A Canon A Day

Thursday, June 11, 2009

2009-06-14T08:21:00.000-07:00

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.

(click on image to enlarge)

Elaborating Dissonant Counterpoint

2009-06-10T18:23:00.000-07:00

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.

(click on image to enlarge)

Reversal of values

2009-06-09T18:05:00.000-07:00

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.

(click on image to enlarge)

A basic canon at the seventh and two elaborations

2009-06-08T07:51:00.000-07:00

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.

(click on image to enlarge)

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.

2009-06-07T16:36:00.000-07:00

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.

(click on image to enlarge)

Saturday, June 6, 2009

2009-06-07T16:27:00.000-07:00

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.

(click on image to enlarge)

More Embellishment

2009-06-05T18:54:00.000-07:00

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

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Embellishment according to Norden

2009-06-04T17:05:00.000-07:00

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.

(click on image to enlarge)

An Application of Double Counterpoint Technique

2009-06-03T17:06:00.000-07:00

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:

(click on image to enlarge)

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

2009-06-02T16:53:00.000-07:00

Here is yesterday's perpetual canon elaborated.

I'll explain these techniques tomorrow.

(click on image to enlarge)

Monday, June 1, 2009

2009-06-02T15:50:00.000-07:00

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.

(click on image to enlarge)

Canon in mensuration and inversion

2009-05-31T19:19:00.000-07:00

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.

(click on image to enlarge)

Saturday, May 30, 2009

2009-05-31T19:12:00.000-07:00

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.

(click on image to enlarge)

Hidden Elaboration

2009-05-29T20:52:00.000-07:00

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.

(click on image to enlarge)

More Pentatonic Inversion

2009-05-28T16:42:00.000-07:00

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.

(click on image to enlarge)

Finally Caught Up Again!

2009-05-27T18:37:00.000-07:00

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.

(click on image to enlarge)

Tuesday, May 26, 2009

2009-05-27T18:26:00.000-07:00

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.

(click on image to enlarge)

Monday, May 25, 2009

2009-05-26T07:34:00.000-07:00

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...

(click on image to enlarge)

Sunday, May 24, 2009

2009-05-26T07:29:00.000-07:00

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):

(click on image to enlarge)

Saturday, May 23, 2009

2009-05-26T07:26:00.000-07:00

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

(click on image to enlarge)

Friday, May 22, 2009

2009-05-26T07:12:00.000-07:00

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:

(click on image to enlarge)

Thursday, May 21, 2009

2009-05-26T07:07:00.000-07:00

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...)

(click on image to enlarge)

Wednesday, May 20, 2009

2009-05-26T07:01:00.000-07:00

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...

(click on image to enlarge)

Tuesday, May 19, 2009

2009-05-23T14:09:00.000-07:00

Elaborated 3-Voice Pentatonic Canon

Yesterday's canon elaborated:

(click on image to enlarge)

Monday, May 18, 2009

2009-05-23T14:00:00.000-07:00

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:

(click on image to enlarge)

sunday, May 17, 2009

2009-05-23T13:58:00.000-07:00

Elaborated Pentatonic Canon at the 4th

Yesterday's canon elaborated.

(click on image to enlarge)

Saturday, May 16, 2009

2009-05-23T13:49:00.000-07:00

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.

(click on image to enlarge)

Friday, May 15, 2009

2009-05-19T19:21:00.000-07:00

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.

(click on image to enlarge)

Thursday, May 14, 2009

2009-05-19T19:18:00.000-07:00

The elaboration of yesterday's canon.

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

(click on image to enlarge)

Wednesday, May 13, 2009

2009-05-19T19:15:00.000-07:00

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

I'll elaborate it tomorrow.

(click on image to enlarge)

Tuesday, May 12, 2009

2009-05-19T19:09:00.000-07:00

...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

(click on image to enlarge)

Monday, May 11, 2009

2009-05-14T19:56:00.000-07:00

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...

