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# Aplications of Group Theory in Granular Synthesis (2007)

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### Aplications of Group Theory in Granular Synthesis (2007)

1. 1. Aplications of Group Theory in Granular Synthesis Renato Fabbri, Adolfo Maia Jr. Núcleo Interdisciplinar de Comunicação Sonora (NICS) UNICAMPSBCM 02/09/2007 1
2. 2. Stimulus and ObjectiveHow can we map geometric and symmetricstructures to the sonic ground?SBCM 02/09/2007 2
3. 3. Tools and Methods●Representation of ● Sound Synthesissymmetric/geometric techniquestructures Group Theory! Granular Synthesis!SBCM 02/09/2007 3
4. 4. Group Theory(1) - Definition if g1, g2 ∈ G, than g1 • g2 ∈ GGroups are sets with a closed binaryoperation satisfying the following threeproperties: 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 acorresponding inverse element. ∀ g ∈ G, ∃ g-1 : g • g-1 = g-1 • g = eSBCM 02/09/2007 4
5. 5. Group Theory(2) - SymmetriesGroup Theory is strongly related to thestudy of symmetry in several areas ofmathematics as well as in physics, andARTS 4 3 3 2 4 2 4 5 3 60º 4 5 1 2 6 5 2 1 5 6 6 1 C6 S6SBCM 02/09/2007 5
6. 6. Permutation GroupsI. C6 and S6 are Permutation Groups.II. Cayleys Theorem states that every groupis 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. 7. Permutations● Used in western music at least since thefourteenth-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. 8. Permutations - Change Ringing● We can trace its origins back toseventeenth-century.● Consists of ringing a set of tuned bells inmathematical 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 CycleSBCM 02/09/2007 8
9. 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. 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 websiteSBCM 02/09/2007 10
11. 11. FIGGS Finite Groups in Granular Synthesis (FIGGS) is the synthesis system that we developed.●Open-source (free usage and developmentand access to source code)●Dedicated to Group Theory application onGranular Synthesis, including PermutationGroupsSBCM 02/09/2007 11
12. 12. FIGGS - Development●Python with WxPython, FloatSpin, NumPy,PyAudioLab, Matplotlib●SAGE (Software for Algebra and GeometryExperimentation)SBCM 02/09/2007 12
13. 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 grainsSBCM 02/09/2007 13
14. 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 writtenSBCM 02/09/2007 14
15. 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 sep3SBCM 02/09/2007 15
16. 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. 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 GrainsSBCM 02/09/2007 17
18. 18. Musical Example ● Reflexões Paradoxais (09:15) ● Texts by Fernando Pessoa ● ABA, A sections use FIGGS structureSBCM 02/09/2007 18
19. 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 othersSBCM 02/09/2007 19
20. 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. 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. 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/SBCM 02/09/2007 22