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Sonic Explorations of Gumowski-Mira Maps

Abstract and Figures

This paper studies the use of Gumowski-­Mira maps for sonic arts. Gumowski-­Mira maps are a set of chaotic systems that produce many organic orbits that resemble cells, flowers and other life forms. This has prompted mathematicians and eventually artists to study them. These maps carry a potential for use in the sonic arts, but until now such use is non-­existent. The paper describes two ways of using Gumowski-­Mira maps: for synthesis and spatialization. The synthesis approach, which runs in real-­time, takes the dynamical system output as the real and imagi-­ nary input to an inverse Fourier transformation, thus directly sonifying the algorithm. The spatialization approach uses the shapes of Gumowski-­Mira maps as shapes across the acoustic space, using the first 128 iterations of each map as audio particles. The shapes can change based on the maps' initial parameters. The maps are explored in live performance using Leap Motion and Cycling '74's MIRA for iPad as control interfaces of audio processing in SuperCollider. Examples are given in two works, Cells #1 and #2.
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195
HEARING THE SELF
Sonic Explorations of Gumowski-Mira Maps
Timothy S. H. Tan PerMagnus Lindborg
Institute of Sonology
The Royal Conservatoire, The Hague
timbretan@gmail.com
School of Art, Design and Media
Nanyang Technological University
permagnus@ntu.edu.sg
ABSTRACT
This paper studies the use of Gumowski-Mira maps for
sonic arts. Gumowski-Mira maps are a set of chaotic sys-
tems that produce many organic orbits that resemble cells,
flowers and other life forms. This has prompted mathema-
ticians and eventually artists to study them. These maps
carry a potential for use in the sonic arts, but until now
such use is non-existent. The paper describes two ways of
using Gumowski-Mira maps: for synthesis and spatializa-
tion. The synthesis approach, which runs in real-time,
takes the dynamical system output as the real and imagi-
nary input to an inverse Fourier transformation, thus di-
rectly sonifying the algorithm. The spatialization ap-
proach uses the shapes of Gumowski-Mira maps as shapes
across the acoustic space, using the first 128 iterations of
each map as audio particles. The shapes can change based
on the maps’ initial parameters. The maps are explored in
live performance using Leap Motion and Cycling ’74’s
MIRA for iPad as control interfaces of audio processing in
SuperCollider. Examples are given in two works, Cells #1
and #2.
1. INTRODUCTION
The Gumowski-Mira map (henceforth called GM map)
was named after I. Gumowski and C. Mira, who were stud-
ying various chaotic maps from the 1960s [1]. Around
1970, while at CERN, Gumowski was studying chaotic in-
stability in accelerators and storage rings, and collaborated
with the Toulouse Research Group led by Mira [1, pp. 129-
131]. What is now known as the GM map was based on a
dissipative perturbation of one such map studied by
Gumowski and then with Mira [1, pp. 121-124]. It com-
prises the following equations:
     
 
   (1)
There are a few variants of GM maps. According to [1,
pp. 180-184], two different F(x) exist:
     

(2)
  
 (3)
The equation (1) also appears in a slightly different form
in [1, pp. 179], with F(x) is similar to (3):
    
 
 
  

(4)
  
 (5)
These variations can produce similar shapes, albeit with
slightly different ranges of usable initial parameters.
GM maps have some benefits not found in other chaotic
systems. Across the phase space, GM maps can produce
intricate, organic images that resemble cells, flowers and
other life-forms. These images can vary a lot in shapes, and
often possess different rotational or reflection symmetries.
Whereas a number of mathematicians [2] and artists [3]
have studied GM maps in detail, until now nobody has
used these maps as sonic parameters. Besides, GM maps
are still less well-known than other chaotic systems such
as the logistic map, Chua circuit, Rössler attractor and
double pendula. After all, chaotic systems have been used
to map onto frequencies [4, 5], rhythm and note durations
[4], dynamic levels (velocities) [5], timbre [6, 7], grain fre-
quencies [6] and grain lengths [7]. Even SuperCollider [8]
has many chaotic UGens that generate sounds based on
audio-rate calculations of chaotic systems, such as
LinCongN and GbmanTrig.
