Saturday, February 28, 2009

Veritas in numeris continetur.

I'm still working on the rhythmic canon sub-project.

I have been finding the act of writing these to be a bit too easy to retain my interest for very long. So, I have decided to explore the ideas of some composer/theorists -- especially those of the latter half of the last century.

First up -- Milton Babbitt, who has worked with several interesting rhythmic techniques.

[I should add a disclaimer here:

I am not claiming any special insight into Babbitt's work (or that of anyone else from whom I garner motivation or inspiration, for that matter.) I simply borrow some ideas to create my own methodology and then proceed. If you want cogent insight into Babbitt's ideas, ask him. I'm sure he's a more reliable source.]

A rhythmic technique that Babbitt used in some of his early works (late 1940s through the '50s) is that of duration rows. The use of these rows parallels the techniques of pitch serialization. The simplest way to look at this isthat the common manipulations of a tone row could be performed on a duration row as well.

These manipulations are retrograde (R), inversion (I) and retrograde inversion (RI). This last is simply the combination of the other two.

R is the backward statement of the original series (O). This poses no problem of comprehension in either the realm of pitch or rhythm. Inversion of melodic phrases is also easy -- where the original goes up, I goes down by the same amount and vice versa. For example, the interval C-F inverts to C-G.

The inversion of rhythm, on the other hand, is counterintuitive. How do you turn a rhythm upside-down?

The answer is found in numbers. Babbitt has long been an exponent of set theory as a way of understanding and manipulating musical materials. Basic to this discipline is an integer model of pitch.

We assign integers 0 - 11 to the pitch classes (here starting with "C."):

0 --C
1 --C#
2 --D
3 --D#
4 --E
5 --F
6 --F#
7 --G
8 --G#
9 --A
10 --A#
11 --B

This model allows us to manipulate pitches and intervals with simple arithmetic operations. To the point: Once the starting pitch is established, inversion of a pitch class can bee seen as its complement, modulo 12.

11 -- 1
10 -- 2
9 -- 3
8 -- 4
7 -- 5
6 -- 6
5 -- 7
4 -- 8
3 -- 9
2 -- 10
1 -- 11

This somewhat simplified (for the sake of this blog entry) definition shows that, as I stated earlier, C-F (5) inverts to C-G (7, the complement, mod 12 of 5)

Or, in terms of simple arithmetic:

to find the inversion of an interval, subtract it from 12 (the octave).

So: 12 - 4 = 8 Again, the inversion of G-F is C-G.

Now back to rhythms. For a duration row, we simply assign integers to the durations using the most basic unit of measure. For example, in today's canon, the duration row is 2, 3, 4, 1 [This row is found in Babbitt's Composition for Four Instruments (1948), but it is used in quite a different way than I use it here.] The lower part uses the unit of an eighth note. Therefore, the row is expressed by quarter, dotted quarter, half, eighth. In this mensuration canon, the upper voice uses a sixteenth note unit and the row is eighth, dotted eighth, quarter, sixteenth.

To find the inversion of this duration row, we simply subtract each number from the largest duration plus one -- in this case, five.

5 - 2 = 3
5 - 3 = 2
5 - 4 = 1
5 - 1 = 4

The inversion of the duration row 2, 3, 4, 1 is 3, 2, 1, 4.

I composed the canon in this way:

O -- RI -- I -- R

or 2 3 4 1, 4 1 2 3, 3 2 1 4, 1 4 3 2

As you can see, this makes a rhythmic palindrome.

Since 1 + 2 + 3 + 4 = 10, I chose to write the canon in 5/8 to allow the row to occupy one measure of the upper part and two measures of the lower part.

Since the upper part moves twice as quickly and therefore, ends in half the time, I repeat it but in the second repeat I use shore notes followed by rests. The overall interval between iterations remains the same.

I've included some annotations on the score to help explicate the row usage.

(click on image to enlarge)

Friday, February 27, 2009

Rhythmic Canons in Compound Meter.

Continuing my rhythmic sub-project...

I used hocket again and much syncopation in the second one.

(click on image to enlarge)

Thursday, February 26, 2009

Rhythmic Canon Project #6

My self-imposed limitations for this one were syncopations and subdivision of the beat.

Simple canon with the comes starting one measure after the dux.

(click on image to enlarge)

Wednesday, February 25, 2009

Rhythmic Canon in Retrograde

The lower part is exactly the reverse of the upper.

