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{\large {\bf Chaotic Interaction Model for Hierarchical Structure in Music}}
�@\\
Yoichi NAGASHIMA
�@\\
Art \& Science Laboratory
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\section{Introduction}
Musical automata and automatic composition are interesting themes in
computer music field.
Many theories and models are discussed, many systems and softwares are
researched or developed by many researchers and composers in the
world \cite{degazio} \cite{beyls}.

Today, software engineering and computer technology are powerful,
so we have good environment for computing musical information
in real time, and the concept of ``real-time composing'' can be used
easily in compact system \cite{chadabe}.

The research called PEGASUS Project (Performing Environment of
Granulation, Automata, Succession, and Unified-Synchronism) has
produced one compact system of real-time Granular Synthesis
\cite{nagasm1} \cite{nagasm2}, and the second step of this project
is aimed at ``automata and unified-synchronism''.

This paper presents three approaches in computer music, (1) Chaos
application for real-time composition in PEGASUS Project, (2) Chaotic
Interaction Model (CIM) for flexible and dynamic generator for music,
and (3) Discussion of CIM application for hierarchical structure
in music.

\section{Applications of ``Chaos''}

\subsection{Logistic Function}

There have been discussed about ``Chaos'' for long time, and
there are many reports of chaos applications for wide fields.

Chaos is easily generated with simple equation :
{\boldmath \[ X_{n} = \mu \cdot X_{n-1} \cdot (1 - X_{n-1}) \] }
and this equation is called ``Logistic Function''.

With the area {\boldmath \( 1 < \mu \leq 3 \)}, the value of
{\boldmath \( X_{n} \)} is only one, but with increasing in the area
{\boldmath \( 3 < \mu \)}, the value of {\boldmath \( X_{n} \)}
is separated into two, four, eight, ... and is going into ``chaos''
zone \cite{devaney}.

In this area, some kind of randomness is generated, but there is
special different from other ``noisy'' randomness.
The parameter {\boldmath \( \mu \)} is very important to control
the random character, and it is possible to control the ``chaos''
with the selecting the value of {\boldmath \( \mu \)}.

With this simple equation, an experimentally software for personal
computer is produced.
This software, named ``Hearing the Chaos'', can be experienced
with downloading the program from Data Library of AI Forum (FAI)
in Nifty Serve.

\subsection{Experimentally Composition}

This interesting Logistic Function is applied for composition.
The piece ``Chaotic Grains'' is composed in 1992-1993, and one
special software is produced for this : development of the
software is some part of the composition in a way.

There are individual eight musical part as parallel running tasks.
Each part has its MIDI parameters : Channel, Program, Volume, Panpot,
...etc, and has its Chaotic parameters : Duration, Delay, Fadeout,
Scale Probability, Note Range, and the value of {\boldmath \( \mu \)}
with 6 mantissa data.

All parameters are controlled individually by conductor or sequence
data from another computer in real time, and the results of the
generation of chaos are different from each other in every
performances [Fig.1].
The characters of ``Chaos'' are changing and metamorphosing every time
in this piece, thus there is one kind of ``real-time composing''.

\section{Chaotic Interaction Model}

\subsection{Chaotic Interaction}
It is very interesting that the result state of chaos cannot be
determined in spite of itsdeterministic definition.
With experiment of chaos simulation, there is more interesting
character as if it lives.

There are many points with finite values of {\boldmath \( X_{n} \)}
in ``chaos zone'' of {\boldmath \( \mu \)} (normally with infinite
values), and called ``Windows''.

The interesting interaction is occured on the edge of this Window.
The finite chaos vibration falls into perfect random state with
wide shift of the value of {\boldmath \( \mu \)}, but if the value
is slight shifted, the chaos vibration may be pushed back in some case.
This reaction is very critical and nervous, but interesting.

\subsection{CIM Generator}
Using this value range (interaction range) of {\boldmath \( \mu \)},
the CIM simulator is produced and experimented.

Both chaos generators are designed as Logistic Functions, and
each chaos outputs are controlled with focusing in ``Window Edge''
area, and each output signal is led into another input of
{\boldmath \( \mu \)} each other.
Then, with some experimentally ``tuning'', this CIM generator acts
more interesting and flexible like human character.

Single chaos generator generates simple chaos random vibration,
but CIM genarator generates more complicated vibration, and
the CIM system generates more interesting reactions with
external interruption or control.

This CIM generator is being tested just now, and the detail is
not found with whole value range of {\boldmath \( \mu \)}.
And it is scheduled to produce CIM musical generator as the next
step of the experiment.

Single chaos generator generates simple chaos random vibration,
but CIM genarator generates more complicated vibration, and
the CIM system generates more interesting reactions with
external interruption or control.

This CIM generator is being tested just now, and the detail is
not found with whole value range of {\boldmath \( \mu \)}.
And it is scheduled to produce CIM musical generator as the next
step of the experiment.

\section{CIM for Higher Level}

\subsection{Hierarchical Structure}
There are many hierarchical levels in music.
For example, time structure in music is important.
The short term of vibration is the basis of the sound, and the
complex of these vibrations is the character of its timbre.
The sound has its own time variant characters (envelopes) :
timbre, loudness, and its duration.
The construction of each sounds generates ``Rhythm'' and ``Beat'',
and these relation of frequencies generate ``Harmony'' and ``Scale''.
Thus, music is constructed with the relation of its elemental sounds
in time scale and in frequency scale.

With the viewpoint of automatic composition, ``fractal'' methods
are applied to this hierarchical character in music.
Fractal structure is based on its self-similarity like musical
self-similarity.
And, there also exists same fractal character in chaos generator.
This is only one hint in this stage, but it will be researched.

\subsection{Compositional Idea}
The experimental composition ``Chaotic Grains'' is the musical
chaos application only with basic elements.
But there is another idea of application for automatic composition,
it is Chaotic Kansei Model for real-time composition.

In the brain or spirit of human composer, there are many characters
acting in many scenes.
Some of them are cooperating, some of them are fighting a battle
in composing.
So, CIM may be adapted to generate this situation for high level
composition.
This model may be act in the intelligent musical environment like
Kansei System \cite{katayose}.

\section{Conclusion}
``Chaos'' is only one approach for composition, and must not
be the main background in music.
But its complicated and dynamic character may be trigger
the new possibility in music, and there remain so many discussions
and experimentally compositions.

\begin{thebibliography}{99}
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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{devaney} Robert L.Devany : 
An Introduction to Chaotic Dynamical Systems (Second Edition).
Addinson-Wesley Publishing Company, 1989.
\bibitem{katayose} H.Katayose et al. : 
Music Interpreter in the Kansei Music System.
Proceedings of International Computer Music Conference, pp147-150, 1989.
\end{thebibliography}

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