Digital Creativity
2005, Vol. 16, No. 2, pp. 79–92
Iterative sound synthesis
by means of cross-coupled
digital oscillators
Nikolas Valsamakis and Eduardo Reck Miranda
University of Plymouth, UK
[email protected]
[email protected]
Abstract
1 Introduction
Musicians have long been interested in using
iterative processes to aid the composition
of musical forms (macrostructure) and to
synthesize sounds (microstructure). This
paper introduces a new sound synthesis
method exploring the non-linear behaviour of
two iterative cross-coupled digital oscillators.
It begins with a brief introduction to iterative
systems followed by background information
on previous attempts at using them for
synthesizing sounds (e.g. feedback frequency
and amplitude modulations). Next, it introduces
our synthesis method and briefly explains how
it has been implemented in a system for realtime composition and performance. The paper
concludes with a discussion on how the system
has been put into practice to compose and
perform a number of works.
An iterative process is the repeated application
of a procedure where each step is applied to
the output of the previous step (Figure 1).
Mathematically, an iterative process is defined
as a rule that describes the action that is to be
repeatedly applied to an initial value x0. The
outcome of an iterative process constitutes an
orbit set and the values of this set are referred
to as the points of the orbit. Thus the orbit O
that rises from the iterated application of a
rule F to an initial value is written as: OF(x0).
For example, consider the following rule:
F : xn+1 = xn + 2. This rule indicates that the
next value of the orbit xn+1 is calculated by
adding 2 to the previous value. If x0 = 0 then
the application of F onto x0 will be OF(0
(00)) =
{0, 2, 4, 6, 8,…}. This is a very simple orbit,
but iterative processes have the potential to
produce fascinating orbits, some of which
can be used to generate interesting sounds if
appropriately mapped onto the parameters of a
sound synthesis algorithm.
Keywords: cross-coupled oscillators, iterative
systems, non-linear behaviour, real-time
computer music, software sound synthesis
Figure 1. An iterative process whereby the output is
fed back to the input.
The outcome of iterative processes
tends to exhibit four types of behaviour: (a)
stability to fixed value, (b) oscillation between
specific values, (c) chaotic behaviour and (d)
explosion to infinity. The two latter cases are
of special interest in computer music research
because they open up new territory for the
1462-6268/05/1602-0078$16.00
Digital Creativity, Vol. 16, No. 2
Valsamakis and Miranda
exploration of new composition methods
and synthesis techniques. There have been a
number of attempts at exploring the behaviour
of iterative processes in music, notably
those exhibiting chaotic behaviour (Pressing
1988; Bidlack 1992). A brief survey on the
application of iterative processes in musical
composition can be found in Miranda (2001).
Less attention has been paid, however, to
the potential of iterative processes for sound
synthesis. The chaotic iterative processes that
have been studied in sound synthesis include
the sine map (F
F : xn+1 = sin(r x xn), where r is
a scaling constant) and the logistic map (F
: xn+1 = r x xn x (1 - xn), where r is a constant
representing the growth rate), both explored
by Di Scipio (1996, 2002). There is also the
Mandelbrot set (F
F : xn+1 = xn2 + c, where c is a
complex constant), which has been explored
by Dobson and Fitch (1995). In these cases,
the orbits are either used to control the
parameters of sound synthesis algorithms or
are relayed directly as sound samples. This
paper, however, presents a slightly different
approach to exploring iterative processes in
sound synthesis: the function F is replaced by
a digital oscillator.
2 Feedback digital oscillators
The digital oscillator is a fundamental
component in many sound synthesis systems
(Miranda 2002). As implemented on a
computer, a digital oscillator often works by
repeating a template waveform, stored on a
lookup table. The speed at which the lookup
table is scanned defines the frequency of the
sound. Although this waveform does not
necessarily need to be a sinusoid, for the sake
of clarity we have chosen to focus solely on
sinusoidal oscillators in this paper.
An oscillator normally requires the
specification of three parameters: frequency,
amplitude and phase. An iterative process is
created if the output of an oscillator is fed
80
back to one of its own inputs. Two types
of iterative processes like this have been
explored in sound synthesis technology,
depending on the input to which the output of
the oscillator is fed back: Feedback Amplitude
Modulation (FAM), if the oscillatorʼs output
is fed back to its own amplitude [1] and
Feedback Frequency Modulation (FFM), if
the oscillatorʼs output is fed back to its own
frequency [2]:
where fx is the frequency of the sine wave
oscillator and I is the modulation index, or in
this case, the feedback factor. In equation [2]
the product (I x fx) is the amount of frequency
deviation from the carrier frequency. In
equation [1] the feedback signal is converted
from bipolar to unipolar. This is a case of
classic amplitude modulation where the
modulating signal takes only positive values
in the range between 0 and 1. To accomplish
this conversion, an offset +1 is added to the
bipolar signal, which is then normalised by
dividing by two.