The limitations were simple meter, syncopation, division of the beat.
I used hocket as a "main feature" of this one.

(click on image to enlarge)

Tuesday, February 24, 2009

More Mensuration

In this one the lower part is twice as slow and begins a measure later. Mensuration canon is quite useful when you want to demonstrate syncopations.

(click on image to enlarge)

Monday, February 23, 2009

Mensuration Canon

Another rhythmic canon.

The limitation on this one was to use subdivision of the beat but no ties or anything too complex. I made it a mensuration canon with the lower part moving twice as slowly as the upper.

(Click on image to enlarge)

Sunday, February 22, 2009

Rhythmic Canon Project

For the next several entries, I'm going to work with rhythmic canons.

Because of the lack of pitch and harmony, rhythmic contrast takes on the utmost importance when it comes to keeping the parts distinct. This is more difficult when the variety of note lengths is limited.

That's exactly what my challenge will be. For another project in which I am currently engaged, I'm writing rhythmic studies for ear training classes to sight read. Since the level will need to progress from easy to difficult, these will necessarily have fairly strict limitations in rhythmic values.

My task is to make enough contrast within these limitations.

Simple canons are fairly easy to manage when working solely in rhythm. Minor complexities are added by the use of hocket (having one part play in the rests of the other in an interlocking manner). Other difficulties arise when working in mensuration canon or retrograde.

Today's canon #1 is a simple canon with no note values shorter than a beat. This was very simple to achieve by just being certain to make the parts contrast each other within a measure.

Canon #2 has some hocketing, mostly in the second half and features rhythmic values that divide the beat in half.

(Click on image to enlarge.)

Saturday, February 21, 2009

Mode wrap-up (for now) and Locrian

To wrap up this segment on modes, I worked with the locrian.

Well...there certainly are issues here. Because there is no perfect fifth above the final (tonic), composers from the middle ages through the 19th century avoided it. Fux didn't cover it for the same reason: no P5 above the tonic = no way to have the fugal answer enter on the dominant.

The standard rotation of the scale to the fifth that I've been using all week to explore the other modes is useless here:


There aren't ANY intervals above the B that are matched above the F:

B-C = m2
F-G = M2

B-D = m3
F-A = M3

B-E = P5
F-B = A4


So -- though it may be interesting to try a canon at the tritone in locrian (and I may some day...), it doesn't fit with the method that I'm using (based on Fux) for the modal canon's of the last several days.

If we take the first, third and fifth degrees of the locrian, we get a diminished triad. Because this chord is made up of two stacked minor thirds, I tried that interval as the interval of imitation.

By aligning the pitches of the scale in thirds, I got this arrangement:

Tonic = B, third = D (half steps in bold)


Better results here. These intervals are the same above both the tonic (B) and third (D):

B - D = m3
D - F = m3

D - G = P4
B - E = P4

and below both:

D - C = M2
B - A = M2

So, unlike the other modes, there are no contiguous groups of notes/intervals that are usable under the restrictions that are imposed but we do have a "gapped" group of pitches:

(Tonic and third in bold)

A B ____ D E
C D ____ F G

An added benefit of this arrangement is that, if I emphasize the minor third (B - D) in the dux, the comes will answer with D - F and we'll hear the tritone B-F in close proximity.

(click on image to enlarge)


By the way, this is certainly NOT traditional. I tried to make it sound as traditional as possible but the composers of earlier times would not have written anything like this.

Friday, February 20, 2009

Dorian Mode

The last of the modes that Fux used. (More about this tomorrow.)

The dorian is an easy mode to use in many ways. I believe I have pointed out before that it is symmetrical around the tonic (in this case D):
(half steps in bold)


Like the ionian (major) and phrygian, it is built from two similar tetrachords:


This feature is exploitable in the construction of a subject suitable to dorian and no other mode, as prescribed by Fux.

Here is the layout (tonic D, dominant A):


The subject can utilize a fifth above tonic and dominant and a whole step below.

(click on image to enlarge)

Thursday, February 19, 2009

My 50th canon and post!

The next mode is the Aeolian. Here is the interval layout (half steps in bold):


With a tonic on A and dominant on E, you can see that the opening phrase would need to be below these two pitches.

(click on image to enlarge)

Wednesday, February 18, 2009

Ionian Canon at the Fifth

Still more work with the Fuxian ideal of modal subject composing. This one in Ionian (major).

