arnold Schönberg, and McCoy Tyner (that's right, the great jazz pianist -- 4th chords are common in some types of jazz and popular music too).

(Click on image to enlarge)

My example is structurally simpler than Bach's. I set mine in 9/8 time, two measures, with the chord tones falling on each of the six beats. The intervening notes are not traditional passing tones which would need to be approached and let by step. Rather, these tones are approached by step and left by leap, when ascending and the opposite when descending This enables me to have another chord of perfect fourths in the unaccented position. In this manner, I'm able to maintain a consistent harmonic language.

The solution is in six voices at the unison.
(Click on image to enlarge)

Ezra Pound once wrote "dichtung=condensare" ("poetry is to condense")

I'm certainly not saying that this canon is poetic. I'm just practicing my condensation.

Inauguration Day

Happy Inauguration Day!

If you follow this blog, my feelings about this are no surprise to you.

I watched the inauguration with a few hundred of my neighbors at the local cinema -- quite an experience. It's been hard to pull myself away from the coverage, even after coming home.

I wrote this canon a few days ago (knowing I wouldn't get much done today). It's a canon on Hail to the Chief. The upper two parts (trumpets) are a canon at the unison on the theme and the two trombone parts are augmentation of the first half of the first phrase.

(click on image to enlarge)


For another musical response to the significance of the day, please check out my friend, Seth Austen's lovely new fiddle tune, 20th of January.

Free at Last

Martin Luther King Day

A few years ago, while working as music director at a Unitarian-Universalist church, I composed an organ work entitled Canonic Variations and Fugue on Free at Last. It had always struck me that almost everyone knew Dr. King's quote but almost no one (at least among the New England communities that I'm familiar with) knew the tune of the hymn that he quoted from. I taught the choir Free at Last and I composed the organ work to serve as a postlude for that Sunday's service.

Near the end of the fugue is a stretto. This is a section in which the subject is played in canon with itself. The two lower voices contain the stretto and the upper voice has the countersubject. I'm just including an excerpt here. If you are interested in the whole work, it is published by Singing String Music Publications: http://www.singingstring.org/publications/compositioncatalog.html

(click on image to enlarge)


But, since the idea of this blog is to write a new canon every day, here is a perpetual canon in inversion. It is also based on the tune Free at Last -- but it is not a part of that earlier composition. Just like the Goldberg Ground within which Bach found so many possibilities, Free at Last is a rich resource for canon.
(click on image to enlarge)

Prelude-canon

Using the same basic technique as yesterday, I wrote today's canon. It was fast and dirty but reasonably successful. If I had more time, I'd edit details and make the melodies more effective.

First, I made up a harmonic progression of the type suggested by C.P.E. Bach in his Essay on the True Art of Playing Keyboard Instruments as a way of improvising preludes. Obviously, writing a canon is different than improvising but the type of harmonic progression is often the same.

Since the canon follows at a sixteenth note distance, the last 16th of each measure in the dux is the common tone. The only case in which there is no common tone is measure 20 where the last note of the dux is a non-harmonic anticipation. In the next measure the same pitch is the first note of the comes and is a chord tone.

(click on image to enlarge)

Harmonic control again

Looking at Bach's canon #9 from the Fourteen Canons on the Goldberg Ground, I started to think about harmonic control again.

Bach takes the Goldberg bass line and, above it, writes a canon at the unison with a time interval of one 16th note. In order to accommodate the harmonies, the last 16th of the dux (leader) becomes extremely significant. This pitch will not only be the last note of the "present" harmony but a split second later it will be the first pitch of the "next" harmony in the comes (follower).

In my analysis of the contrapuntal techniques in this canon, you can see how Bach handled this. When the two adjacent chords have a tone in common, that tone is used as the last note of the dux (and first of the comes). When there is no common tone Bach must use either a non-harmonic tone or a rest -- both of which he does, one time each.

