ï~~The Application of Differential Equations to the
Modelling of Musical Change
J L Leach & J P Fitch,
The Media Technology Research Centre & School of Mathematical Sciences
University of Bath, Bath, BA2 7AY, United Kingdom
E-mail: jll@maths.bath.ac.uk, jpff@maths.bath.ac.uk
ABSTRACT: In this paper we examine differential equations as a means of creating likeable temporal forms,
since differential equations are often used as models of natural processes. To account for the great diversity
in music composed to the present day, the use of differential equations must remain generic. The method
chosen here was to use many interacting units in a network. Each unit was governed by a set of differential
equations describing a relaxation oscillator, which depending upon the particular choice of parameter values
could exhibit a steady state or oscillatory response.
Introduction:
In its most tangible form music can be defined as patterns of sound changes organised temporally such that
together, they affect the emotions in a certain way. Here, we consider music which does not include other
sound phenomena (such as words, language and certain specific sounds), and hence can be described as a
"closed system". This is a useful distinction because there is a large body of music that contains no references
to the outside world. It is this property that separates it from other art forms, such as literature and fine art
(non abstract) painting (for further discussion see Meyer,L.B.). We can deduce from this that whatever it is
that affects the human emotions must be internal to the music.
The way in which music affects the emotions is not easily described. It has often been said that music
can convey happiness, sadness, etc. (Sloboda,J.A.), and also has the potential to excite or relax the listener.
However, distinct from these emotive forces is music's ability to invoke pleasure. Whatever the particular
emotion conveyed by a good piece of music, it is agreed that people enjoy the experience of listening to it.
The important word here is "good". What is it about a piece of music that makes it good? This question
already presupposes that we have an exact definition of the word "music" - which we do not. To be more
objective and at the same time more general, we ask the question "What is it about certain sound sequences
(without external reference) that make them pleasing to listen to?"
We know that the factors determining whether or not a particular sound sequence will be liked cannot be
explained merely by individual preference. Otherwise, we might expect to find sizeable populations of people
who like random noise just as there exist populations of people who like the Beatles, Mozart or Chinese music.
This would seem to suggest that groups of sound sequences liked by humans possess a type of temporal
organisation to which we are mentally receptive. This organisation must be a function of how the brain has
evolved. The individual's preference within these groups would thus be determined by his or her social background. It would also suggest that the more a certain sequence of sounds contains organisation to which we
are biologically predisposed, the greater the number of people that will like it.
Since our brain has evolved to process temporal information, it makes sense to suggest that pleasing sound
sequences might abstractly mimic the way in which the world changes with respect to time. We believe this
is the case and so studied how differential equations could be used to create abstract temporal forms which
bear similarities to the organisation found in music. We found that the most appropriate and useful category
of equations are those termed relaxation oscillators.
Relaxation Oscillators:
Phase and frequency locking has been observed in many biological mechanisms. One such example is that
of the membrane response of the squid axon to electrical stimulus (Matsumo,G. et a!). The membrane was
stimulated with periodic trains of current pulses, and it was found that the response could exhibit temporal
periodicity or chaos.
The different types of responses that can be obtained are quite varied. The membrane potential on some
occasions increases rapidly to form a 'spike' and then drops to a resting potential for each current pulse that
stimulates it. At other times the membrane potential fires similarly in response for each 1st and 2nd stimulating
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