This layout of whole and half steps shows that the Ionian nature is best expressed (for this type of thing anyway) in a passage above the tonic and dominant. (half steps in bold)


(click on image to enlarge)

Tuesday, February 17, 2009


Similar to the last few...

The layout of half and whole steps makes writing below the tonic (G) and dominant (D) more "mixolydian."


(click on image to enlarge)

Monday, February 16, 2009

Following Fux's Dictum

Today's offering is in the same spirit and with the same intent as yesterday's -- exploring the relationship between Fux's fugue subject dictum (to write something that suits that mode and no other) and canon writing.

Today, the phrygian.

consider this alignment:


The tonic is E and the dominant is B. there is a half step (marked in bold) right above each and the rest of the pitches listed are in whole steps. You can see that there are possibilities of exact imitation that encompass the range of a major sixth that could emphasize the half step placement that is so characteristic of this mode.

That's what I did. It is in the style of Fux.

(click on image to enlarge)

Sunday, February 15, 2009

Today, while riding the train to a gig in Boston playing the banjo for Gershwin's Rhapsody in Blue, I was preparing for a class in fugue that I'll be teaching tomorrow. (There's a sentence you don't hear every day!) I was reading what Fux had to say about modal fugue subjects and decided to apply some of those principles to canon writing.

Essentially, Fux says to construct a subject that suits the mode at hand and no other based on the spacing of the half steps within that mode. Since the fugue imitates at the dominant (fifth), that must be considered as well.

Here are the modes with the half steps in bold:

Ionian: C D E F G A B C

Dorian: D E F G A B C D

Phrygian: E F G A B C D E

Lydian: F G A B C D E F

Mixolydian: G A B C D E F G

Aeolian: A B C D E F G A

For today's canon, I used the lydian mode. I have chosen to make it a canon at the fifth so that I can relate what I do to Fux's fugal precepts as much as possible.

Because there is a half step below both the tonic and dominant in this mode, I've chosen to have the opening of the melody proceed down from the tonic in the dux and down from the dominant in the comes. This guarantees that the characteristic raised 4th of the lydian will be featured early in the piece.

(click on image to enlarge)

Saturday, February 14, 2009

My Canonic Valentine

For Valentine's Day:

A canon at the octave for tenor sax and trumpet over the bassline for "My Funny Valentine."

It was fairly easy to put together -- somewhat like improvising over the chord progression but with an eye (ear) toward having it work for both "this measure and the next." The only necessary adjustments were some chromatic inflections to suit the chords.

Rather fun to write, actually.

(click on image to enlarge)

Friday, February 13, 2009

Irrational Fear of Canons at the 13th

Anyone suffering from Triskaidekaphobia should skip today's entry and come back tomorrow.

In honor of Friday the 13th, I've written a canon at the 13th. It's scored for two Theremins, the traditional instrument of spooky movie scores.

For those who still believe that foolishness about the tritone being the "devil's interval," I've included copious diminished fifths and augmented fourths.

(click on image to enlarge)

Thursday, February 12, 2009

Canon as Accompaniment

This is a canon at the octave in the two bassoon parts in the whole tone scale . The canon acts as an accompaniment to the lyrical melody of the oboe which is in the other whole tone scale.

If you have been following the last few days' entries, you will have noticed that I find the technique of using two different whole tone scales more congenial than using only one.

I should probably address that in technical terms one of these days.

Here it is--sketched out on the commuter train this morning and massaged into its current form at home this evening:

(click on image to enlarge)

Wednesday, February 11, 2009

Chorale canon for low voices

This is another that I wrote quickly. The bass and tenor are in canon both in the same whole tone scale and the alto is a free voice in the other.

I tried to give the impression of moving quickly through keys. I start in C and end in Bb with some others in between.

(click on image to enlarge)

Tuesday, February 10, 2009

Balance and Brevity

Short and to the point on this very busy day.

Here is another whole tone canon. In this one, I simply wrote without too much pre-planning. It's good to balance the cerebral with a freer approach now and then.

Well... off to do some teaching...

(click on image to enlarge)

Monday, February 9, 2009

Escaping the "Whole -Tone Effect" while using the Whole Tone Scale

This one took a bit more effort to work out. The intent was to "go tonal" as much as possible using the whole tone scale as the only melodic resource. By making this a double canon and using the two different whole tone scales -- one in each canon--I was able to essentially alternate between I and V. This is impossible to achieve using only a single WT scale due to the lack of perfect 4ths and 5ths.