This article presents how the two authors have used GM
maps for synthesis and spatialization, with their methodol-
ogies, results and discussions.
2. GM MAPS FOR SYNTHESIS
Lindborg developed a program in Max [9] for concurrent
real-time synthesis and visualization of a GM map. The
state of the GM map is updated for each audio sample (Fig-
ure 1) and its output (x, y) is mapped to the real and imag-
inary input of an inverse Fourier transform (Figure 2).
Used as an experimental audio synthesizer, the output
has been employed in two sonic artworks in the past year
[10] [11].
As pointed out by Dean et al. [12], the concurrent audio
visual output of the system correspond fully on a synthesis
level but this does not necessarily mean that they corre-
spond perceptually. Future work aims to evaluate experi-
mentally the extent to which auditory and visual modalities
of the outputs correspond.
Copyright: © 2017 Timothy S. H. Tan and PerMagnus Lindborg. This is
an open-access article distributed under the terms of the
Creative
Commons Attribution License 3.0 Unported
, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original
author and source are credited.
196 2017 ICMC/EMW
Figure 1. Max implementation of the dynamical system
feedback.
Figure 2. Max implementation of the GM map’s two-di-
mensional output as input to an inverse Fourier transform.
3. GM MAPS FOR SPATIALIZATION
To date, chaos has been applied to spatialization in [13]
and fractals in [14], but for algorithmic spatialization, there
has been some progress. Some have implemented algorith-
mic spatialization with OpenMusic, as in [14]–[16],
whereas Schacher et al. have composed with swarm algo-
rithms and flocking, both controlling spatialization among
other parameters [17]. Lindborg has utilized particle colli-
sions registered in bubble chamber images [18] as well as
the collision of particles [19] for algorithmic spatialization.
Using GM maps for spatialization can prove interesting
and innovative. The numerous patterns of GM maps tend
to be highly distinctive and thus can provide equally dis-
tinctive spatial shapes across the acoustic space. The
shapes also change unpredictably and very quickly even if
there are tiny changes to the initial parameters. For Tan,
this meant a vast choreography of spatial shapes. In fact,
Deleuze and Guattari, while discussing philosophy, had
even reflected on the speed at which these shapes change:
Chaos is not so much defined by its disorder
as by the infinite speed with which every
form taking shape in it vanishes. It is a void
that is not a nothingness but a virtual, con-
taining all possible particles and drawing out
all possible forms, which spring up only to
disappear immediately, without consistency
or reference, without consequence. Chaos is
an infinite speed of birth and disappearance
[20].
Their last point makes spatializing chaotic maps like GM
maps a highly efficient tool in building up tension and in-
tensity, furthermore engaging listeners through a given
acoustic space. They also easily broach the “conflict / co-
existence” behavioral relationship addressed by Smalley
[21]: Tiny changes to initial parameters create small errors
that accumulate and later cause great deviations to the or-
bits. These orbits can conflict and even tangle each other.
Moreover, using chaotic maps for spatialization is more
robust than for other parameters. When other parameters
rely on streams of numerical outputs, the heard results of-
ten correspond poorly with the original map used. For in-
stance, the timbres of the gingerbread man map in
GbmanTrig from SuperCollider may sound interesting, but
listeners cannot identify its timbre specifically with
GbmanTrig. Besides, scaling, rotation, translation and
skewing of these maps easily preserves their visual percep-
tion, but distorts their listening experience when heard as
streams of output. On the other hand, spatialization of cha-
otic maps rely on graphical representations of these maps,
thereby preserving the shapes across the acoustic space.
Because GM maps can create captivating visual shapes,
Tan has focused on recreating these shapes through spati-
alization of GM maps in the acoustic space.
3.1 Implementation
When chaotic maps exhibit their sensitivity to initial con-
ditions, they imply the need to compare between close
neighboring values of initial parameters. Plotting the di-
vergent orbits one iteration at a time inadequately provides
a fuller picture of these maps, and worse still when the pre-
vious iterations disappear as the sound travels along the
orbit. This makes the implication of comparison too long
and weak, thereby poorly demonstrating sensitivity to ini-
tial conditions. Instead, Tan uses all the iterations of each
GM map as simultaneous audio particles for the acoustic
space, similar to Fonseca’s audio particle system in his
“Sound Particles” software [22] [23]. These iterations are
updated instantaneously whenever the initial parameters
change. Both methods easily highlight the comparisons
and preserve the picture of the maps, and thus the maps
can continue to demonstrate sensitivity to initial condi-
tions.