(click on image to enlarge)

Sunday, May 10, 2009

2009-05-14T19:54:00.000-07:00

Another very busy day.

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

(click on image to enlarge)

Saturday, May 9, 2009

2009-05-14T19:51:00.000-07:00

A three-voice canon in symmetrical enharmonic pentatonic.

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

(click on image to enlarge)

Friday, May 8, 2009

2009-05-13T09:18:00.000-07:00

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...

(click on image to enlarge)

Thursday, May 7, 2009

2009-05-13T09:15:00.000-07:00

The promised elaboration:

(click on image to enlarge)

Wednesday, May 6, 2009

2009-05-13T09:10:00.000-07:00

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

I'll elaborate it tomorrow...

(click on image to enlarge)

Tuesday, May 5, 2009

2009-05-13T09:06:00.000-07:00

Elaboration of Yesterday's Canon

Simple elaboration using tones of the symmetrical enharmonic pentatonic:

(click on image to enlarge)

Monday, May 4, 2009

2009-05-13T09:00:00.000-07:00

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:

(click on image to enlarge)

Sunday, May 3 2009

2009-05-13T08:38:00.000-07:00

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.

(click on image to enlarge)

Saturday, May 2, 2009

2009-05-05T13:05:00.000-07:00

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.

(click on image to enlarge)

Lá Bealtaine 2009

2009-05-05T12:51:00.000-07:00

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.

(click on image to enlarge)

Thursday, April 30, 2009

2009-05-05T08:45:00.000-07:00

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.

(click on image to enlarge)

Wednesday, April 29, 2009

2009-05-03T16:56:00.000-07:00

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

(click on image to enlarge)

Tuesday, April 28, 2009

2009-05-03T16:51:00.000-07:00

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

(Click on image to enlarge)

Monday, April 27, 2009

2009-05-03T16:41:00.000-07:00

I have been keeping up with my canon writing even during the very busy end-of semester time. I have been falling behind with the posting though. Bear with me and I'll be caught up soon.

I have kept the canons pretty simple, though -- just a function of the schedule...

This one is a canon in augmentation based on a rhythmic canon that I wrote a month or so back.

(click on image to enlarge.)

Sunday, April 26, 2009

2009-04-28T04:31:00.000-07:00

Using the same expedient again...

I simply chose another of my earlier rhythmic canons and selected my pitches with care.

Another matinee of Showboat...
It's good to have these working methods when free time is so scarce.

(click on image to enlarge)

Saturday April, 25 2009

2009-04-28T04:02:00.000-07:00

Very busy days...

Today I have two shows (2 & 8 pm) of Showboat (I'm playing banjo and guitar in the pit orchestra) -- so time is tight.

What I did for today's canon:

Continuing with the Liadov cantus firmus, I selected a rhythmic canon that I wrote back in March and used that for the basis of this one. I simply chose pitches that would work. It's pretty easy to proceed this way -- I'll likely do something similar tomorrow.

(click on image to enlarge)

Canon Below the Liadov Cantus Firmus

2009-04-24T08:09:00.000-07:00

This one imitates at the sixth above -- with both canonic voices below the cantus.

(click on image to enlarge)

Thursday, April 23, 2009

2009-04-24T07:48:00.000-07:00

Canon "Above & Below" the Liadov Cantus

Just as the title says -- this one imitates at a ninth below -- one voice above the C.F. and one below.
(click on image to enlarge)

Wednesday, April 22, 2009

2009-04-24T07:28:00.000-07:00

A very busy week -- nearing the end of the semester, teaching during the day and playing in a pit orchestra every evening -- I'll take an easy route and write more canons based on the Liadov cantus firmus.

This one is a canon in two voices above the C.F. The interval of imitation is the fourth above.

(click on image to enlarge)

when short in time, expand on earlier work...

