Aplications of Group Theory in Granular Synthesis (2007)
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Transcript of Aplications of Group Theory in Granular Synthesis (2007)
Aplications of Group Theory in Granular Synthesis
Renato Fabbri, Adolfo Maia Jr.
Ncleo Interdisciplinar de Comunicao Sonora (NICS)UNICAMP
SBCM
02/09/2007
1
Stimulus and Objective
How can we map geometric and symmetric structures to the sonic ground?
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02/09/2007
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primeira figura:http://www.sciencemusings.com/blog/blogarchive/2007_03_01_blogarchive.html
segunda figura:http://www.flickr.com/photos/luciddrifter/5506452/segunda figura: http://www.stefangeens.com/2005_03.html
Tools and Methods
Representation of symmetric/geometric structures
Sound Synthesis technique
Group Theory!
Granular Synthesis!
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02/09/2007
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Group Theory(1) - Definition
Groups are sets with a closed binary operation satisfying the following three properties:1. The operation must be associative. 2. There must be an identity element.
3. Every element must have a corresponding inverse element.
e G : g e = e g = g
g G, g-1 : g g-1 = g-1 g = e
if g1, g2 G, than g1 g2 G
g1 (g2 g3) = (g1 g2) g3
SBCM
02/09/2007
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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
6
2
1
5
4
3
6
2
1
5
4
3
60
6
2
1
5
4
3
2
5
4
C6
S6
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02/09/2007
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Permutation Groups
C6 and S6 are Permutation Groups.
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!
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02/09/2007
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Permutations
Used in western music at least since the fourteenth-century.
Music of India
Folk music of Africa.
('talea and color' of Ars Nova)
J. S. Bach
I. Xenakis
K. Stockhausen
A. Prt
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02/09/2007
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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
Peal
Cycle
Position of the bell
etc...
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02/09/2007
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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)
a * b = c = (1, 4, 3)
For a given set S of permutations, there is a relatedGroup = { g | g = an * bm * co ... a, b, c, ... S, n, m, p, ... N }
SBCM
02/09/2007
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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
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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
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FIGGS - Development
Python with WxPython, FloatSpin, NumPy, PyAudioLab, Matplotlib
SAGE (Software for Algebra and Geometry Experimentation)
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02/09/2007
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FIGGS Current Version
On
Grain Input Panel
Group Action Panel
Some Permutation Groups
Regions of Actions
Regions of played grains
OFF
The GS Composition Panel
Non Trapezoidal envelopes
Waveform Options
Pan/Reverberation
SBCM
02/09/2007
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FIGGS Making Sounds(1)
Input parameters for each grain involved, as well as the number of grains in an ordered sequence
Specify which part of the sequence is going to be played, and the number of cycles
Specify which parameters are going to be permuted by groups
Choose groups to act, period of action, and on which part of the ordered set
Command the sound to be written
SBCM
02/09/2007
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FIGGS Making Sounds(2)
Timbre Creation
(Duration + Separation < 50)
Granular Synthesis
Musical Structure
(Duration + Separation > 100)
Melodic Patterns in fixed scales
dur3
sep3
freq3, amp3
fade3
freq1
freq2
freq4
freq5
(1,3,4) on frequencies
freq1, amp3
freq3
freq4
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02/09/2007
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Sound Examples(1)
Played Grains
Grains Permuted
Set: 5 Grains
Permuted Parameter: Frequencies
Permuted Set: All 5 grains
Played Set: Last 2 Grains
Played Grains
Grains Permuted
(Freqs)
By the Action of: a Symmetric Group
SBCM
02/09/2007
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Sound Examples(2)
Set: 30 Grains
Permuted Parameter: Set Dependent
Permuted Set: Set Dependent
Played Set: last 5 Grains
By the Action of: Set Dependent
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02/09/2007
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Played Grains
Grains Permuted
Musical Example
Reflexes Paradoxais (09:15)
Texts by Fernando Pessoa
ABA', A sections use FIGGS structure
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02/09/2007
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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
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02/09/2007
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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
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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
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Contact
http://cortex.lems.brown.edu/~renato/sonic-art/nics
www.nics.unicamp.br
www.nics.unicamp.br/renato_pessoal/
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02/09/2007
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