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ILMLI2012NON-COCHLEAR SOUND
____ LJUBLJANA _9.-14. SEPTEMBER
A STIGMERGIC MODEL FOR OSCILLATOR
SYNCHRONISATION AND ITS APPLICATION IN MUSIC
SYSTEMS
Andrew Lambert
ABSTRACT
Non-linear and chaotic dynamics, predominantly used in
engineering, have become a pervasive influence in
contemporary culture. Artists, philosophers and
commentators are increasingly drawing upon the richness
of these systems in their work. This paper explores one
area of this territory: the synchronisation of a population
of non-linear oscillators used for the generation of
rhythm as applied in musical systems.
Synchronisation is taken as a basis for complex
rhythmic dynamics. Through the self-organisation notion
of stigmergy, where entities are indirectly influenced by
each other, the notion of local field coupling is
introduced as a qualitatively stigmergic alternative to the
Kuramoto model and noise, distance, delay and influence
are incorporated.
An interactive system of stigmergic synchronised
oscillators was developed, that is open to be used across
many fields. The user is allowed to become part of the
stigmergy through influencing the environment. The
system is then applied to the field of music, generating
rhythms and sounds by mapping its state.
1. INTRODUCTION
Oscillator synchronisation is a potential biological root
of musical creativity. Through oscillation, interesting
musical behaviour can be achieved.
In section 2, stigmergy, a notion where entities are
environmentally influenced by each other, is used as the
mode of exploration into self-organisation. The
Kuramoto model is introduced as a powerful and elegant
mathematical formula describing the phenomena of
oscillator synchronisation in the natural world. However,
since synchronisation has its roots in self-organisation,
the Kuramoto model encounters a problem and falls short
of complete plausibility. An alternative model, local field
coupling, derived from Kuramoto and other methods of
oscillator synchronisation taken from biology and
neuroscience, is described to solve this problem.
Section 3 discusses theories from the fields of
chronobiology and biomusicology, which use oscillator
synchronisation phenomena to explain many forms of
behaviours in living systems. A clearly rhythmic, but not
necessarily creatively musical behaviour is achievable
through stigmergic synchronisation, termed protomusical
behaviour.
An interactive system developed by the author,
Crickets, is detailed in section 4. Crickets is an
environment in which low-level creativity is achievable
through biologically inspired protomusical behaviour.
The protomusical behaviour generated by the system is
able to be used in many applications across disciplines.
2. SELF-ORGANISATION AND OSCILLATION
2.1. Stigmergy
A self-organising system is a system that forms a pattern
or order without a central control mechanism or external
influence. The pattern is formed instead via interactions
on a local scale, with each part of the system knowing
nothing of the global effect of these interactions. Selforganisation is interlinked with two other related terms,
emergence and stigmergy, which seek to encapsulate
self-organisation from differing viewpoints.
In emergent behaviour, a set of properties or rules are
defined through which a sophisticated pattern not present
in the design of these rules is revealed [2], [16]. The
main criticism of emergence is that an observer must be
present. It is only via external observation that emergent
behaviour is defined. Agents within the system, by their
very nature, cannot intend to produce emergence as that
will defeat the point. Furthermore, it is the observer that
labels that outcome of the process a 'pattern' prior to
being an emergent pattern. This leads to the area being
difficult to study with great accuracy.
Stigmergy on the other hand circumvents this problem
through its own definition. It is another term that has its
roots in the natural sciences, being devised to explain the
control of collective behaviour of social insects such as
ants and bees [21]. It is a notion common today in many
agent based simulations, in that the agents remain
independent entities. Their interactions with the
environment affect the behaviour of the other agents,
which in turn affects them. Stigmergy is therefore
defined as pattern formation in a collective via an
interaction with an environmental mediator.
A common example of Stigmergy is an ant following a
pheromone trail to a food source. The ant is merely
following a trail it senses in the environment. The ant in
turn leaves behind a trail of its own, thus strengthening
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