Risset was probably the first to employ
a feedback oscillator in sound synthesis in the
late 1960s at Bell Telephone Laboratories in
the USA. In 1969 he published an algorithm
where the output of the oscillator was fed
back to its own amplitude input, resulting in
FAM (Risset 1969). Various implementations
of feedback oscillators have appeared since
then, including an FFM scheme patented by
Yamaha (Chowning et al. 1986).
3 Cross-coupled oscillators
We have extended the concepts of FAM
and FFM by employing two cross-coupled
oscillators instead of a single oscillator. In this
Iterative sound synthesis by means of cross-coupled digital oscillators
where, fx and fy are the frequencies of the two
sine-wave oscillators, and Ix and Iy are the
feedback factors, one for each oscillator.
The output scaling factor controls
the amplitude of the signals relayed by each
oscillator to the output as follows [6]:
where S is the scaling factor: -1 ≤ S < 1. If S =
0.5 then the signals from both oscillators are
heard with equal contribution to the resulting
sound. Values equal to 0.0 or 1.0 output the
signal of only one of the oscillators.
4 Auditioning nonlinear
phenomena
Cross-coupled oscillators are interesting
because they allow for the exploration
of nonlinear phenomena to synthesise
new sounds with interesting microtemporal properties that are very difficult,
if not impossible, to produce using other
synthesis methods. In order to gain a better
understanding of these phenomena with
respect to sound synthesis, we have tested
a number of settings on a trial-and-error
basis, but in a systematic fashion. We looked
primarily for initial values that produced
unstable behaviour.
The system displayed the typical
nonlinear phenomena that one would normally
expect to observe in a dynamic system.
Three out of the four types of outcomes for
iterative processes mentioned earlier could be
observed here: (a) stability to a fixed value,
(b) oscillation between specific values, and (c)
chaotic behaviour.
From our own subjective assessment,
the most interesting of the three configurations
of cross-coupled oscillators proved to be
CFFM. CFHM also produced interesting
sounds, probably due to its asymmetrical
structure. On the whole, CFAM produced
less interesting sounds. From here on
our discussion will focus on the CFFM
configuration.
The CFFM configuration proved
to have a strong dependency on the initial
conditions, which is a typical characteristic
of dynamic systems. There are four different
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Digital Creativity, Vol. 16, No. 2
way, one oscillator functions as the modulator
of the other and vice-versa. Two feedback
factors, referred to as cross-modulation
indices, control the amount of output signal
that is fed from one oscillator to the input of
the other. At the output stage a scaling factor
controls the contribution of each oscillator to
the synthesised sound.
There can be three possible
configurations of cross-coupled oscillators:
(a) Cross-Feedback Amplitude Modulation
(CFAM)
(b) Cross-Feedback Frequency Modulation
(CFFM)
(c) Cross-Feedback Hybrid Modulation
(CFHM)
Whereas in CFAM the output of an oscillator
is fed to the amplitude input of the other,
and vice-versa [3], in CFFM the output of
an oscillator is fed to the frequency input of
the other, and vice-versa [4]. Finally, there is
CFHM, where the output of the one oscillator
is fed to the amplitude input of the other, while
the output of the second oscillator is fed to the
frequency input of the first [5]:
Digital Creativity, Vol. 16, No. 2
Valsamakis and Miranda
initial variables: the central frequencies, fx
and fy, of the two oscillators and the feedback
factors, Ix and Iy. The two frequencies played
a less significant role in the exhibition of
nonlinear phenomena. Most important in this
algorithm are the feedback factors Ix and Iy,
where even subtle changes (e.g. ~0.001 %)
are capable of producing completely different
sounds.
Stability to a fixed value produced
monotonous buzz-like sounds commonly
unexpectedly to a fixed spectrum. The values
producing this sound were: fx=60Hz,, fy= 60Hz,
Ix= 10.58, Iy= 18, S = 0.5.
Oscillatory and chaotic behaviour
produced sounds with various micro-temporal
properties ranging from micro-fluctuations
and wild turbulences to ‘coloured noises’ (i.e.
white noise confined to a narrow frequency
band). Some sounds, exhibiting microfluctuations at lower rates, were reminiscent
of various natural phenomena, such as sounds
Figure 2. The
spectrogram
of a sound
exhibiting
stability to
fixed value
behaviour.
found in standard AM and FM synthesis.