2009-04-21T04:29:00.000-07:00

An elaboration on yesterday's canon using simple passing tones and suspensions.

(click on image to enlarge)

Modal Shift

2009-04-20T06:16:00.000-07:00

By changing the Liadov cantus firmus from major to phrygian, I was able to make a first-species type canon at the 7th below.

Simple -- but begging for elaboration...

(click on image to enlarge)

Sunday, April 19, 2009

2009-04-20T05:49:00.000-07:00

Another Canon on the Liadov Cantus Firmus

Actually, it is an elaboration of yesterday's canon.

(click on image to enlarge)

Saturday April 18, 2009

2009-04-20T05:43:00.000-07:00

Since this is a long weekend for me (Patriots Day), I've decided to take it easy. I'm still writing the canons but they'll all be pretty straightforward.

Here's one on a cantus firmus by Russian composer, Anatoly Liadov (1855-1914). He wrote 12 canons on it. I'll use it for a few, as well.

(click on image to enlarge)

Friday, April 17, 2009

2009-04-18T06:36:00.000-07:00

Another elaboration on the canon from Wednesday.

The effort here was to "sound different" than either Wednesday's or yesterday's canons.

(click on image to enlarge)

Elaborating Yesterday's canon at the 7th

2009-04-16T18:20:00.000-07:00

Simple elaboration by passing tones and figuration.

Compare it to yesterday's to see how I did it. There's no mystery to this process...

(click on image to enlarge)

Wednesday, April 15, 2009

2009-04-16T18:11:00.000-07:00

A "Simple Form" Canon for Tax Day

The pitches of this canon are derived from the word "tax" and my two favorite parts of the process: deductions and refunds.

Here is how ir works:

tAx DEDuCtion tAx rEFunD

Giving a theme of:

A D E D C A E F D

Today, I'm keeping it simple -- in the form of 1st species counterpoint. It imitates at the 7th below.

Tomorrow, I'll elaborate it.

(click on image to enlarge)

Tuesday, April 14, 2009

2009-04-16T04:16:00.000-07:00

Double Canon on a Hungarian Folksong

This folk tune was used by Brahms in his Opus 21 #2, Variationen über ein ungarisches Lied. He set it in alternating measures of 3/4 and 4/4. I considered that but really believe that the tune is in 7. Well, that and the fact that changing meters every measure is a pain in Finale...

Anyway, the tune is in the 1st and 3rd parts and works with only a tiny inflectional difference from Brahms' setting as a canon against itself. My added parts are in voices 2 and 4.

(click on image to enlarge)

Canon at one eighth note

2009-04-13T19:03:00.000-07:00

Similar in style and intent to yesterday's, this one is based on a minor key diatonic circle of fifths.

(click on image to enlarge)

Lessons from Mozart

2009-04-12T15:07:00.000-07:00

I was reading through some Mozart again and found a canon in part of his K. 576 Piano Sonata.

Here is the excerpt:

(click on image to enlarge)

I've decided to do likewise, basing my canon, as Mozart did, on scalewise passages and arpeggios.

(click on image to enlarge)

Another Omnibus Canon

2009-04-11T08:54:00.000-07:00

This one is a three-voice elaboration of the progression that I used yesterday.

I resolved the last chord of the omnibus differently at the cadence.

The descending bassline from yesterday's canon is incorporated into the other parts -- so this one works without the non-canonical bass.

(click on image to enlarge)

100th canon and post

2009-04-10T18:10:00.000-07:00

This canon is a straightforward version of a voice leading paradigm that theorist Victor Fell Yellin has called the omnibus.

It's an elaborate voice exchange that takes advantage of enharmonic reinterpretation.

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Thursday, April, 9 2009

2009-04-10T04:54:00.000-07:00

Octatonic Inversion

The octatonic scale, being a series of alternating half and whole steps, is perfectly suited to inversion canons.

This one is a canon at the fifth in inversion.