(click on the image to enlarge)


You may also want to look back to my discussion of Bach's F major Invention on
January 4 (The Gift of Plurisignificance) for other ideas about controlling the chord progressions in canons.

For my canon, I used this same technique. For a harmonic progression, I chose a complete diatonic circle of 5ths in D minor. (i - iv6 - VII - III6 - VI - iiº6/5 - V7 - i) Since diatonic chords a 5th apart have common tones, I simply chose that common tone to be the last note of the measure for the dux each time. My time interval is an 8th note.

This one was fun to write. I may do a few more of this kind before moving on to other ideas.

(click on the image to enlarge)

augmentation / diminution

Feeling a bit guilty for trying to make more of a political than musical statement, I get myself back on track today.

There are potential problems when writing canons in diminution or augmentation:

1) Fast passages in the "long-note voice" of the canon can sound neurotic or at least awkward in the "short-note voice."
2) The short-note voice finishes in half the time (assuming that the ratio is 2:1).
3) If the solution to #2 is to simply repeat the short-note voice again, the canon could sound too repetitious.

I tackled these problems in the manner described below.

For #1, I simply kept the note values within a limited range. If I wanted to be more adventurous (and had the time to work it out), I could have been more bold. There's plenty of time for that (350 canons to go...).

For #2 & #3:

Well, this is a bit more complicated. First, you'll note that I do the long-note part twice. -- once in the lower part and then, starting in measure 13, in the upper part.
By making this part forty six beats long (11 measures of 4/4 and one of 2/4), I insured that the short-note part (23 beats) would start its repeat in a different metrical position (on a different part of the measure). This gave a certain balance between repetition and contrast.

Starting in measure 13, the parts switch. Simple ways of keeping interest despite the repetition...


Oh, the reason I used "augmentation/diminution" in the title instead of choosing one or the other is that with both parts starting simultaneously, neither description seems more appropriate than the other.

But you probably knew that.

(click on image to enlarge)

Don't let the door hit you on the way out.

Tonight I watched a certain public figure's farewell address.

This canon in diminution is more symbolic than skillful -- just a simple commentary. If the repetition gets annoying...well, that seems to be in the spirit of the proceedings.

I'll be in a better frame of mind tomorrow (certainly by Tuesday).

(Click on image to enlarge.)

Making the voices more distinct

                                                  (click on the image to enlarge)

For this three voice canon, I simply elaborated yesterday's canon with non-harmonic tones. Just a bit of trial and error (or trial and success?) and attention paid to keeping the parts rhythmically distinct. In yesterday's canon the parts moved at the same half-note rate. This could cause it to be heard as a series of chords rather than as distinct melodies.

This canon has much clearer counterpoint.

If you are new to contrapuntal writing, you will probably learn more by comparing this one to yesterday's than you would by any amount of explanation on my part.

Serendipity and/or attention to the possibilities

For this 13th movement of my 366 part Ode to Autodidacticism I had an interesting experience.

A bit pressured by mundane tasks, I found myself multitasking. I was glancing through a book on counterpoint (Fundamental Counterpoint by Hugo Norden) while having lunch. I was looking for some different ideas to present to my students this coming semester when we go into 4th species -- but, in the back of my mind, I knew I hadn't written my canon yet.

As my eyes fell onto a B minor cantus firmus that Norden gives for an end-chapter assignment, I saw quite clearly that the first 8 notes worked as a canon at the octave. I think that this daily canon writing is already having an advantageous effect.

Upon closer examination, I found that it worked in three parts as well.

Here it is, unadorned. Perhaps I'll elaborate upon it for another day but, for now, it helped me to accomplish today's entry with negligible effort.

What a nice thing to have happen on a busy day.

(Click on image to enlarge)

Another Canon in Inversion

To explore symmetrical inversion further:

Yesterday's canon in inversion required each hand to be in a different key/scale. As I pointed out, this is a feature of strict intervallic inversion. When you invert this way (using the tonic as the axis of symmetry), major inverts to phrygian, mixolydian inverts to aeolian, lydian inverts to locrian and vice versa.

