Zach Grove

12/3/08

Chaos Under Control

Dr. Frey

For five bongo and talking drums

-All seven performers will have a die

-Five will hold a drum at all times, rolling the die to determine what drum part to play

-Sides of the die correspond to notated rhythms as follows:

Roll Action

-The composition will begin with five performers rolling their dice. A beat will then come in alone for four measures—all performers will come in at the conductor’s cue on the fifth measure.

-After playing (and repeating) their parts, performers will roll again, switching drums if necessary. Players begin their next parts immediately after rolling, but in line with the drum beat. Concentration is a point of emphasis! As performers begin their parts at different points, things may become busy. It is ideal to maintain your part as written, but it is crucial that you always keep on playing.

-This cycle of perpetual rolling and re-entering the composition as soon as possible will continue for 3 minutes. Its end will be signaled by a bell. Hearing the bell, performers will finish their line (without repeating) to end the piece.

Before we drum out some thought-provoking jams, one final word from the

Frog…

“With a 1/6 chance of rolling a switch-necessitating 6 on the die, x 5 players, there should be a good amount of drum giving and taking. Nonetheless, there will always be two players not performing. While waiting for that promising 6 roll, use these intervals as opportunities for brief listening and observation. Croak.”

On what levels in this performance do you hear self-similarity? Is there any point at which the rhythms sound predictable to you?

How is the music’s final result similar to that of the Sierpinski Gasket, another iteration of dice-rolling? Is it comparable?

What characteristics of your performance, if any, were reminiscent of the music you might hear on the radio or your ipod?

Reflect on the sound of dice: can it contribute aesthetically to the music?

How does the meter/beat evolve?

Remember:

-Roll a 1, play the 1 part, etc. through 4.

-5 is improvise

-6 is give your drum.

-Roll after playing (and repeating!) each line, and come back in on the beat as soon as you can. Make observations when you relinquish your drum.

-We all start after the 4 count on the drum beat and play ‘til the bell tolls.

It is important to realize that the example we just performed is NOT chaotic music…

…not strictly. Peak and Frame emphasize and reiterate one of the broadest concepts of Chaos Under Control to begin Chapter Six, revisiting our beginning definition of chaos:

“Deterministic chaos is very different from random occurrence, where even short-term prediction is impossible. Deterministic chaos…is filled with order. The challenge is to detect that order and, once it is detected, to harness it for prediction and control” [1].

“The Omnipotent Frog,” then, is more appropriately labeled as an aleatoric piece. Aleatoric music is “music in which some element of the composition is left to chance, and/or some primary element of a composed work’s realization is left to the determination of its performers. In fact, ‘alea’ translates literally as ‘dice’ in Latin. The term is most often associated with procedures in which the chance element involves a relatively limited number of possibilities…” [2].

Remember, we define chaos as “irregular output from a deterministic source”—that is, that the chaos we’re concerned with is, though at times infinitely complex, determined completely by its past.

So how can we reconcile our music as chaotic or otherwise using these definitions?

A case can be made that this music, like other so-called “chaos” composition, is essentially chance or aleatoric. There are, however, qualities of “The Omnipotent Frog” that are undeniably chaotic, even by the narrow definition we’ve dealt with thus far.

While our method is founded on randomness, so is that of the Sierpinski Gasket—in both cases, initial conditions restrict chance. Ours has an uncountably infinite number of possibilities, though, whereas the Chaos game

always yields

the same

result.

One reason for our straying from determinism is the incorporation of choice (the 5 die.)

Another is offered by Peak and Frame, who comment on music’s unique, inherent quality of ephemerality. “Ephemeral” here is synonymous with “episodic” or “transitory.” They point out that in visual art, used to depict the Sierpinski fractal, we see an impression of the whole creation at once. [1]

A child could recognize the underlying pattern—ever-dividing triangles.

