2015 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP'15)
Kuala Lumpur, Malaysia, February 27 - March 2, 2015

Sound Creation based on Multi-oscillation Dynamics
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 presents sound creation with nonlinear differential equations including frequencies generated from several
kinds of attractor. An attractor has own frequency characteristic. Then, by using the various kinds of attractor frequencies, many kinds of sound including various fluctuation
can be expected to create. Specifically, the method is tocombine the note-depending Fourier series and the no-notedepending Fourier series where the no-note-depending means
corresponding to the attractor frequencies. Attractor frequencies are generated from the circuit simulator SDP (SoundDynamics Program). Step size, determined in that process,seems to be one of the important parameters to change thequality of new sound. There are various parameters and coefficients in this research, especially chaos parameters. The
created sounds can be evaluated by listening and comparing
frequency spectrum. Harmonic overtones are created and thesound difference based on each attractor can be confirmed.
1. Introduction
Recently there are many types of electronic musical instrument, such as synthesizers. Synthesizers are classified
into the analog synthesizers and the digital ones. Analog
sounds are created based on recording wave data and the digital sounds are based on simple waves those were created artificially such as sin waves, then these basic waves are manipulated by signal processing. There are so many functions
to modulate the sounds, AM (Amplitude Modulation), FM(frequency Modulation), delay, phaser equalizer and so on,these functions are also created by signal processing. On theother hand, traditional instruments such as grand pianos andviolins have their own specific shapes, The tone colors corresponding to the waveforms are effected by harmonic overtones, harmonic overtones are created by reflecting the each

instrument’s construction, so the difference of instrument’s
shape corresponds to the difference of tone color. Moreover,ways of wave’s reflections are slightly changed by the musician’s delicate movement such as timing, strength of fingers,arms, breath and so on. Music performed by musicians, especially professional musicians has comfortable, unique or
artistic shaking that is created by musician’s feelings andemotions. Of course, the way of vibration is different every
performance even the same songs, even by the same person,that is often called 1/f fluctuation. However to realize suchdelicate movement by only signal processing is limited. Thenthe new methods to realize unique movement may be kind of
beating or vibration are needed, so it is the main purpose ofthis research. As mentioned above, music has some kinds offluctuation such as harmonic overtones, timing, strength andso on. In this time, modulations of amplitude and frequency
are focused on, that correspond strength and kind of roundfrequencies. We pursue the suitable and various vibration byusing frequencies which are generated from some circuits.
In our previous research [1] , several kinds of chaos suchas Nishio chaos [2] , Roessler chaos and Lorenz chaos areused to create FM, frequency modulation and AM, amplitude
modulation as Eq.(1)−tfb [t] = (e length − e−1 ) × 213 × g(t) sin 2πf t(1)

By this equation, round frequency can be created, furthermore about amplitude, in other words, vibration can be also
appeared. Nevertheless, by this method, harmonic overtones
or higher harmonics cannot be created, so the difference between the chaoses is hard to feel. The spectrum is shown inFigure 1. The left one is the spectrum in case of using Nishiochaos, and the right one is of Lorenz chaos.
Figure 2 shows the attractors of Nishio and Lorenz, these
corresponds to the each spectrum of Figure 1. The left attractor is nishio chaos, and the right one is Lorenz chaos. Though- 409 -

Table 1 is in case of a single melody. The first line means toplay the notenumber 52 for 0.25 seconds. In this time, since aquarter note is defined to 1 seconds, 0.25 seconds means 1/16note.
Table 1: MIDI information
MIDI number length
52
0.25
57
0.25
57
0.25
59
0.25
60
0.5
62
0.5
64
160
0.5
57
0.5
71
1

Figure 1: Spectrum of Nishio and Lorenz for Eq.(1)Figure 2: Attractors of nishio and lorenz
the shapes of attractors are different so much, the tone color,
spectrum for Eq.(1) are almost the same. To consider thattone color depends on harmonic overtones, sound creationfrom only one sine wave such as Eq.(1) is almost unrealistic. Harmonics contain so many kinds of frequencies, thento use the Fourier series can be expected to solve the lackingof harmonics. The detail of new method [3] is to combine
the note-depending Fourier series and the no-note-depending
Fourier series from circuit frequencies, as Eq.(2)B(x) = fb (x) + fc (x).

(2)

Moreover in this time, frequencies generated from circuit are
not limited only chaos, a circuit in which diodes are removed
from Nishio circuit is tried. We find the suitable FFT (FastFourier Transform) coefficients from the methods of solving
the circuit equations, which are the kinds of the differential
equations.
2. Data and Method

In this time, we realize to play several notes simultaneously, it means that not only the melody line, but with chords.
By reflecting different kinds of chaos, to each layer, thesounds, harmony comes to be like an orchestra, each part are
played by each musicians. The following MIDI informationin Table 1 is in case of melody with chords, and this MIDI
information corresponds to the score as shown in Figure 3. In
addition, when 3000 is substituted as notenumber in Eq.(3),the frequency is much higher, it’s above the range that humancan hear, so note number 3000 can play the role of rest.
Table 2: MIDI information in chord
52
57
57
59
60
62
64
60
57
71
67
64
67
67
3000