The stabilisation of the system emerged
from either the first iteration or after some
cycles. In the latter case, the number of
iterations the system needed to stabilise was
unpredictable. Stabilisation often occurred
when experimenting with small values for the
feedback factors, Ix and Iy, but also occurred
unexpectedly with larger values. This is,
however, a general property of dynamic
systems: regions of order can be found within
chaos as well as chaos emerging within areas
of order. The spectrogram of a sound from
this class of behaviour is shown in Figure 2.
This sound exhibits initial chaotic behaviour
and after around 700 milliseconds it stabilises
82
of burning, air or water streams, rustling trees
and insect-like sonorities (Figures 3 and 4).
The spectrum of an insect-like sonority is
shown in Figure 3. The values to produce this
sound were: fx=107Hz, fy= 3.21Hz, Ix= 12214,
Iy= 6.12, S = 0.5. The spectrum of a water
stream-like sonority is shown in Figure 4. The
values producing this sound were: fx=93Hz, fy=
104.16Hz, Ix= 13.16, Iy= 7, S = 0.5
Special attention had to be paid to the
role of the feedback factors, Ix and Iy. Higher
values forced the model to produce various
ʻcoloured noisesʼ. The spectrograms of sounds
from oscillatory and chaotic behaviours are
shown in Figures 5 and 6 respectively. The
sound of Figure 5 oscillates between different
Iterative sound synthesis by means of cross-coupled digital oscillators
5 Making interactive music with
iterative sound synthesis
In the following paragraphs we introduce
a case study where the coupled-feedback
oscillators technique has been implemented
in a system for real-time composition and
performance. The system has been used
extensively by Valsamakis to compose and
perform at a number of international festivals
and concerts (Figure 7).
Figure 3. The
spectrogram of
an insect-like
sound.
Figure 4. The
spectrogram
of a water
stream-like
sound.
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Digital Creativity, Vol. 16, No. 2
spectral configurations. The values producing
this sound were: fx=60Hz,, fy= 30Hz, Ix= 42,
Iy= 9.5, S = 0.5.
The sound of Figure 6 exhibits chaotic
behaviour and has a coloured noise spectrum.
Interestingly, there are short moments of order
when the sound settles to a specific harmonic
spectrum before returning to chaos. The values
producing this sound were: fx=420Hz,, fy=
420Hz, Ix= 1.7, Iy= 1.16, S = 0.5.
Digital Creativity, Vol. 16, No. 2
Valsamakis and Miranda
Composition with iterative sound
synthesis belongs to the territory of nonstandard synthesis where the sound-generating
process does not rely on the simulation of a
pre-existing acoustical model, as is the case
with Physical Modelling, Additive, FM,
AM and other standard techniques, but on a
network of relations between a set of machine
instructions. It is a ‘bottom-up’ composition
approach where the morphological properties
emerge from an abstract model of sound
construction. The compositional process is
applied directly to the microstructure level of
sound. Historically, non-standard synthesis has
been used by composers such as H. Brün, G.
M. Koening and I. Xenakis, and more recently
by P. Berg, A. Di Scipio, A. Chandra and M.
Hamman.
Only CFFM was implemented in
this system and the chosen platform was the
Max/MSP programming environment running
on a Macintosh PowerBook, under Mac OS
X. There is no specific technical reason for
choosing this platform apart from personal
preference. The core of the instrument is
shown in Figure 8.
A few extra features were added to the
core CFFM algorithm in order to add extra
Figure 5. The
spectrogram
of a sound
exhibiting
“oscillatory”
behaviour.
Figure 6. The
spectrogram
of a sound
exhibiting
“chaotic”
behaviour.
84
Iterative sound synthesis by means of cross-coupled digital oscillators
capabilities, such as band-pass filtering, the
ability to move the sounds in an octaphonic
circular space, and a morphing facility. Bandpass filtering made it possible to focus on a
specific spectral band and highlight interesting
micro-temporal activities otherwise masked
by other bands of the spectrum. Spatial
movement in a multi-speaker set-up with
carefully defined trajectories of individual
sounds gave perspective to the listening
experience and highlights the dynamic
morphology of the sounds. The morphing
facility allowed for making transitions from
one set of parameter values to another within
a previously specified time interval. This
expanded the preset sonic palette in the sense
that it allowed for the production of gradual
transitions between sounds.