Music, by contrast, presents the “whole” over the duration of its playing. Identifying specific rules demands an attentive ear and an acute memory.

Our example composition displays some self-similarity in its repetition and rhythmic symmetry. Vague and inconsistent, the nature of our performance is best classified as a “natural fractal” (its parts reminiscent of the whole) rather than of exact, mathematical self-similarity.

The Butterfly Effect is also vaguely present: a nuance on a drum from one performer may affect when another performer reenters the performance, affecting when another looks up to roll his die….and the whole mood of the piece changes.

Despite these and other parallels to fractal characteristics and deterministic chaos, “The Omnipotent Frog” does not progress solely from its past. Random occurrence is commonly mistaken among ambitious musicians to be a staple of chaos.

Chapter Six points out a few different, deterministically chaotic compositions following from time series. Peak and Frame suggest listening for correlations in note strings generating from:

-Uniform white noise

-Brownian noise

-1/f noise

“Set a range of note durations (for example, whole, half, quarter, eighth, sixteenth), and a range of tones.Then use a random number generator to select the duration and tone of each note in sequence” [3].

http://classes.yale.edu/fractals/Panorama/Music/OneOverFMusic/WhiteScaling.html

“Strings of notes originating from uniform white noise tend to have so little correlation—so little organization—that such sounds quickly become irritating” [1].

Brownian noise is generated with a random number generator that, instead of producing tones and durations, produces changes in such tones and durations. [3]

Intensity= some constant/ f^2

http://classes.yale.edu/fractals/Panorama/Music/OneOverFMusic/BrownianScaling.html

“A Brownian motion time series is filled with correlation; its ‘music’ wanders up and down like someone playing scales. These repetitive meanderings are too boring to hold our attention for any length of time” [1].

Intensity= a constant/ f

http://classes.yale.edu/fractals/Panorama/Music/OneOverFMusic/OneOverFScaling.html

“The intermediate correlations of 1/f noise actually harbor audio interest…somehow (balancing) familiarity with surprise, regularity and novelty” [1].

Ahh…this porridge is just right!... Almost.

http://members.tripod.com/~paulwhalley/

Small sites like these illustrate the interest chaos theory and fractal mathematics has instilled in the general public. Everyday people are learning about and readily applying them in their own experiments with music.

http://www.youtube.com/watch?v=gzXoKBqFlZQ

Following our performance, all the examples we’ve listened to have been created using algorithms. These songs (though some people argue that these sounds are not appropriately called music at all) are extensions of the mathematics of fractals and chaos. They are deliberate.

Perhaps more interesting, though, is the study of chaos as an element of music universally, non-intentionally, indeed even in music long before the invention of Calculus.

In Chapter Six, Peak and Frame articulate one of the more fascinating aspects of chaos and determinism: implications that seem to explain why certain stimuli are appealing to humans [1].

When we use spectral analysis—a method useful for deciphering the underlying order of sound—on classical music pieces, “often the resulting spectra vary with frequency like 1/f” [1].

Why exactly does 1/f noise appeal to us?

As we explored, people like the “perfect medium” of surprise offered by 1/f. Chaos Under Control delves into other possibilities, too. Among them:

“There is speculation about a mechanistic, physiological reason why 1/f noise is more pleasing. (Mandelbrot says) that noise at the point of sensory reception—the eyes, ears, and fingertips—is roughly white, whereas the closer it gets to the brain, the more closely it resembles 1/f noise. In other words, the nervous system seems to act as a filter to remove any part of the signal that is not 1/f. Voss speculates that this signal is more pleasing than others because it resembles the 1/f of our sensory experience” [1].

1] Chaos Under Control: The Art and Science of Complexity

Peak and Frame

2] Wikipedia– Algorithmic Composition.

Wikipedia.org/wiki/Algorithmic_composition.

3] Yale.edu– 1/f Aspects of Music

Classes.yale.edu/fractals/Panorama/Music/oneoverfmusic.html