One challenge is the need to balance between providing
an adequate image of the GM map and minimizing CPU
processing usage. As such, only 128 iterations per map are
used as audio particles, and only three GM maps are used.
Too many particles can overload the CPU and disrupt real-
life performances involving GM maps.
Additionally, whenever the GM maps change their
shapes, the maps’ particles skip directly to their new posi-
tions, without the need to consider the Doppler Effect. Un-
like a particle system in which a particle can have a life of
its own, here the audio particles live and die together
whenever their respective maps are switched on or off.
A visualizer of the particle systems ensures that the per-
former can see where the particles are; such a feature also
doubles as visuals for the audience to observe the particles’
positions and orbits. The size of the speaker space, which
197
HEARING THE SELF
Figure 1. Max implementation of the dynamical system
feedback.
Figure 2. Max implementation of the GM map’s two-di-
mensional output as input to an inverse Fourier transform.
3. GM MAPS FOR SPATIALIZATION
To date, chaos has been applied to spatialization in [13]
and fractals in [14], but for algorithmic spatialization, there
has been some progress. Some have implemented algorith-
mic spatialization with OpenMusic, as in [14]–[16],
whereas Schacher et al. have composed with swarm algo-
rithms and flocking, both controlling spatialization among
other parameters [17]. Lindborg has utilized particle colli-
sions registered in bubble chamber images [18] as well as
the collision of particles [19] for algorithmic spatialization.
Using GM maps for spatialization can prove interesting
and innovative. The numerous patterns of GM maps tend
to be highly distinctive and thus can provide equally dis-
tinctive spatial shapes across the acoustic space. The
shapes also change unpredictably and very quickly even if
there are tiny changes to the initial parameters. For Tan,
this meant a vast choreography of spatial shapes. In fact,
Deleuze and Guattari, while discussing philosophy, had
even reflected on the speed at which these shapes change:
Chaos is not so much defined by its disorder
as by the infinite speed with which every
form taking shape in it vanishes. It is a void
that is not a nothingness but a virtual, con-
taining all possible particles and drawing out
all possible forms, which spring up only to
disappear immediately, without consistency
or reference, without consequence. Chaos is
an infinite speed of birth and disappearance
[20].
Their last point makes spatializing chaotic maps like GM
maps a highly efficient tool in building up tension and in-
tensity, furthermore engaging listeners through a given
acoustic space. They also easily broach the “conflict / co-
existence” behavioral relationship addressed by Smalley
[21]: Tiny changes to initial parameters create small errors
that accumulate and later cause great deviations to the or-
bits. These orbits can conflict and even tangle each other.
Moreover, using chaotic maps for spatialization is more
robust than for other parameters. When other parameters
rely on streams of numerical outputs, the heard results of-
ten correspond poorly with the original map used. For in-
stance, the timbres of the gingerbread man map in
GbmanTrig from SuperCollider may sound interesting, but
listeners cannot identify its timbre specifically with
GbmanTrig. Besides, scaling, rotation, translation and
skewing of these maps easily preserves their visual percep-
tion, but distorts their listening experience when heard as
streams of output. On the other hand, spatialization of cha-
otic maps rely on graphical representations of these maps,
thereby preserving the shapes across the acoustic space.
Because GM maps can create captivating visual shapes,
Tan has focused on recreating these shapes through spati-
alization of GM maps in the acoustic space.
