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

2. Granular Synthesis and Analysis Very Short History 1)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 Frequency ∆t∆t ∆f∆f ∆t ∆f ≥1 Time-frequency Uncertainty Relation Characteristic Cells

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 Let be an arbitrary discrete set Ex 1) Ex 2) Let Ω = B(R) the sphere of radius R u(x)= 1/r Denote r=|x|

5. The Fuzzy Grain Matrix ω i j = j-th frequency of the i-th grain a i j = j-th amplitude of the i-th grain α i j = 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 X n depends only on the previous X n-1 Transition Matrix P Probability Condition The Process Probability Condition

7. Algorithm: Diagram for FuzzKov 1

8. Parameters Input for FuzzKov 1 The Markov Chain N = number of states (grains) n = number of steps of Markov Chain v 0 = initial vector The Grain fs = sample frequency dur = duration of the grain r = number of Fourier partials grain_type = type of grain (1 -3) Fuzzy Parameters alpha_type = type of vector (to generate the Membership Matrix) memb_type = type of Membership Matrix

9. Walshing the Output 1.Walsh Functions are Retangular Functions 2.They form a basis for Continuous Functions 3.They can be represented by Hadamard Matrices H(n) 4.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