The one challenge I had with this one was that, since I had decided not to cross voices, I left myself with a fairly restricted range. If I were to do this again, I'd make it a canon at the 12th instead.

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Wednesday, April 8, 2009

2009-04-09T04:03:00.000-07:00

I'm working with the octatonic scale for a few days. This is a scale with many possibilities.

Today -- a simple canon at the octave.

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Elaborating yesterday's canon

2009-04-07T03:39:00.000-07:00

OK -- so I didn't elaborate it very much. I wanted to simply take yesterday's very simple voice leading paradigm and make it a bit more musical.

It's a busy week -- so I'm inclined toward keeping things simple.

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Monday, April 6, 2009

2009-04-07T03:36:00.000-07:00

Previously, I've worked with the whole tone scale in ways that avoided the overly-augmented-chord sound. Working "against the grain" so to speak.

In today's canon, I attempt the same with the octatonic (diminished) scale. As I have already demonstrated, this scale can very easily generate diminished seventh chords. This canon is written to cycle through the four major and four minor triads that exist in this scale.

The lower two voices are in an exact canon at one beat temporal distance and a minor third above.

The upper voice can either be though of as a doubling of the comes (starting a fourth above) or as an additional canonic voice that is inexact in intervals because of its position above the dux.

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Chromatic/Diminished

2009-04-05T07:33:00.000-07:00

Yesterday I mentioned that the diminished seventh chord divides the octave into four equal parts. This means that symmetrical passages involving the chromatic scale can outline the chord as well.

In the same spirit as yesterday's offering, today's canon is a demonstration of that fact.

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Saturday, April 4, 2009

2009-04-05T07:28:00.000-07:00

The diminished chord and the octatonic (or diminished) scale are both symmetrical structures.

The diminished seventh chord divides the octave into four equal parts:

E (m3) G (m3) Bb (m3) Db (m3) E
The diminished scale is made up of two diminished seventh chords.
In this case:

E G Bb Db and B D F Ab

It (the scale) divides each minor third of the chord into a half step and whole step.

Todays canon is in the spirit of J.S. Bach's Trias Harmonica -- not really an artistic canon but a demonstration of the facts outlined above.

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Friday, April 4, 2009

2009-04-04T14:53:00.000-07:00

Canon on a Chord from Bartok.

The chord in question is this:

B E F# G A D

It lays out symmetrically :

P4, M2, m2, M2, P4

I've used it as a scale for this canon in inversion because of that symmetry.

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3:1 Mensuration Canon

2009-04-02T16:47:00.001-07:00

Another mensuration canon. In this one, the upper part moves at three times the rate of the lower.

Again the scale is the ancient Greek enharmonic pentatonic.

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Canon in Inversion on the enharmonic pentatonic

2009-04-01T19:42:00.000-07:00

For this canon, I used the enharmonic pentatonic
E C B A F E

and its inversion
E G# A B D# E

to create a canon in inversion in 7/8 time.

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Canon at the fifth on a blues scale

2009-03-31T16:14:00.000-07:00

This is the scale that I used for today's canon:

C Eb E F G Bb B C

Since this scale has two equal tetrachords (m3, m2, m2) it is conducive to a canon at the fifth. However, when the range of the melody exceeds the fourth that contains that tetrachord, the intervals are no longer exact.

Rather than considering that to be a liability, I've taken advantage of it to provide a bit more variety than I would have otherwise gotten.

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Canon at the Second on the Lydian Dominant Scale

2009-03-30T17:44:00.000-07:00

The scale that I used in today's canon is derived from the overtone series.

C D E F# G A Bb C

"Lydian" refers to the #4 and "dominant," to the b7.

I made it a canon at the second so that most of the intervals of the opening phrase would be exactly replicated in the comes.

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A Canon after Mozart

2009-03-29T16:12:00.000-07:00

Today, while sight reading Mozart's Piano Sonata in c minor (K. 457), I noticed a little canon at the beginning of the coda to the first movement. It's only a few measures long and descends harmonically in thirds from I to VI to viiº7/V.