Oh, I didn't mention dorian...

This is a special case. Dorian inverts to itself or, to phrase it a different way: dorian is symmetrical around the tonic. See the examples.

The last example shows the chords that I arpeggiated as measures 12-15.




                          (Click on image to enlarge)







The violin is the dux until measure 10. I have the cello lead after that. The strict canon at 2 beats lasts until measure 15. From there to the end I use strict mirroring. Technically, this ending is still a canon in inversion -- but the time interval has changed to "zero."

Harmonic control was fairly easy using dorian in inversion.

"D" inverts to itself (tonic).

"A" and "G" invert to each other -- so I was able to use a seventh interval as a dominant for the middle section (starting in measure 10).

"B" and "F" invert to each other -- I exploited this by coming to this tritone at the end of the strict canon. At that point, I moved in contrary motion -- first to the seventh A-G (functioning as dominant and then, with more rhythmic interest, converging on the tonic at the end. The "stinger" is a Dsus4 chord -- symmetrical around the middle D axis.

                                      (Click on image to enlarge)

Symmetrical Inversion

Inspired again by Bartok's Mikrokosmos, I've written a canon in inversion at the octave.

As I mentioned two days ago, Mikrokosmos contains a wealth of compositional ideas. Though not a canon, #29 (book one) Imitáció tükörképben (Imitation Reflected) uses a strict inversion which causes some interesting pitch material. At this point in the book, Bartok is still using five-finger positions. The strict inversion of the first five notes of E major (in the right hand) inverts to the last five (or the first five below the tonic) pitches of E phrygian in the left hand. See the notated examples for clarity.

(click on image to enlarge)






As you can see, using the tonic as the axis of inversion, a major scale will always invert to a phrygian.

For my example, I used the phrygian in the right hand which reflects as major in the left. This time, though I still kept it short and simple, I didn't limit myself to a five finger position. I did, however, contain it within an octave in each hand.

I kept to mostly consonances and approached and left dissonances by step. The result certainly isn't 18th century in style...


(Click on image to enlarge)

Butch Baldassari (Dec. 11, 1952 - Jan. 10 2009)

Just as I was sitting down to write today's canon, my friend Seth Austen informed me of the passing of Butch Baldassari, musician (mandolinist) and a true gentleman.

I met Butch at Cape Cod Mandolin Camp and had the pleasure of performing several duets with him at the faculty concert. This was just before he became ill. I wish I had the chance to know him better.

In Butch's honor, I offer this double canon at the octave on the refrain to the hymn Rest for the Weary from the Sacred Harp. It is set for 2 mandolins and guitar. The original melody is in the 2nd mandolin.

My condolences to the Baldassari family on this sad day.

(click on image to enlarge)

Canons in the style of Bartok #1

As I promised, I'll be leaving the Fux for a little while (though it is difficult to ignore him when discussing any kind of counterpoint.). I think it might be interesting to leave Bach alone for a while, too, though I might make the same comment about him -- they both pervade our contrapuntal discourse...

I spent some time looking at canons in book 1 of Bartok's Mikrokosmos. For those who aren't familiar with it, Mikrokosmos is a six volume piano method / anthology of progressive pieces / compendium of Bartok's pianistic and compositional style etc. It is difficult to fully assess its importance. As you might guess by my description, there are compositions that range from the simplest five-finger position beginner pieces to concert repertoire that Bartok, himself performed.

It is a remarkable resource for Bartok's compositional style. Even the easiest pieces in book one are well-crafted gems with many of the stylistic features that you'd find in his large scale orchestral works.

Of course, here we'll focus on canons. There is a great deal of canonic influence and imitation of various kinds throughout these volumes. The first one actually called a canon is #28 (Vol.1) Kánon oktávában (Canon at the Octave). I won't violate copyright to reproduce it, but it's easily found in libraries and music stores. If you are serious about music, you should probably own it anyway!

