Chaotic Behavior of the Double Pendulum as Applied to Music Composition

Chaotic Behavior of the Double Pendulum as Applied to Music Composition

Prof. Bruno Degazio

Sheridan College, School of Animation

Oakville, Ontario, Canada

degazio@sheridanc.on.ca

ABSTRACT: 

The double pendulum is perhaps the simplest example of a physical system that exhibits chaotic behavior . This paper describes a computer simulation of the Double Pendulum with friction. The friction component complicates the equations considerably but allows for a natural evolution of the pendulum motion.  The non-realistic extension of friction to negative numbers allows for the possibility of a naturally increasing chaotic  dynamism in the motion of the two pendula. The specific applications of the pendula’s dynamic parameters to musical composition are discussed as applied to the author’s computer music composition environment, The Transformation Engine. 

 MUSICAL ASPECTS OF PENDULUM MOTION

1. Simple Pendulum 

a. Musical pulse

The regular motion of a simple pendulum is musically equivalent to the “beat”. Because pendular motion is inherently binary or duple (due to the back-and-forth nature of the swinging) there is also a larger grouping which is analogous to musical duple metre, i.e.  2/4 or 2/2.  

b. Sine Wave

Pendular motion has also been important in the study of musical sound because it is a model for the simplest form of periodic motion, the sine wave. For this reason it has been a preliminary model in the study of musical instrument acoustics since the inception of the field in the 18th century. 

2. Double Pendulum

The double pendulum is a simple extension of the simple pendulum system which exhibits surprisingly complex behavior. A double pendulum system can be easily created by simply hanging a second pendulum from the end of a simple pendulum as shown in figure 1. 

The basic parameters of the physical system are: 

Inner (Upper) Pendulum - Length L1

- Mass M1

- Damping D1 

Outer (Lower) Pendulum - Length L2

  - Mass M2

  - Damping D2 

Gravity - G = 9.81

When set in motion, this system produces the following dynamically changing parameters: 

Inner Pendulum - Position 1 (angle measured from the vertical)

- Velocity 1 (change of Position 1 per unit time)

Outer Pendulum - Position 2

- Velocity 2

MUSICAL INTEREST

Chaotic Dynamics have a “natural” quality 

Blend of predictability and unpredictability 

Long-term Dynamic unfolding due to positive or negative Friction

This produces a pleasing sense of increasing or decreasing intensity, depending on whether the friction (damping) parameter is set to a negative (increasing intensity) or positive (decreasing intensity) value.