3.1 Implementation
When chaotic maps exhibit their sensitivity to initial con-
ditions, they imply the need to compare between close
neighboring values of initial parameters. Plotting the di-
vergent orbits one iteration at a time inadequately provides
a fuller picture of these maps, and worse still when the pre-
vious iterations disappear as the sound travels along the
orbit. This makes the implication of comparison too long
and weak, thereby poorly demonstrating sensitivity to ini-
tial conditions. Instead, Tan uses all the iterations of each
GM map as simultaneous audio particles for the acoustic
space, similar to Fonseca’s audio particle system in his
“Sound Particles” software [22] [23]. These iterations are
updated instantaneously whenever the initial parameters
change. Both methods easily highlight the comparisons
and preserve the picture of the maps, and thus the maps
can continue to demonstrate sensitivity to initial condi-
tions.
One challenge is the need to balance between providing
an adequate image of the GM map and minimizing CPU
processing usage. As such, only 128 iterations per map are
used as audio particles, and only three GM maps are used.
Too many particles can overload the CPU and disrupt real-
life performances involving GM maps.
Additionally, whenever the GM maps change their
shapes, the maps’ particles skip directly to their new posi-
tions, without the need to consider the Doppler Effect. Un-
like a particle system in which a particle can have a life of
its own, here the audio particles live and die together
whenever their respective maps are switched on or off.
A visualizer of the particle systems ensures that the per-
former can see where the particles are; such a feature also
doubles as visuals for the audience to observe the particles’
positions and orbits. The size of the speaker space, which
covers the audiences in a concert hall, is similar to the size
of the visualizer, so as to maintain close correspondence
between audio and visuals. Audio particles can exceed the
boundaries of the visualizer and sound behind the speakers
(often more softly), so as not to overcrowd within the
speaker space. This allows better localization of the parti-
cles (Figure 3).
Figure 3. The visualizer of the particles’ positions based
on the three GM maps. Dashed square lines at the center
indicate the size of the speaker space, which in real life sur-
rounds the audiences in a concert hall.
Not too many particles are allowed to crowd within the
speaker space, so as to make the spatial shape of the GM
maps clearer. Other than using three GM maps with a max-
imum of 128 audio particles per map, second-order 2D
Ambisonics is used. The spatial image of the GM maps can
be zoomed in and out, so as to reveal the intricate inner
shapes of the maps. The speaker setup, which surrounds
the audiences, is octophonic. This is used along with the
visualizer mentioned just above.
In order to differentiate which particles belong to which
GM map, each GM map has a different synth and thus tim-
bre. This is to prevent all the audio particles from different
GM maps from blending together as if they are generated
from the same map. Normally one map contains some ad-
ditive synthesis UGens, another some ChaosGens, and the
last granular synthesis. In turn, their particles have varying
parameters based not on their iteration index, but their po-
sitions in space. The particles’ positions can affect the par-
ticles’ frequencies and other audio effects. This is im-
portant in varying the sounds of each particle, because if
the same sound is applied to every particle in a particle
system, it sounds almost similar to that sound used in one
particle whose size covers a far larger space (e.g. directly
from one speaker), thereby making the particle system use-
less. For the visualizer, each GM map is assigned a color,
as a visual aid to differentiate the GM maps.
The form is based on the rate of change of the initial pa-
rameters. The music starts slowly, exploring and scrutiniz-
ing the different spatial shapes of each map, before speed-
ing up from the middle. Often the shapes are repeated at
the start, so that listeners can realize that they are not lis-
tening to a random system, but a chaotic system of unpre-
dictable, but specific shapes.
Each GM map can be switched on or off. Usually only
one GM map starts the work, then is joined by the second,
and finally the third.
The range of iterations is confined within the first 128
iterations. This can be narrowed, so as to scrutinize the
movements of that number of particles. With modulo op-
eration, the iteration index can be filtered with divisors of
1-7 and remainders of 0-6. At times where the GM maps
often look similar, this modulo filter reveals that the same
iteration index of each map occupies very different places
(Figure 4). As such, the modulo filter is often used to play
with the spatiality of the GM maps.
Figure 4. An instance of three GM maps played together
in Cells #2. Left: without modulo filter; right: with modulo
filter of divisor 4 and remainder 2.
The GM maps are performed with MIRA on iPad (with
Max 7) by Cycling ’74 [9], as well as Leap Motion [24]
(with Processing 3 [25]). These two interfaces enable the
performer to play with speed: MIRA for slow changes, and
Leap Motion for fast changes. Leap Motion communicates
with MIRA via Open Sound Control (OSC), and likewise
for MIRA with SuperCollider.