A quick description of the Bartok piece: It is in the Phrygian mode in a five-finger position (this is a piano pedagogy term meaning that the fingers are set on five consecutive keys without shifting). The pitch material is E, F, G, A, B --right hand in the octave above middle C and left hand in the octave below. Most of the intervals are consonant (6ths, 10ths and 8ves). Dissonances are approached and left by step.

For today's canon, I've also written a five-finger position, modal canon. Mine is in the Lydian mode (F, G, A, B, C). The hands are space an octave apart. The dissonances are all approached and left by step. I tried to give tonal variety by having the focus move to G for the middle section and back to the tonic F again at the end.

Over the next few days, I'll look at other techniques of canon found in Bartok.

(click on image to enlarge)

One more canon on Fux...

Busy day today-- errands, lessons, rehearsals...

So, I gave myself a relatively easy task for my canon. I used the Fux cantus as, of all things, a cantus firmus and wrote a little trio for two violins and cello with the violin parts in canon. It's not in the style described in Fux's Gradus but rather more like a trio sonata in style and texture. I suppose that I could have figured the bass...

The one license that I allowed myself was the C# in measure 3 in the comes. I needed it for the harmony.

Tomorrow, we get away from Fux for a while.

(click on image to enlarge)

Another Canon from Fux

This one needed a bit of tweaking to make it work. It is a canon at the 4th below and came about this way:

I found the harmonic and rhythmic interval by testing options (yesterday, on the train). I discovered that all the intervals between the two voices were consonant except for the seventh in measure 5. Since the seventh resolves down, this proved to be usable as well. The one remaining issue was the proper approach to the dissonance. Fortunately, a little rhythmic shift made the seventh work as a suspension. See the example -- I've included my sketches and working out for the sake of clarity.

The corresponding rhythmic shift in the comes, a measure later also worked advantageously, keeping the canonic imitation exact.

I did, however, depart from a strict canon by adding additional suspensions in the approach to the cadence.

The final step was embellishment with passing and neighboring tones to enliven the texture.

Inversion Canon on a Fux Cantus

I decided to make a canon from the first cantus firmus (the assigned melody against which one writes a countermelody) in Fux's Gradus ad Parnassum, one of the most famous melodies in the history of counterpoint pedagogy.

Maggi and I had a couple of business meetings in Boston -- so, the half-hour train ride from Salem became my "limiting factor" today.

Here's the process:

1) I wrote out the Fux cantus and its inversion.
2) I tested the canonic possibilities at various time intervals (how much later the following voice enters) and harmonic intervals (the pitch distance between the voices)
3) I completed the simple canon below and sketched some other ideas that may bear fruit later.

The canon that I offer today is a canon in inversion at the 6th. I gave the inversion to the dux (Latin for "leader") and the original cantus to the comes ("attendant" or, if you wish "follower").

The result is a simple canon in a Fuxian 1st species (voices moving in the same rhythm). I've annotated the score to explain how I wrote it.


--------------------------------------------------------------
A word about languages: You may notice a fair amount of Latin here. Most of the early counterpoint treatises were in Latin, including the Fux, which, after all, was the most influential of them all. Therefore much Latin terminology creeps into the discourse. Eventually, I'll put together a glossary but, for now, I'll try to define them as we go.

Gradus ad Parnassum -- "Steps to Parnassus" Fux's text is notable for the gradual steps that make up his method of teaching.

Even the word "counterpoint" comes from Latin originally:
punctus contra punctum -- "point against point" (point=note, or in some usages"melody")

The word "canon" however, comes from the Greek KANÓN meaning "law."

----------------------------------------------------------------
Tomorrow, I'll look at another canon derived from Fux's cantus.

An Eye/Ear for Canons

If you don't know Bach's Fourteen Canons on the Goldberg Ground, treat yourself to a look at them. They are quite interesting, very short and deceptively masterful.