I used only a portion of each scale and employed the contrapuntal device of voice exchange to facilitate the common practice style voice leading.

A voice exchange is simply having two parts exchange pitch classes. For example, the simplest form is to exchange two notes. In a C chord, one voice can move from C to E and the other from E to C.

Another form is to have one voice pass from C to D to E while the other does E-D-C, both treating D as a passing tone.

The next level would be to again use the C-D-E and E-D-C exchange but to have the D harmonized by its own chord. This last one is certainly not the most complex form possible, it is simply the one I chose to use here.

There are voice crossings which enable some other melodic touches that would be impossible with a single WT scale. I've indicated these in the score with dotted lines. I use these to improve the melody and the bass line. The crossings create an aural illusion of a continuous line that is actually passing from one player to another.

I first wrote the parts within the repeat signs and then took notes away to create the more spare opening section.

The alert, diligent or theory geeks among you will spot the fact the underlying structure of the piece if a voice exchange, too, as exemplified by the stripped down opening measures.

(click on image to enlarge)

Sunday, February 8, 2009

Whole Tone/ Twelve Tone

Today's canon is another approach to solving the "interval problem" in a whole tone piece.

In the WT scale, the intervals M2, M3, tritone, m6, m7 and octave occur on every scale degree. This is a feature that manifests both melodically and harmonically, as I have pointed out before.

This time I decided to use one whole tone scale in the upper part and the other in the lower part. This creates the unusual circumstance of having different intervals available harmonically than melodically.

Now, those intervals that don't occur at all melodically are the ONLY ones available harmonically -- m2, m3, P4, P5, M6, M7 (no 8ves!). This combination of scales also makes all 12 chromatic pitches available -- 6 in each part.

So, we get the melodic effect of whole tones scales but a very different harmonic effect.

Though I still focused on consonances, harmonically, I made no effort to conform to "common practice" or traditional tonality at all. The rapid turnover of pitch classes contributes to the overall atonal Gestalt.

(click on image to enlarge)

Saturday, February 7, 2009

Further Harmonic Adventures in Whole Tone

In an attempt to squeeze more variety from the whole tone scale than I originally thought possibly, I tried something different today.

This is precisely the reason I chose to do this particular canon/blog project -- to push myself into little experiments here and there.

Here's the deal:

Since the whole tone scale has no perfect fifths or perfect fourths in it, I realized that I could solve that problem by the addition of another part that wasn't restricted to the scale. It was a short cognitive leap to realizing that all the pitches that I needed were in "the other whole tone scale." If I kept each individual voice in just one of the two scales, I would increase my resources but still maintain the melodic features of WT scale.

My first canon of this type is a brief fanfare that moves through an implied circle of fifths.

Interestingly, when all voices are in the same WT scale, there are a limited number of intervals and the same intervals are used both melodically and harmonically. In this case, by the technique described above, I have greatly expanded the number of available harmonic intervals while keeping the limited melodic, interval resources peculiar to the WT scale.

I used the WT scale C, D, E, F#, G#, Bb in the two trumpet parts and the WT scale 1/2 step higher in the two trombone parts.
N.B. I did try to create as "common-practice" a sound as possible in this canon.

(click on image to enlarge)

Friday, February 6, 2009

Harmonic Progression and the Whole-Tone Scale

Today I endeavored to find a way to create a sense of chord progression in a whole-tone environment. I also hoped to avoid too much of the sound of augmented triads.

To illuminate the problem, we need to consider the harmonic resources of the scale:

--There are no major or minor triads.

--There are augmented triads on every scale degree.

--There are no perfect fourths or fifths or minor thirds.

My solution:

--Imply Major triads by using major third and minor sixth intervals (They should sound like the root and third of a major chord sans fifth.)

--Use much contrary motion. Parallel motion tends to allow those major thirds to aurally "stack up" into augmented triads and other structures that could undermine the sense of progression.

--Keep it short. It's easy to run out of resources in a whole tone scale.

--Imply VII to I as a cadence form at the end.

I have included chord symbols below the staff as a kind of analysis.

(click on image to enlarge.)

Thursday, February 5, 2009

Elaborating the WT canon.

This is an elaboration of yesterday's canon.

As I have done before, I simply added some passing and neighboring tones and a few suspensions. Of course, I added some rhythmic interest to make it more than just a species exercise.

There really is no harmonic progression per se. It still sounds like a static harmony -- though much more interesting than Der übermassige Dreiklang...

