2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Sound Creation Based on Different Chaos Attractors
Naruno Fujii1 ,Masaki Bandai1 , Yoshifumi Nishio2
and Mamoru Tanaka1
1

Sophia University
7–1, Kioi-cho, Chiyoda-ku, Tokyo 102–8554, Japan
Phone/FAX:+81-3-3238-3878 / +81-3-3238-3321
E-mail: nanno0917@sophia.ac.jp, mamoru.tanaka@gmail.com
2Tokushima University
2–1, Minami-Josanjima, Tokushima 770–8506, Japan
E-mail: nishio@ee.tokushima-u.ac.jp
Abstract
This paper describes sound creation without signal processing but with nonlinear differential equations. Usually soundcreation is realized by using signal processing. However tounderstand the signal processing completely is difficult for
amature users. And the quality of sounds is artificial than
the sounds of real instruments like grand pianos and violins.
So the purpose of this research is to create more natural andartistic sounds. So as to consider about natural and artistic
sounds, 1/f fluctuation is also important element. Sound thatgives good feelings for human’s ears includes kind of shaking. To realize that, using chaoses is so effective method.
Then considering about to create various kinds of sound byusing chaoses, it comes to necessary to think about frequency
characteristics and other elements each chaos has.
1. Introduction
Music should be artistic. There are many kinds of electronic instrument like a synthesizer and those instruments
have so many functions as modulation, delay, equalizer andso on. Those functions are usually realized based on signal
processing. However actually, such sounds are quite different
from those of traditional instruments as grand pianos, violins, drums and so on. Traditional instruments can express
the feelings of players, because of these instrument’s composition. For example piano has keys, strings and hammers,then they are connected. When a player strikes a key of a piano, the connected hammer strikes the corresponded strings.
So, the delicate movement of fingers affects the quality ofsounds as players emotions and skills. Violins or other string
instruments are also same. In case of wind instruments, ofcourse we can say the same thing. Player’s breath affects thesound quality as their feelings. It is impossible for any musicians to play completely same nuance again. That is the art.
So we want to realize such a real instrument.
In our previous research, by using the nonlinear differential
equation, we succeeded to play the music. However the soundwas generated from only sine waves so quality of that was likea electronic one. Then in order to make the sound to be morenatural or artistic one, we tried to add kind of shaking. Kindof shaking can be realized by using chaos, since the chaos hasvarious random frequencies. Therefore, it can be expected towork as moderate beats and make the better quality of sound.
By the emotion, sensitivity, something of feelings, or interpretation of the performer, the quality of sound changes. Thenonce a music has been performed, it is quite impossible to do
the same performance again, even by the same person. When
we pay attention to the waveform by signal processing, they
contain several frequencies these are kind of harmonic overtone or so on. And the rate of each frequency also changeswith the passing of time. Furthermore, as for amplitude ofvibration, it’s clearly changes. It means that if changes of frequency and amplitude of vibration were added to the basic
sine wave, quality of sound and envelope will be changed, As
a result, music comes to more artistic. This research is to create sound that is from non-linear differential equation, and tomake the sound more artistic.
We would like to challenge to create the music based on
the nonlinear differential equations without signal processing. Especially, each note sound should be generated fromthe chaos equations to insert the basic equation with the notefrequency made by MIDI. The MIDI note frequencies andits time length are generated from the EXCEL. And in thisresearch 1/f fluctuation is also important key point. There
are many natural phenomena those are related 1/f fluctuation in the world. For example, the intervals of our heart’s
beating, tremor of candlelight, jolting of cars, the murmuring- 689 -

of a stream, the movement of the pupil of the eyes and so on. 3. MIDI
They came to be the strong hints in case of music and in thisresearch.
First of all, information of songs must be input to the program. Usually players use the scores as follws photo.
2. Basic Equation−ty[t] = (e length − e−1 ) × 213 × sin 2πf t(1)

By using this equation(1), sin waves of each note are created.
However to create it, frequencies of each note are needed.
′ ′In case of this research, MIDI information as follows plays
f in the last of equation(1) means ’frequency’. frequenciesof each note can be calculated by using other equation(2) as the role of score.
follows.
f requency = 440.0 × 2.0

notenumber−69.0
12.0(2)The table.1 shows the relation between MIDI note number
and frequency that is result of the calculation.