There are two good resources for them (both with scores and realizations):
http://www.flagmusic.com/work.php?r=BWV_1087
and
http://jan.ucc.nau.edu/~tas3/fourteencanonsgg.html
The second one has a clever animation (click on the score) to show how the canons are derived.

Bach takes a given melody and, because of all of his contrapuntal experience, is able to see (that is, hear, internally -- as I keep telling my Ear Training classes) a number of options for canonic settings of that melody against itself at different pitches, in retrograde, inversion etc.

I won't belabor this now, nor will I dwell on the Fourteen Canons at this point -- though I reserve the right to return to them later.

For my humble offering today, I used a source of hymn tunes different than the Lutheran chorales that were the grist for Bach's mill. I turned to the quintessentially American hymnal, The Sacred Harp.

Because today has several other demands on my time, I needed to work quickly -- so, I decided to make it easy on myself and try to find a hymn tune that would work as a canon at the unison -- this being easier to see/hear at a glance.

Luck was with me because I only examined a handful of tunes before finding one that worked. This was Holy Manna (page 59 in the 1971 edition of the Original Sacred Harp), a wonderful, lively, pentatonic, folk-hymn that I found would work with only two rhythmic adjustments: the ties in measure 16-17 and 18-19 and the rests in measures 27 & 29. These two simple elongations were enough.

(click on image to enlarge)



Until tomorrow...

The gift of plurisignificance

Yesterday's canon exercise left me pondering other issues of harmonic control in canons.  My thoughts leapt to J.S. Bach's Two-part Invention in F major -- a piece that is certainly familiar to anyone who ever got beyond a certain point in piano lessons.  Though it isn't a canon throughout the entirety of the piece, it does have some interesting canonic usage.

We do know that Bach intended these as compositional models as well as pieces to study and perform.  On the title page of the inventions, he states that one of the purposes of the inventions is to "acquire a strong foretaste of composition."

Bach begins this invention with a canon at the octave which continues to the first beat of measure 8.  I've included an annotated excerpt to show what happens harmonically.  Bach first establishes he key with five measures of arpeggios of the tonic chord and scalewise passages that simply elaborate the I chord.  If you follow the 16th notes of the right hand part of measures 4 through 6, you'll see essentially descending thirds (A-C, F-A, D-F) -- which are, of course, answered in the left hand a measure later.

Bach takes advantage of what music theorist Bernhard Ziehn (1845-1912) later called plurisignificance and herein lies the genius of the passage.

The F-A pair first serves as the root and third of an F major chord and then as the third and fifth of D minor. 

The D-F pair is first the root and third of D minor and then the fifth and seventh of G7.

By exploiting the plurisignificance of these diads, Bach is able to move the harmony along rather than have it stagnate in any one chord. See my annotations to the Bach example for clarification.

(Click on image to enlarge)

--------------------------------------------------------

For my canon, I've also worked with the plurisignificance of thirds in a harmonic environment of descending third progressions:

A

F# -- F#

D   -- D   --  D

         B   --  B  --  B

                  G  --  G  --  G

                           E  --   E

                                    C#

I        vi       IV     ii       viiº  ---then resolution to I

This is, to be sure, a very typical tonal progression.-- a simple elaboration of tonic --subdominant -- dominant -- tonic.

Following Bach's example, I, too, begin with a few measures of elaborated tonic harmony.

With the addition of a few well-placed accidentals, my canon could also modulate and serve as part of a larger work like a two-part invention.  Perhaps another time...

Here's mine:

(click on image to enlarge)

Dealing with the harmonic boredom issue

I delved into my extensive library (no exaggeration- we have 48 overflowing bookcases at home.) again today. The impetus behind today's canon comes from Fourteen Canonic Inventions by Meyer Kupferman [1926-2003] (NY: General Music Publishing, 1966).  It's a useful and interesting volume in which the composer, himself, analyzes the compositions that he offers as models of canonic technique.