Here's an excerpt:
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I decided to pattern my entry for today after this Mozart model but to carry it on for several more measures. Mine descends in thirds fro I to VI to IV to N (the Neapolitan chord -- bII then cadences -- viiº7/V -- I 6/4 -- V7 -- I.

It is a canon at the fifteenth until measure 7 and then contracts to an octave for m. 7-8.

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Canon at the third (over a bass)

2009-03-28T16:54:00.001-07:00

Using a bass adapted from the Handel lessons for Princess Anne, I've written a canon at the third.

This type of thing is very easy --just the thing when I need a bit of a break.

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Another Canon on the Same Bass

2009-03-27T11:48:00.000-07:00

I've used the same bass (from Handel) as yesterday-- but, this one is a bit unusual.

It is a canon at the third but the rhythmic distance begins at two beats and, at the end of the 3rd measure, contracts to one beat. At the fifth measure, the pitch canon ends but rhythmic imitation at two beats preserves the unity of the piece.

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Thursday, March 26, 2009

2009-03-27T11:41:00.001-07:00

Another Canon over a Handel Bass.

Back on January 30th, I took a bass from the lessons that Handel gave to Princess Anne of England and set a canon over it.

I've done the same today with a different bass from the same lessons.

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Pushing this scale a bit farther

2009-03-25T16:54:00.000-07:00

So, I'm still using the same scale and trying to wrestle more from it.

To recap -- it is made of two major triads a tritone apart. C and F# for example.

Today I wrote a canon at the fifth. One scale (the lower) is C and F#, the other is G and C#.

A simple canon that wears its tritone on its sleeve--starting in C major and ending in F#.

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Another Take on Inversion

2009-03-24T19:12:00.000-07:00

This scale that I have been using has an interesting property. Its complement is also its inversion.

This feature inspired today's canon.

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Perpetually Descending Chorale/Canon

2009-03-23T20:07:00.000-07:00

Using a technique that I worked with way back on January 3rd, I have written another chorale/canon. This one is on the same scale I've been using for the last few days.

The title refers to the fact that this canon can descend indefinitely -- of course, depending on the range of the instruments and the limits of good taste.

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Sunday, March 22, 2009

2009-03-23T03:16:00.000-07:00

Using the same scale again, I worked with implied harmonies.

There are only two complete triads in this scale: C and F#.

I've implied G minor and C# minor using contrary motion and, in the case of the G minor, by means of the traditional resolution of a tritone.

I have included chord names below the staff to clarify my intent.

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Spring (at last)

2009-03-21T20:21:00.000-07:00

Another canon on the same scale as yesterday.

This one is at the octave.

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Major chords and tritone

2009-03-20T14:32:00.000-07:00

This canon is composed on a scale that is built on two major chords a tritone apart:

E and Bb.

The tritone divides the octave exactly in half and therefore gives a great resource for symmetry.

I start by arpeggiating the chords -- one in each part, right at the beginning to kind of stake out the territory.

It's really quite straightforward.

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Mensuration Canon on the Enharmonic Pentatonic

2009-03-19T08:57:00.000-07:00

This is a mensuration canon in which the lower part plays the first two measures worth of the upper melody in dotted quarters while the upper part moves in sixteenths.

It is in the same enharmonic pentatonic as yesterday's canon.

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Enharmonic Pentatonic

2009-03-18T18:46:00.000-07:00

Here is a canon on an ancient Greek scale.

The enharmonic has a number of interesting forms. The one I used today is its pentatonic.

It works like this:

Two tetrachords each spanning a perfect fourth (the ratio 4:3) and connected by a whole step (9:8).

Each tetrachord contains a descending Major third and a half step.