Leap Motion is very sensitive to the performer’s hand
positions. This fits well for performing chaotic works,
since chaotic systems are also sensitive to initial condi-
tions, and accuracy is not so important in this context. This
enables the resultant shapes to transform quickly, but not
slowly, because it is difficult to control hand positions un-
der Leap Motion’s high sensitivity. The hand parameters
are chosen and mapped based on ease and comfort. Only
the ranges of α = [-0.25, 0.25], σ = [-0.25, 0.25], μ = [-1.0,
1.0], x0 = [-2.0, 2.0] and y0 = [-2.0, 2.0] are used as the
initial parameters, as in these ranges, generally the maps
are conservative, do not enter infinity too soon, and do not
settle into a stable attractor at the origin within the first 128
iterations. The last feature tends to create an unwanted cen-
tral ringing tone that overwhelms other audio particles.
In contrast, MIRA allows the performer to play with one
or two parameters at a time on the iPad. This mode allows
better accuracy and slower changes than the Leap Motion
mode. One can switch between the MIRA mode and the
Leap Motion mode, in which only one of the two can affect
the parameters. For both cases, MIRA will track the
changes in the parameters as a visual feedback to the per-
former. (MIRA also tracks the parameters changed by
Leap Motion through OSC.)
198 2017 ICMC/EMW
Figure 5. Three GM maps for Cells #1. Top: (1) with
(2) for F(x), middle: (1) with (3), bottom: (4) with (5).
α = -0.03125, σ = -0.125, μ = -0.765625, x0 and y0 = 0.25,
and n = 128. Ranges of both axes are [-12.5, 12.5].
Figure 6. Three GM maps for Cells #2, using (1) with (3)
for F(x), α = -0.03125, σ = -0.125, x0 and y0 = 0.25, and
n = 128, but μtop = -0.765625, μmiddle = μtop + 2-12 and
μbottom = μtop + 2-11. Ranges of both axes are [-12.5, 12.5].
199
HEARING THE SELF
Figure 5. Three GM maps for Cells #1. Top: (1) with
(2) for F(x), middle: (1) with (3), bottom: (4) with (5).
α = -0.03125, σ = -0.125, μ = -0.765625, x0 and y0 = 0.25,
and n = 128. Ranges of both axes are [-12.5, 12.5].
Figure 6. Three GM maps for Cells #2, using (1) with (3)
for F(x), α = -0.03125, σ = -0.125, x0 and y0 = 0.25, and
n = 128, but μtop = -0.765625, μmiddle = μtop + 2-12 and
μbottom = μtop + 2-11. Ranges of both axes are [-12.5, 12.5].
3.2 Cells #1 and Cells #2
Tan composed Cells #1 (2016, rev. 2017) and Cells #2
(2017) with Leap Motion and Max with MIRA for perfor-
mance interface, and SuperCollider 3.8.0 for audio pro-
cessing. Both works are to be performed live for a two-
dimensional 8.1 speaker setup. Usually visuals of the GM
maps are projected to a large screen in front of the audi-
ences, because GM maps remain unknown to many.
Cells #1 involves three GM maps, formed by coupling (2)
and (3) with (1), and (5) with (4), and using the first 128
iterations (Figure 5). Tan wants to study how the shapes
will look and sound if the same set of initial parameters to
these three maps is applied. Cells #2 also involves three
GM maps. This time, all three use the same equation (1),
with (3) for F(x), but the first and the second had μ that
differs by 2-12, and likewise for the second and third (Fig-
ure 6). In both works, all these maps exist alongside each
other and contest for spatial prominence and likeness and
differences on their orbits in the same acoustic space. Both
works are meant for an octaphonic speaker setup with sec-
ond-order Ambisonics, for greater spatial clarity.
