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)

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