So:

C Ab G F Db C

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Dlí Port

2009-03-17T20:29:00.000-07:00

Beannachtaí na Féile Pádraig agaibh!
(Blessings of the Feast of Patrick to you)

Here is a little canon/jig for St. Patrick's day.

I wrote about half of it in my head while walking to town to do some errands. I finished up when I got home.

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a quicky...

2009-03-16T18:18:00.000-07:00

Busy day, today. Business meetings and spring cleaning.

So, I took the easy way out today. I took the rhythms from an earlier rhythmic canon and filled in pitches for a canon at the unison. I took care to make the voice crossings smooth.

That's about it.

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Rendering unto Caesar ...

2009-03-15T18:20:00.000-07:00

Since today is the Ides of March, I render unto Caesar -- a lament.

It is based on the letters of his name

C, A, E, Eb ("es" or "s" in German note names), A, and for "R" -- a rest.

Not much else to say about it.

Interesting... you can't spell Marcus Brutus using notes...

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Fanfare Canon for Pi Day 3.14

2009-03-14T16:29:00.000-07:00

In celebration of Pi Day, here is a fanfare-canon.

The rhythms and pitches are represented by the following numbers (of course):
3 1 4 1 5 9 6 5 and then 8 to enable an ending on the tonic.

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Friday, March 13, 2009

2009-03-14T05:44:00.000-07:00

I was thinking about the Balinese Kecak (also called the Monkey Chant) today and I remembered that it has canonic features as well. These are quite difficult to hear because of the complexity of the resulting texture and the number of parts.

Here is one of a number of versions on you tube:

http://www.youtube.com/watch?v=q8ZT-xzHVno&feature=related

It is a bit easier to see on my transcription (of another version -- there are many).

The canons are as follows (all based on a sixteenth note unit):

--the first three parts are based on 3, 3, 2

--parts 5 & 6 are based on 3, 3, 3, 3, 2, 2
part 4 is very closely related to these two with 3, 3, 3, 3, 4

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My canon for today is a cak in 9/8 time which results in a kind of Irish Slip Jig-Cak.

There are two canons in the cak parts:

--the first 3 parts are based on an eighth note unit -- 2, 1, 5, 1

--parts four through six are based on sixteenth notes -- 3, 7, 6

I've also included a vocal ostinato canon in four voices to create a dorian mode harmonic accompaniment. This is a canon with two voices in unison and another with two a fifth above the first. Both of these imitate at one measure distance.

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Thursday, March 12, 2009

2009-03-13T03:15:00.000-07:00

A technique from Elliot Carter that he called tempo modulation but is commonly called metric modulation.

His term makes more sense. Through a series of meter changes the tempo of a piece is made to change (either faster or slower) in a controlled manner.

In this canon, I have written a tempo modulation that speeds up. First I establish the dotted quarter note beat in 6/8 and then speed up the pulse by moving to a quarter note beat in 2/4. Then I maintain that beat rate but change the beat back to a dotted quarter.

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A different kind of retrograde.

2009-03-11T17:52:00.000-07:00

A second canon on the row from Webern's Symphony Op. 21.

For this one, I made an unusual kind of retrograde canon.

I made the upper part from O and I and the lower from RI and R. All this seems to create the the retrograde in the lower part. However, the retrograde in time points ends up being quite a different thing. Here the retrograde is 4 measures lone, for instance. (I repeated the first for bars of the upper part...)

So, even though I used the exact numerical retrograde, I have an entirely different type of musical retrograde.

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Canon after Webern

2009-03-10T16:52:00.000-07:00

Using time points again, I made this rhythmic canon in three parts from the row that Webern used in his Symphony op.21.

0, 3, 2, 1, 5, 4, 10, 11, 7, 8, 9, 6

Of course, Webern used the row as pitch classes, not time points but he was fond of canon...

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Canon on two different rhythmic techniques

2009-03-09T17:22:00.000-07:00

I'm trying to find less common ways to make canons. In this one I've used another rhythmic technique from the arsenal of Milton Babbitt. This one is the time-point system.