3.3 Artistic Results
The spatial shapes can generally be identified, especially
by applying the modulo filter, but more work is needed to
make spatial shapes even clearer. The modulo filter and the
limit on the range of iterations work best for Cells #2, es-
pecially when the maps often overlap each other but the
particles with the same iteration indices are at different or
even opposite places. Timbre-wise, the chosen timbres for
each GM map remain distinct and do not blend, and this is
required. One listener had even highlighted the timbral va-
riety of Cells #2. Additionally, some listeners had found
the performance with MIRA and Leap Motion very engag-
ing, and noted the close correspondence between the
sounds and visuals. Meanwhile, Cells #1, originally com-
posed in 2016, is now being revised to be on par with Cells
#2.
3.4 Discussion
Chaotic maps, such as GM maps, can be a powerful tool
for spatialization. However, it can be difficult trying to rep-
licate the success of visualizing GM maps (or other chaotic
maps) across the audio spatial realm, when creative moti-
vations conflict with technical constraints.
Attempts to produce complete shapes of GM maps au-
rally with a lot of audio particles will cause overcrowding
inside the listening space, thus blurring the choreography
of the spatial shapes. As such, just the first 128 iterations
for one map are already adequate enough as audio parti-
cles. Besides, having a lot of particles enhances and does
not blur the visual experience, but blurs and distorts the
listening experience. While visual particles do not diffuse
light themselves and blur their own shapes, audio particles
can at best only represent the ideal visual particle. Breg-
man elaborates that:
This way of using sound has the effect, how-
ever, of making acoustic events transparent;
they do not occlude energy from what lies
behind them. The auditory world is like the
visual world would be if all objects were
very, very transparent and glowed in sput-
ters and starts by their own light, as well as
reflecting the light of their neighbors. This
would be hard world for the visual system to
deal with [26].
Despite efforts to reproduce the shapes of the GM maps
as faithfully as possible, the acoustic space has some re-
strictions. One implication of Bregman’s is that while
anyone can replicate the directionality, scale and distribu-
tion of particles across the space, the sharpness in the edges
and corners remains poorly defined. This can easily blur
the shapes’ intricate designs and corners, as well as cover
the holes in between the particles. As such, having thou-
sands of audio particles like of visual particles do not en-
hance the listening experience similarly to the visual expe-
rience. By this approach, one can increase the number of
speakers for better spatial definition and clarity, though
perhaps not so cost-effectively.
One drawback of performing with real-time audio parti-
cle systems is that as the number of particles increase, both
the demand for audio processing power and the chance of
a breakdown also increase. Whereas visual implementa-
tions of GM maps tend to involve hundreds or even thou-
sands of particles, doing likewise for audio easily over-
loads the CPU and thus handicaps real-life performances
of the works. Currently Fonseca’s “Sound Particles” soft-
ware still cannot perform real-time rendering and play-
back. Cells #2 went smoothly during rehearsals and sound-
check, but halfway through the actual performance at the
Institute of Sonology, crashed just after the three GM maps
have been introduced. This was likely due to the audio in-
terface (Focusrite Scarlett 18i20) unable to handle Super-
Collider’s immense processing power, thereby causing the
audio interface to disconnect from SuperCollider. The per-
formance had to be aborted. Work is underway reducing
CPU usage in one laptop, perhaps by splitting the pro-
cessing power into more than one laptop and inviting one
extra performer, for both Cells.
While tempo is used to build form, listeners often per-
ceive fast changes in spatial shapes as random instead of
being chaotic. These intricate shapes are hence no longer
perceived as specific to GM maps, even if they indeed are.
As such, the tempo will be reduced across both Cells. Cer-
tain shapes may be paused, so that the listeners can per-
ceive these shapes fully.
Another problem is the need to adjust the amplitudes of
all the audio particles, since audio particles can cancel each
other a bit, possibly due to phasing. This occurs especially
with larger number of audio particles. As such, the sum of
the audio particles can sound less loudly than the ideal to-
tal, and needs correction.
200 2017 ICMC/EMW
4. CONCLUSION
Using GM maps for sonification remains a promising sub-
ject to explore. GM maps’ captivating shapes have in-
spired both authors to sonify them. Tan makes all the first
128 iterations of each GM map sound together as a spatial
shape to play with spatialization in Cells #1 and #2,
whereas Lindborg uses audio synthesis with GM maps for
two sound installations. More research is needed for other
effective sonifications of GM maps, plus synthesis and
spatialization with other chaotic maps.
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