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{\Huge \bf PEGASUS-2 : Real-Time Composing Environment with Chaotic
Interaction Model \\}
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\vspace{5mm}
\begin{center}
{\Large {\bf Abstract}}
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{\small {\sf
This is the second report of the PEGASUS (Performing Environment of
Granulation, Automata, Succession, and Unified-Synchronism) project.
In the first report at ICMC92, compact / portable / real-time Granular
Synthesizer was installed into the system, and used in some pieces.
This paper focuses on ``automata'', not only simple generated
randomization but also ``chaos'', in particular focusing on
its dynamic characteristics.
This paper describes (1) Chaotic Interaction Model (CIM) to generate
flexible and dynamic musical elements,
and (2) CIM experimental application for hierarchical structure in music.
We chose simple Logistic Functions for that of chaos generators,
but the generated output has much variety with well-tuned combination
of parameters.
In addition, CIM generator is applied to automatic composition systems.
Considering the hierarchical structure in music, the system is constructed
with three hierarchical modules (tempo / rhythm / note), and each module
is controlled by an individual CIM generator. \\
}}
]
\section{Background}
``Musical automata'' and ``automatic composition'' are interesting
themes.
Many theories and models have been discussed, and many systems and software
has been developed \cite{beyls}.
Today we have good tools for computing musical information
in real time.
PC-based ``real-time composing'' and ``interactive composing''
has been developed recently \cite{chadabe}.
One of authors has produced one compact system of real-time Granular
Synthesis.
This research project was called the PEGASUS (Performing Environment
of Granulation, Automata, Succession, and Unified-Synchronism) project
\cite{nagasm2}.
The target is not only producing a hardware / software system, but also
experimentally composing with this system.
The piece called ``Chaotic Grains'' was performed on Feb. 11th in
Tokyo \cite{nagasm4}.
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\paragraph{}
\section{``Chaos'' in Music}
\subsection{Logistic Function}
``Chaos'' is easily generated with the following simple function :
{\boldmath \[ X_{n} = \mu \cdot X_{n-1} \cdot (1 - X_{n-1}) \] }
this function is called ``Logistic Function''.
There are many complicated and two or three dimensional chaotic functions.
But we chose the most simple one, because we wanted observe
chaos dynamics.
With the area {\boldmath \( 1 < \mu \leq 3 \)}, the value of
{\boldmath \( X_{n} \)} is converged into only one value, but on
increasing in the area {\boldmath \( 3 < \mu \)}, the value
of {\boldmath \( X_{n} \)} is branched into two, four, eight, ... and
into the ``chaos zone'' \cite{devaney}.
In the ``chaos zone'', some kind of randomness is generated, but there is
a particular difference from other ``noisy'' random systems.
The parameter {\boldmath \( \mu \)} is very important to control
the random characteristic, and it is possible to control the ``chaos''
dynamics with the value of {\boldmath \( \mu \)}.
\subsection{Around the ``Window''}
We were interested by the fact that the resulting state of chaos cannot be
determined in spite of its deterministic definition.
Many critical points in the ``chaos zone'' were observed in our
previous work \cite{nagasm3}.
Even branching many values of {\boldmath \( X_{n} \)} in the ``chaos
zone'' of {\boldmath \( \mu \)}, it is impossible to obtain finite values
normally.
But there are many points with finite values of {\boldmath \( X_{n} \)}
in the chaos zone of special {\boldmath \( \mu \)}, which is called
``Window''.
If the value of {\boldmath \( \mu \)} is varied widely,
the finite ``chaos vibration'' (sequence of values
of {\boldmath \( X_{n} \)} with calculation cycle) falls into
perfect random state.
When the value is slightly varied on the edge of the ``Window'',
the ``chaos vibration'' is shifted somewhere in short term.
It may return back to the finite state in some cases, as if it were
pushed back by an active something.
This reaction is very critical and sensitive for
the value of {\boldmath \( \mu \)}.
\subsection{Musical Replacement}
As a simple replacement of ``chaos vibration'' to music,
it is easy experimentally to assign the value of {\boldmath \( X_{n} \)}
to the ``pitch'', and the calculation cycle to the ``duration''.
For example, the MIDI notes which are generated by chaos simulator
with the value of {\boldmath \( X_{n} \)} in ``chaos vibration'',
are perceived as ``arpeggio'' or ``trill'' performance in music.
When we add a heuristic tuning of parameters, a ``slight shift of
the value into the chaos zone'' may cause some kind of variations
or improvisation of the musical phrase.
\section{CIM and Application}
\subsection{CIM}
In order to use this effect, two individual chaos generators are
connected to each other.
The output of one generator is led to the other input as the value of
the parameter offset.
We call this ``Chaotic Interaction Model (CIM)'', and use the value
range (interaction range) of {\boldmath \( \mu \)}.