As mentioned earlier, it is almost
impossible to predict the behaviour of an
iterative system. We have learned the hard
way that the most convenient way to operate
our system was to adopt an experimental and
explorative attitude. In general, the process
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Digital Creativity, Vol. 16, No. 2
Figure 7. Iterative sound synthesis performance by Valsamakis at the
Ionian Academy in Corfu, Greece, in October 2003.
of creating musical pieces with the system
involved three stages.
At the first stage, in-depth
experimentation was carried out and various
combinations of initial parameter values
were tested. A meta-feedback loop was
set, involving the composer, the synthesis
algorithm, and the musical result. This
feedback loop involved three tasks: (a) setting
initial values of the algorithms, (b) listening
to the result and (c) adjusting the values. The
real-time aspect of the systems has contributed
to speeding up the creative process by offering
direct audible results and the possibility to
quickly try out various combinations of sound
materials. This first exploratory stage resulted
in various sets of initial values for the CFFM
algorithm and their respective sounds.
At the second stage, the sounds were
sorted and grouped into different categories,
according to their morphological properties
and micro-temporal development. The lack
of well-established taxonomies for such
sounds led to the use of labels using adjectives
reminiscent of various natural phenomena
such as boiling, sparkling, whistling, fizzing,
rumbling, crackling, scratching, and so forth.
At the third stage, a MIDI keyboard
was prepared for performance. We decided to
limit the polyphony to only two independent
voices because of the spectral richness of the
sound repertoire produced in the first stage.
Each key of the keyboard was programmed
to trigger different sets of initial values for
the CFFM algorithm. That is, the keyboard
was not ‘tuned’ to play at specific pitches,
but rather to produce different CFFM sets of
values, each of which produced a different
sound. The sounds were allocated to keyboard
regions according to compositional and
performance criteria. Two different layers of
sounds were arranged, each associated with
one of the two polyphonic voices. A MIDI
fader box was used to give the opportunity to
coarse-tune the algorithm by controlling each
parameter value through a fader.
Digital Creativity, Vol. 16, No. 2
Valsamakis and Miranda
Figure 8. The core instrument as implemented in Max/MSP.
6 Final remarks
Our iterative sound synthesis technique using
cross-coupled digital oscillators has proved to
be a fruitful tool for exploring the dynamics
of non-linear systems in musical composition
and performance. Our experiments yielded a
large number of different sounds characterised
by interesting micro-temporal structures. The
morphological properties of these sounds are
unique in the sense that they are not likely to
be produced by other synthesis techniques.
The implementation discussed in
this paper employed only two sinusoidal
oscillators. A natural further step in this
work would be to expand the iterative idea
by adding more inter-modulating oscillators.
However, the addition of extra oscillators
will increase the number of possible feedback
paths and therefore the complexity of the
86
system and the sounds as a whole. Such
complexity would certainly require more
sophisticated control methods, possibly using
artificial intelligence technology supported
by powerful numerical methods to analyse
complexity.
References
Bidlack, R. (1992) Chaotic systems as simple (but
complex) compositional algorithms. Computer
Music Journal 16(3) 33–47.
Chowning, J. and Bristow, D. (1986) FM theory
and applications for musicians. Yamaha Music
Foundation, Tokyo.
Di Scipio, A. and Prignano, I. (1996) Synthesis
by functional iterations. a revitalization of
nonstandard synthesis. Journal of New Music
Research 25(1) 31–46.
Iterative sound synthesis by means of cross-coupled digital oscillators
Nikolas Valsamakis is a composer and
Lecturer in the Music Technology and
Acoustics Department of the Technological
Educational Institute of Crete, Greece. He has
an MSc in Music Information Technology
from City University (London, UK) and is
currently studying for a PhD in Computer
Music at the University of Plymouth’s
School of Computing Communications and
Electronics (UK). His research interests are
live electroacoustic music composition and
performance, sound synthesis and complex
systems.
Eduardo Reck Miranda is a composer and
Reader in Artificial Intelligence and Music at
the University of Plymouth (UK), where he is
head of Computer Music Research. He has an
MSc in Music Technology from the University
of York (UK) and a PhD in Music from the
University of Edinburgh (UK). His research
interests are composition, sound synthesis,
new musical interfaces, artificial intelligence
and evolutionary computation. Dr Miranda is
a member of the editorial boards of Leonardo
Music Journal and Contemporary Music
Review, and is the regional editor for Latin
America of Organised Sound
Sound.
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Digital Creativity, Vol. 16, No. 2
Di Scipio, A. (2002) The synthesis of
environmental sound textures by iterated
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