Table 2: MIDI information
MIDI number length
640.25
620.25
64
3.5
630.5
30000
364
0.25

Table 1: Relation between notenumber and frequency
name of note
CC#/D♭
DD#/E♭EF
F#/G♭
GG#/A♭
AA#/B♭BC

MIDI number frequency
60
261.625661
277.182662
293.6648
63311.127064
329.627665
349.228266
369.9944
67391.9954
68415.3047
6944070466.1638
71493.8883
72523.2511

Table 3: MIDI information in chord
MIDI number
64
62
64 30000
length
0.25 0.25 0.5
3MIDI number
57
58
MIDI number
55
56
MIDI number
53
54
MIDI number
52
length0.5
3.5

The Table.2 is in case of single melody, and the Table.3
is in case of melody with chord. MIDI consists of both notenumber and its time length. Note number starts from 0 to127. The lowest note of piano keys is note number 21 andthe highest key is note number 108. So MIDI can cover awide area of notes. It s the definition of note number. From
inputtingthis MIDI information, simulation of this research
MIDI note number is corresponding to each note and thestarts.
Eachnote number has its frequency, and the frequency
frequency of each note is determined completely, for example
isalsobasedon temperament. What is temperament? You
the frequency of the middle A is determined to 440Hz liketable.1. That enables to work as a score. The next section is know, before orchestra starts the performance, they usually dotuning with the note of ’A’ so, everyone possess in the samethe property of MIDI.
frequency. But as for piano, it’s too difficult to do tuning in ashort time. That’s why turning of piano is finished based ona certain frequency previously. It’s called ”temperament”. Incase of piano, one key is corresponding to one hammer, and- 690 -one hammer strikes several strings. The several strings are
tuned to little different each other, for example the middle one
is 440Hz, the right one is 442Hz, and the left one is 438Hz.
That’s reason why piano can play the correct tone withoutorchestra tuning. This idea came to the hint that we added theround frequency by using chaos.

4. 1/f Fluctuation

And the equation of Lorenz is as follows.









dx
dt = −px + pydy
dt = −xz + rx − ydz
dt = xy − bz






p = 10.0 r = 28.0 b = 8.0
3.0

Lorenz circuit equation

In this research, 1/f fluctuation is one of the important keywords. After all, what is the 1/f fluctuation? the f of 1/ffluctuation means frequency, so it’s so small value. It indiFollowing figure.1 is the ring attractor generated fromcates the delicate changes in much small range. That can be nishio circuit.
discovered in natural phenomena. For example, the intervalsof our heart’s beating, tremor of candlelight, jolting of cars,
the murmuring of a stream, the movement of the pupil of theeyes and so on. Recently 1/f fluctuation is said to cause thephenomenon that strawberry grown with Mozart music tastes
good. As for music, 1/f fluctuation is the most importantelement. The emotion, sensitivity of performer are the point
what people feel comfortable. In other words, by means ofthat irregular movements are mixed to regular movement, wecannot expect the next movement or something of law, it gives
good feelings for human’s ears. In this research, we are expecting that chaos can work as 1/f fluctuation.
Figure 1: Nishio attractor
5. Circuits and Chaoses
In this research two kinds of chaos are used, Nishio chaosand Lorenz chaos. Both chaoses have characteristic shapeof attractor. In case of Nishio attractor, it has smooth ringshape. Compared to that, the shape of Lorenz attractor is likea butterfly. Figure.1 and Figure.2 are the attractors of eachchaos. The following equation is Formalized Nishio circuitequation.

And the figure.2 is the attractor generated from Lorenz equation.

 dxk
∑4

j=1 xjdτ = β(xk + yk ) − zk − γ




 dyk = αβ(xk + yk ) − zk − f (yk )dτ
dzkdτ = xk + yk

Figure 2: Lorenz attractor
Figure.3 is the circuit model of Formalized Nishio circuit.

(k = 1, 2, 3, 4)

wheref (yk ) = 0.5(δyk + 1 − |δyk − 1|)
Formalized Nishio circuit equation- 691 -

Figure 3: Modified Nishio Circuit model

Figure 5: Nishio attractor parameter changed

6. Simmulation and Result

7. Conclusion

To create various kinds of sound by using several chaoses
is very difficult. In case of Nishio chaos and Lorenz chaos,the shape of attractors is different completely. But as for thefrequency characteristics of both chaoses, it can be thoughtthat they are similar. So to change the sound within the rangethat human can recognize the difference is difficult. In case of
only Nishio chaos, by changing the parameters for example
as figure 4, it is possible to create different sounds.

The difference between sounds of different chaoses depends on several elements as the envelope waveform, characteristics of each chaos has and so on. However by changing
the parameters of circuit, to create various kinds of sound canbe realized. We expect that to create more various kinds ofsound by considering several elements each chaos has. Moreover by mixing several chaoses, the possibility of sound creation will be growing.
References[1] Y.Nishio and A.Ushida, ”Chaotic Wandering and its
Analysis in Simple Coupled Chaotic Circuits ”, IEICE
Transactions on Fundamentals, vol.E85-A,no.1,pp.248255,Jan.2002.

Figure 4: parameter code
The following figure.5 is the attractor that was generated from parameter changed Nishio circuit. Ring attractor
changed to shell shape attractor, and the sound also changed.

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