0.25
0.25
0.25
0.25
0.5
0.5
1
0.5
0.5
1
0.5
0.25
2.25
00

3000
3000
3000
3000
3000
3000
3000
3000
3000
3000
52
52
52
55
59

4
00
0
0
0
0
0
0
2
0.25
0.25
0.5
0.5
0.5

57
3000
3000
3000
3000
3000
3000
3000
3000
52
3000
3000
3000
3000
3000

40000000040000060
3000
3000
3000
3000
3000
3000
3000
3000
55
3000
3000
3000
3000
3000

400000000
40000064
3000
3000
3000
3000
3000
3000
3000
3000
59
3000
3000
3000
3000
3000

First of all, information of songs, such as which note toplay and each note’s duration time, must be input to the program. Usually music player uses the scores. In case of thisresearch, MIDI (Musical Instrument Digital Interface) information made by excel as follows plays the role of score. MIDI
note number is defined based on note-A4 of 440Hz, each freAs this ’Amagigoe score’ shows music score has manyquency can be calculated from the Eq.(3)cells, and the performance time of each cell must be fixed
to the same. In this score, it takes 4 seconds for each cell.
notenumber−69.0
12.0
f requency = 440.0 × 2.0(3) However in case of MIDI information, ’cell’ cannot be seen,.

- 410 -

400000000400000

ics Program) is used to solve each circuit, the construction ofeach circuit can be expressed in net list. Figure 7 is the net listof nishio circuit, and Figure 8 is of the circuit removed diodefrom nishio circuit.

Figure 3: Amagigoe scorethat means once the duration time is mistaken, time lag is kept
to the end of the song. It’s the one of the problems but is not
solved yet.
Next, the attractors used in this time are as follows. Figure
4 is a circuit model of Nishio circuit, and the attractor is Figure 5. Figure 6 is in which diodes are removed from Nishiocircuit. By the comparison of the two sounds and spectrum,
how effects the difference of the construction of circuit canbe expected to appear.
In addition, SDP (Sound DynamFigure 7: SDP net list of A

Figure 4: Circuit model with diodeFigure 5: Attractor with diodeFigure 8: SDP net list of B

Figure 6: Attractor without diodeBy using these attractors generated from each circuit, andcontrol the parameters such as step size and so on, characteristics of each circuit are expected to be reflected to eachsound faithfully. The method is proposed by expressing eachnote waveform as- 411 -B(x) = fb (x) + fc (x)wherefb (xb ) = b20 +fc (xc ) = c20 +∑m(4)ˆi=1 (bi cos ixb + bi sin ixb )∑mi=1 (ci cos ixc + cˆi sin ixc ).

m = 4096 × 4(5)xb = 2π fˇb t
xc = 2π fˇc t

Figure 9: Spectrum AHere, the coefficient bi and bˆi are selected appropriately togenerate the basic form 1/f for note frequency as
notenumber−69.0
12.0fˇb = 440.0 × 2.0,

(6)

The coefficient ci ,cˆi are

ci = π1cˆi = π1∫π−π∫πg(t) cos ixc dt, (i = 0, 1, 2, 3, ...)
(7)g(t) sin ixc dt, (i = 1, 2, 3, ...)−π

can be derived by using FFT algorithm, for the circuit variableg(t) from SDP.
Figure 10: Spectrum B

3. Result
The created sounds are evaluated by listening them and frequency spectrum as follows. Figure 9 is the sound spectrumof Nishio circuit, and the Figure 10 is that of removed diodes
from Nishio circuit. It can be described that the differenceof the circuit’s construction effects the way of sound fluctuation and the creation of harmonics. However in case of using the same original circuit, the created sounds are similar,
because the frequency characteristics are also similar. Multioscillation mode can be generated, shown in the attractor ofFigure 6 and the multi-frequency characteristic of Figure 10
for all cells with different capacitances and without diodes in
the Nishio Circuit.
4. Conclusion
There are many types of attractor. Each attractor is generated from differential equations of a circuit which has many
characteristic circuit frequencies. It is important to change theoriginal circuit frequencies to sound frequencies from about
100Hz to 10000Hz by mapping virtually time step 2.26757E5 corresponding to 44100Hz. It is found that multi-oscillation
dynamics from a circuit including chaos are used to create
new sounds because of having many frequencies.

References[1] N. Fujii, M. Bandai, Y. Nishio and M. Tanaka, ”Sound
Creation Based on Different Chaos Attractors”, RISPInternational Workshop on Nonlinear Circuits, Communications and Signal Processing NCSP’2014, Honolulu,
Hawaii, USA. February 28 - March 3, 2014, pp.689692.
[2] 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.248-255, Jan. 2002.
[3] N. Fujii, M. Bandai and M. Tanaka, ”Sound Creation
based on Several Chaos Attractors”, IEEE Workshop on
Nonlinear Circuit Networks, pp.55-58, December 1213, 2014.

- 412 -