The second canon in the book is called "Canon on a Choral."  [Out of respect for the copyright owner, I will not reproduce it here.]  Kupferman wrote a one-measure chorale phrase based on the harmony i, iv, V in the key of A minor.  He then took the four parts of the chorale "linked together in a linear fashion" to create the 4 measure melody.

It's an interesting technique to generate a canon but it does have its limitations.  The most obvious of these is that the harmony will repeat for the duration of the canon.  This is a perennial problem for many types of canon.

The task that I set myself was to use this particular technique but to try to overcome the harmonic boredom issue, at least for this type of small form.

I wrote a two-measure chorale phrase with the harmony i - III - N6 - i6/4 - V.

I chose the Neapolitan chord because of its function (pre-dominant) and because of its pitch content, as you will see below.  The Neapolitan chord is a major chord built on the flatted second degree of the scale (bII) -- in the key of E minor it is an F major chord.

By making the rest of the chorale diatonic and saving the F natural for the end of the melody, I was able to delay the chromaticism and the piquant Neapolitan chord until late in the piece despite the repetitious nature of the canon.

The first beat of each of the even-numbered measures has the pre-dominant function. In measure 6, that function is handled by only two pitches ("A" and "C") -- ambiguous, perhaps --but in the diatonic environment, we'll likely hear it as iv.  In measures 8 & 10, the added F natural makes that pre-dominant function a very unambiguous Neapolitan.

To add interest, I allowed myself the license of a change in the bass melody in the antepenultimate measure.  This half note "C" creates a deceptive cadence which adds just enough interest to allow one more repetition of the two-measure "chorale" which brings the canon to a close on an authentic cadence.  This simple (and common) rhetorical device also, in this case, brings the violin around to the beginning of the canon again to recap the first two-measure phrase.

I've included an outline of my procedure and a harmonic analysis.  Compare, if you will, the analysis of the chorale as I first wrote it and the analysis of the completed canon.

(click on image to enlarge)

A Canon at the Eleventh

I took a look at a well-known counterpoint text, the appropriately-named Counterpoint by Kent Kennan (my copy is pre-ISBN, 1959).  It's one of a large number of such books in my collection.  It has a very useful section on canons that I will probably use several times in the course of this project.

My motivation, this time, was a quote from Kennan in which he says "The 12th is not uncommon as a harmonic interval, but the 11th is seldom encountered." 

By "harmonic interval" he is referring to the distance between the starting pitches of the dux (leader) and the comes (follower).

Well, it may be "seldom encountered" but it certainly isn't any more difficult to write than a canon at the octave, such as the one that I wrote yesterday.  I used essentially the same technique of writing a measure at a time and copying it into the other part. (see yesterday's post) 

Again, I've kept it short, simple and to the point.  

                                                           (click to enlarge)

Beginnings...

January 1, 2009 )

Athbhliain faoi mhaise duit.
"Happy New Year to you." (in Irish, which I am learning.)

About a year and a half ago, I wrote a short canon on a New Year's text written by my wife and creative partner, Maggi. To represent the mythical Janus, facing both forward and back, I wrote a canon that was also a palindrome. Since today is New Year's Day, I humbly submit it here.

                                                            (Click to enlarge)


But, since this blog is about writing a new canon every day, I've done that as well.

I'm starting with a very simple canon of the type that is first taught to counterpoint students. I first wrote a skeletal version.. It's pretty much a Fuxian 1st species counterpoint, except for the minor 7th in measure four which I resolve down by step, in the traditional manner.

Then, to make it a bit more musical, I embellished the melody by adding rhythmic figures and non-harmonic tones. The idea was to make the simultaneous rhythms different in each part, render the melody more interesting and differentiate the phrasing.

                                                             (click to enlarge)

It's simple and straightforward -- a nice measured beginning to this project.