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Fuzzkov 1

Fuzzkov 1. Fuzzy Granular Synthesis through Markov Chains. Eduardo Miranda (SoCCE-Univ. of Plymouth) Adolfo Maia Jr.(*) (NICS & IMECC –UNICAMP). (*) Supported by S ã o Paulo Science Research Foundation( FAPESP) / Brazil. Granular Synthesis and Analysis Very Short History. D. Gabor (1947)

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Fuzzkov 1

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  1. Fuzzkov 1 Fuzzy Granular Synthesis through Markov Chains Eduardo Miranda (SoCCE-Univ. of Plymouth) Adolfo Maia Jr.(*) (NICS & IMECC –UNICAMP) (*) Supported by São Paulo Science Research Foundation( FAPESP) / Brazil

  2. Granular Synthesis and AnalysisVery Short History • D. Gabor (1947) • Uncertainty Principle (Heisenberg) and Fourier Transforms • 2) I. Xenakis (~1960s) • Granulation of sounds (clouds) (tape) • 3) C. Roads (1978) • automated granular synthesis (computer) • 4) B. Truax (1988) • Real time granular synthesis (granulation) • 5) More recently: • R. Bencina  Audiomulch • E. Miranda  ChaosSynth • M. Norris  MagicFX • ………………..

  3. Microsound Gabor Cells Time Characteristic Cells ∆t Frequency ∆f Time-frequency Uncertainty Relation ∆t ∆f ≥1

  4. Fuzzy Sets (Zadeh – 1965) To model vagueness, inexact concepts Membership Function u Let A be a subset of a universe set Ω u: A→[0,1] , where 0≤u(x) ≤1, for all x in A Ex 1) Let be an arbitrary discrete set Ex 2) Let Ω = B(R) the sphere of radiusR Denote r=|x| u(x)= 1/r

  5. The Fuzzy Grain Matrix ωij = j-th frequency of the i-th grain aij = j-th amplitude of the i-th grain αij = membership value for the j-th Fourier Partial of the i-th grain

  6. Markov Chains 1) Sthocastic processes Random variables X(t) take values on a State Space S 2) Markov Process The actual state Xndepends only on the previous Xn-1 Transition Matrix P Probability Condition Probability Condition The Process

  7. Algorithm: Diagram for FuzzKov 1

  8. Parameters Input for FuzzKov 1 The Grain fs = sample frequency dur = duration of the grain r = number of Fourier partials grain_type = type of grain (1 -3) • The Markov Chain • N = number of states (grains) • n = number of steps of Markov Chain • v0 = initial vector Fuzzy Parameters alpha_type = type of vector (to generate the Membership Matrix) memb_type = type of Membership Matrix

  9. Walshing the Output • Walsh Functions are Retangular Functions • They form a basis for Continuous Functions • They can be represented by Hadamard Matrices H(n) • They can be used to sequencing grain streams 11111111 10101010 11001100 10011001 11110000 10100101 11000011 10010110 H(8) =

  10. Walshing Crickets Click to listen

  11. Future Research • Asynchronous Sequency • Modulation • Glissand Effects • New Probability Transitions for Markov Chain • Include Fuzzy Metrics • New applications of Walsh Functions and Hadamard Matrices

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