(click on image to enlarge)

Wednesday, February 4 2009 -- more whole tones...

Continuing with my exploration of canons in the whole-tone scale, I made a quasi-Fuxian, 1st species canon.

For this one, I did it the "hard way" --rather than working out the canon a few notes at a time, I wrote the whole melody (cantus firmus). I based it on a Fux cantus but set it in whole tone which, of course, Fux, himself, never did.

Then I set about trying canonic possibilities. This particular way of working doesn't often yield success. I fully expected to have to tweak the cantus to make it work --but, I was lucky. It took only a few moments to line the parts up in a way that gave me all consonant intervals and no voice leading faults.

The fact that almost all the intervals are M10ths (M3rds+8ve) and minor sixths is a peculiar (and well-documented) feature of whole tone scales. I'll discuss this more in a day or so.

The octave in the penultimate measure results from my compunction to end with contrary motion. I ended the canon and move in opposite directions to close on the 10th.

(click on image to enlarge)

Tuesday, February 3, 2009

Der übermassige Drieklang

As I've mentioned before, many symmetrical musical structures are rich in canonic possibilities. For the next few days, I'll explore one of these structures -- the whole tone scale.

This scale, to state the obvious, is made up of all whole steps. Its extremely symmetrical nature makes it easy to have exact imitation and exact inversions -- on any scale degree -- without leaving the scale.

However, this nature giveth and it taketh away. What we gain in imitative facility, we lose in intervallic variety. There are plenty of whole steps, major thirds, tritones, minor sixths and minor sevenths. In fact each of these intervals exists on each scale degree but there are absolutely no half steps, minor thirds perfect fourths, perfect fifths, major sixths or major sevenths. Not just a paucity of these intervals -- none, period.

It is easy to create harmonies of augmented triads but very difficult to imply much else. There are no major or minor triads, at all and progression by fifths is impossible.

So, to today's canon...a puzzle canon.

(click on image to enlarge)

This is a strange little canon patterned after Bach's Trias Harmonica but using the whole-tone scale instead of the major scale. Instead of the major triad, it generates the augmented triad. Since Bach, a German, named his canon in Latin, I feel compelled to name mine in German: Der übermassige Drieklang.

This is offered in the same spirit as Bach's little canon -- a simple demonstration of contrapuntal/harmonic facts about triads and traditional voice leading.

Here is the solution:
(click on image to enlarge)

Generating a gently fluctuating augmented triad in a whole tone environment is no great feat. Tomorrow, and for the next few days we'll see what else I can squeeze out of this scale.

Monday, February 2, 2009

Invertible Counterpoint at the Tenth

So, I'm two days into my second month of writing a canon every day and it has yet to become a chore.

Perhaps this makes me a theory geek. (Maggi says "Tell me something I don't already know!")

Today's entry is a canon in invertible counterpoint at the tenth. "Invertible" means that either part can be the bass. "At the tenth" refers to the relationship between the two parts as they are inverted.

This has implications for the manner in which you write the parts. Inversion won't necessarily work simply because the original orientation does. If you don't take into consideration, both orientations, bad voice leading or dissonance treatment may occur. Inversion at the tenth has more strictures than the other common inversions --octave, twelfth (5th plus an 8ve), and fifteenth (two 8ves).

The fun part about this particular type of inversion is that you can convert a two-voice inversion at the tenth into a three-part texture by doubling one of the parts at the tenth.

In my canon, the original orientation is stated first. Then I double the lower part a tenth higher so that the outer parts move in parallel (measures 9-16). Finally, I invert the parts so that the original upper part is now in the bass (a tenth lower). Instead of stating the inversion in two-parts, I double the bass at the tenth above as well here which gives a satisfactory ending.

(click on image to enlarge)

Sunday, February 1, 2009

Perpetual Canon

Welcome to the one-month anniversary post for this blog.

I'm still keeping up the pace of one canon per day.

Today, I wrote a canon on the octatonic scale (alternating whole and half steps). The inherent symmetry of this scale is felicitous for canon writing -- this is something I will certainly explore again.

I stuck to traditional consonance and dissonance usage but the chord progression is certainly not "common practice."

The canon ends a half step higher than it begins. At this point the players should play it again a half step higher, proceeding in the same way at the end of each repeat. Thus, if the ranges of the instruments would allow, the canon could go on to infinity.

That would make his the longest piece I have written so far.

(click on image to enlarge)