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1. Aplications of Group Theory
in Granular Synthesis
Renato Fabbri, Adolfo Maia Jr.
Núcleo Interdisciplinar de Comunicação Sonora (NICS)
UNICAMP
SBCM 02/09/2007 1

2. Stimulus and Objective
How can we map geometric and symmetric
structures to the sonic ground?
SBCM 02/09/2007 2

3. Tools and Methods
●Representation of ● Sound Synthesis
symmetric/geometric technique
structures
Group Theory! Granular Synthesis!
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4. Group Theory(1) - Definition
if g1, g2 ∈ G, than g1 • g2 ∈ G
Groups are sets with a closed binary
operation satisfying the following three
properties:
1. The operation must be associative.
g1 • (g2 • g3) = (g1 • g2) • g3
2. There must be an identity element.
∃e∈G:g•e=e•g=g
3. Every element must have a
corresponding inverse element.
∀ g ∈ G, ∃ g-1 : g • g-1 = g-1 • g = e
SBCM 02/09/2007 4

5. Group Theory(2) - Symmetries
Group Theory is strongly related to the
study of symmetry in several areas of
mathematics as well as in physics, and
ARTS
4
3 3
2
4
2
4
5 3
60º
4
5 1
2
6 5
2
1
5
6 6
1
C6 S6
SBCM 02/09/2007 5

6. Permutation Groups
I. C6 and S6 are Permutation Groups.
II. Cayley's Theorem states that every group
is isomorphic to a Permutation Group.
∀ (G, *) ∃ (Gp, @), ∃ f: G → Gp ∀ u, v ∈ G :
f (u * v) = f (u) @ f (v)
Permutation Groups!
SBCM 02/09/2007 6

7. Permutations
● Used in western music at least since the
fourteenth-century.('talea and color' of Ars Nova)
J. S. Bach I. Xenakis K. Stockhausen A. Pärt
● Music of India
● Folk music of Africa.
SBCM 02/09/2007 7

8. Permutations - Change Ringing
● We can trace its origins back to
seventeenth-century.
● Consists of ringing a set of tuned bells in
mathematical patterns.
Plain Hunt Minimus
1 2 3 4 Position of the bell
2 1 4 3
2 4 1 3
4 2 3 1
4 3 2 1 Peal
3 4 1 2 etc...
3 1 4 2
1 3 2 4
1 2 3 4
Cycle
SBCM 02/09/2007 8

9. Groups and Permutations
We have Permutation Groups, whose elements are
permutations. But what is the connection between a
given set of permutations and group theory?
a = (1, 4, 3, 2) b = (2, 3)
For a given set S of permutations, there is a related
Group = { g | g = an * bm * co ...
a, b, c, ... ∈S, n, m, p, ... ∈N }
a * b = c = (1, 4, 3)
SBCM 02/09/2007 9

10. Granular Synthesis
“Granular synthesis [...] is based on the
production of a high density of small
acoustic events called 'grains' that are less
than 50 ms in duration and typically in the
range of 10-30 ms.”
- B. Truax in his website
SBCM 02/09/2007 10

11. FIGGS
Finite Groups in Granular Synthesis (FIGGS)
is the synthesis system that we developed.
●Open-source (free usage and development
and access to source code)
●Dedicated to Group Theory application on
Granular Synthesis, including Permutation
Groups
SBCM 02/09/2007 11

12. FIGGS - Development
●Python with WxPython, FloatSpin, NumPy,
PyAudioLab, Matplotlib
●SAGE (Software for Algebra and Geometry
Experimentation)
SBCM 02/09/2007 12

13. FIGGS – Current Version
On OFF
● Grain Input Panel ● The GS
● Group Action Panel Composition Panel
● Some Permutation
● Non Trapezoidal
Groups envelopes
● Regions of Actions
● Waveform Options
● Regions of played
● Pan/Reverberation
grains
SBCM 02/09/2007 13

14. FIGGS – Making Sounds(1)
1) Input parameters for each grain involved, as
well as the number of grains in an ordered
sequence
2) Specify which part of the sequence is going to
be played, and the number of cycles
3)Specify which parameters are going to be
permuted by groups
4) Choose groups to act, period of action, and on
which part of the ordered set
5) Command the sound to be written
SBCM 02/09/2007 14

15. FIGGS – Making Sounds(2)
Timbre Creation Musical Structure
(Duration + Separation < 50) (Duration + Separation > 100)
Granular Synthesis Melodic Patterns in
fixed scales
(1,3,4) on frequencies
freq2 freq3
freq4
freq4
freq1
fade3
freq5
freq1,
freq3, amp3
dur3
sep3
SBCM 02/09/2007 15

16. Sound Examples(1)
Set: 5 Grains
Played Set: Last 2 Grains
Permuted Parameter: Frequencies
By the Action of: a Symmetric Group
Permuted Set: All 5 grains
Grains Permuted Played Grains
Grains Permuted Played Grains
(Freqs)
SBCM 02/09/2007 16

17. Sound Examples(2)
Set: 30 Grains
Played Set: last 5 Grains
Permuted Parameter: Set Dependent
By the Action of: Set Dependent
Permuted Set: Set Dependent
Grains Permuted Played Grains
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18. Musical Example
● Reflexões Paradoxais (09:15)
● Texts by Fernando Pessoa
● ABA', A sections use FIGGS structure
SBCM 02/09/2007 18

19. ToDo
● The OFF list in “FIGGS – Current Version”
slide
● New ways for applications of permutation
groups (Composition)
● Find and apply systematic orderings in
which elements of a group acts on a
given set.
● Explore other related structures like
Grupoids and others
SBCM 02/09/2007 19

20. Conclusions
● FIGGS is dedicated to group actions in
audio, which can be very useful to
composers in electronic music
● It is an open source software
● Its interface is friendly
● Sounds created within current FIGGS
methods ranges from simple structures
to complex clouds , which were already
used musically.
SBCM 02/09/2007 20

21. Conclusions
● Sounds created within current FIGGS
methods ranges from clouds to melodies.
● Its usefulness as a compositional tool was
already verified in a musical piece.
● We created an open source software
dedicated to group actions in audio.
● This software can be a real exchange
medium of related musical concepts
between composers and other interested
people.
SBCM 02/09/2007 21

22. Contact
● renato.fabbri@gmail.com
● adolfo@nics.unicamp.br
● http://cortex.lems.brown.edu/~renato/son
ic-art/nics
● www.nics.unicamp.br
● www.nics.unicamp.br/renato_pessoal/
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