Essentially, the system sets up a series of time points upon which the rhythm of a piece is built. A series of numbers is again used for the rhythmic materials but this series refers to points on a time continuum. I've chosen six points (0 to 5) in each measure of 6/8 time.

The numbers are then used as they occur in the series. Babbitt has used this technique with 12 time points as a parallel to the twelve pitches of the chromatic scale. It enables him to perform the same operations on rhythm as he does on pitch.

Again, I'm not trying to imitate Babbit's sound or technique here -- just borrowing an interesting idea to use in my own way...

I've decided to make my canon from both the time-point system and duration rows (introduced a few days back).

I've chosen a rhythmic row 1, 5, 3, 4, 2 and turned it into a palindrome by using O. I, RI, R:

15342, 51324, 42315, 24351

I treated this series a duration row in the upper part and in time-points for the lower part.

So, the same numbers form both parts but they are quite different. I have a feeling that this type of idea may be fertile ground for me when I reintroduce pitch to my canons in a few days.

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Another "hidden" canon

2009-03-08T16:53:00.000-07:00

This one uses the canon from February 26 and hides it in a manner similar to yesterday's entry.

I altered the same notes in both parts but I altered them in different ways. Ithe measure one I added grace notes before the second half note but in the lower part (meas. 2), I replaced that note with a rest.

In places that I altered notes by dividing them in one part, i divided the other with different note values.

I find this to be an interesting way of working and may explore it further.

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Saturday, March 7, 2009

2009-03-08T16:26:00.000-07:00

Hidden Canon.

I made this canon by taking the same duration row (in fact, the same canon) as February 28th and disguising it in the following manner:

In the upper part, I changed every third number in the row into a rest of the same duration. In the lower part, I articulated each unit of every third number.

Fairly simple, really but it hides the fact that it is still a mensuration canon.

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Friday, March 6, 2009

2009-03-07T02:13:00.000-08:00

Tomorrow, Maggi and I are giving a workshop on folk hymns to the Merrimack Valley Chapter of the American Guild of Organists. We'll be presenting hymns from the shape note, Shaker, and Celtic traditions and talking about issues of style and performance practice.

I always feel compelled to try to combine my various disparate projects together in some way.

So....

Here's a canon on a old Shaker hymn that we will be doing in tomorrow's workshop.. In the spirit of the title (and first line), I've kept it simple.

It's a canon at the unison -- the pentatonic nature of the tune (and the general strength of the melody) makes even the mild dissonance of A against G sound good and traditional.

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palindrome, retrograde and paradox

2009-03-05T03:35:00.000-08:00

In this canon, I use the same duration row as I did for Monday's. This time I used only the original (O) and inversion (I) for the upper part. In this way I could make a canon in retrograde by using RI followed by R in the lower part.

If I had kept the palindromic structure of Monday's canon, I wouldn't have been able to make a canon in retrograde...or would I?

Since a palindrome is its own retrograde, perhaps every simple canon on a palindrome is also a retrograde canon.

This puts me in mind of Messiaen's non-retrogradable rhythms. These are simply rhythms that read the same left to right as they do right to left -- in other words palindromes. They are non-retrogradable because they are their own retrograde.

A nice little paradox...

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Cage's square root form

2009-03-04T19:16:00.000-08:00

Thinking about square root day, yesterday, I remembered John Cage's square root form.

It works like this:

The overall form is the short form (phrase level) squared. So, to use today's canon as an example. Five measure opening phrase -- 25 measure overall form.

But the real key is that the features that are present in the short form are also exemplified by the large form and each of the other parts of that form.

Again, this canon is 5 squared or 25 measures long. The overall form divides 10 + 10 + 5.

Each 5 measure section divides 2 + 2 + 1 -- This structural rhythm is also stated twice at the beginning by the dux and then, of course by each comes.

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