If want to add expression to the music using only one chaos generator,
we must control the dynamics of it interactively.
But it is possible to control or interface each chaos dynamics in CIM,
and we can observe and consider the actions from outside.
Like the CIM system, the edge area of chaos (Window) is considered a
somewhat ``intelligent system'', and its application may be useful for
composition \cite{bidlack} \cite{pecora}.
\subsection{Real-Time Composition}
CIM is not considered a perfect musical generator, but it is useful tool
in musical composition.
With the ``Sensor Fusion'' system and musical recognition system,
CIM is applied in the musical environment with its real-time composing
characteristics.
In this system, there are two compositional job streams.
As ``composition'' is an interactive process, both streams contain
an information loop.
One loop, ``non real-time composition'' means the traditional style
of composition.
Composers write scores, input sequence data, and repeat test-listening
with arrangement many times.
The other loop, ``real-time composition'' has two
meanings : (a) improvisation and (b) real-time generation.
Improvisation is real-time control of musical parameters by performers
with many sensors.
Real-time generation is constructed with CIM, and parameters of
``chaos vibration'' are interactively controlled by performers
in real-time.
\subsection{Musical Information Processing}
\paragraph{Note : Scale and Tonality}
In some pieces, notes are selected from some scales with 12 equal
temperaments.
Each probability of note assignment is described in the table.
It is controlled with special MIDI command, computer console, and sensors
of the performer.
Global scale and tonality are frequently metamorphosed.
\paragraph{Rhythm : Event Timing}
Percussive sounds are used for most of the chaotic notes in the piece.
The interval of each note event makes a form of ``Rhythm'',
and are also generated by CIM.
\paragraph{Performer / Conductor}
There are many sensors to detect controls and cuing of performers
and the conductor.
All actions are detected and converted into special MIDI signals,
and transferred in order to control all parameters, the value, and
mapping tables.
The conductor can send the message of beat control as ``Tempo Rubato''.
\section{Conclusion}
The second step approach of the ``PEGASUS'' project, focusing on
fractal and chaos,
and the CIM application system has been described in this paper.
There still remains many areas for experimentation.
More discussion and a demonstration of CIM is planned,
and an experimentally composed piece called ``CIS (Chaotic Interaction
Show)'' is to be performed at IAKTA Workshop (16th Sep.) in Osaka.
\begin{thebibliography}{99}
{\footnotesize
{\bf
% \bibitem{degazio} B.Degazio :
% Musical Aspects of Fractal Geometry.
% Proceedings of International Computer Music Conference, pp.435--442, 1986.
\bibitem{beyls} P.Beyls :
The Musical Universe of Cellular Automata.
Proceedings of International Computer Music Conference, pp.34--41, 1989.
\bibitem{chadabe} L.Chadabe :
Interactive Composing.
Proceedings of International Computer Music Conference, pp.298--306, 1983.
% \bibitem{nagasm1} Y.Nagashima :
% Neural Network Control for Real Time Granular Synthesis.
% Proceedings of 6th Annual Conference of JSAI, vol.1, pp.381--384, 1992.
\bibitem{nagasm2} Y.Nagashima :
Real-time Control System for ``Psuedo Granulation''.
Proceedings of International Computer Music Conference, pp404--405, 1992.
\bibitem{nagasm4} Y.Nagashima :
Musical Concept and System Design of ``Chaotic Grains''.
IPSJ SIG Notes Vol.93, No.32, pp9-16, 1993.
\bibitem{devaney} Robert L.Devany :
An Introduction to Chaotic Dynamical Systems (Second Edition).
Addinson-Wesley Publishing Company, 1989.
\bibitem{nagasm3} Y.Nagashima :
Chaotic Interaction Model for Hierarchical Structure in Music.
Proceedings of 46th Annual Conference of IPSJ, vol.2, pp.319--320, 1993.
% \bibitem{aihara} K.Aihara, T Yoshikawa :
% Ordered and Chaotic Systems and Information Processing.
% Journal of JSAI, vol.8, no.2, pp.179--183, 1993.
% \bibitem{isabelle} S.Isabella, A.Oppenheim, G.Wornell :
% Effects of Convolution on Chaotic Signals.
% Proceedings of 1992 IEEE ICASSP, vol.4, pp.133--136, 1992.
\bibitem{pecora} L.Pecora, T.Carroll :
Synchronized Chaotic Signals and Systems.
Proceedings of 1992 IEEE ICASSP, vol.4, pp.137--140, 1992.
\bibitem{bidlack} R.Bidlack :
Chaotic Systems as Simple (bit Complex) Compositional Algorithms.
Computer Music Journal, vol.16, no.3, pp.33--47, 1993.
}
}
\end{thebibliography}
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