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

Granular Synthesis and AnalysisVery 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 AudiomulchE. Miranda ChaosSynthM. Norris MagicFX………………..

MicrosoundGabor Cells

Time

Frequency

∆t

∆f

∆t ∆f ≥1

Time-frequency Uncertainty Relation

Characteristic Cells

Fuzzy Sets (Zadeh – 1965) To model vagueness, inexact concepts

Membership Function uLet 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 setEx 1)

Ex 2) Let Ω = B(R) the sphere of radius R

u(x)= 1/r

Denote r=|x|

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

Markov Chains1) Sthocastic processes Random variables X(t) take values on a State Space S2) Markov Process The actual state Xn depends only on the previous Xn-1

Transition Matrix PProbability Condition

The ProcessProbability Condition

Algorithm: Diagram for FuzzKov 1

Parameters Input for FuzzKov 1

The Markov Chain• N = number of states (grains)• n = number of steps of Markov Chain• v0 = initial vector

The Grainfs = sample frequencydur = duration of the grainr = number of Fourier partialsgrain_type = type of grain (1 -3)

Fuzzy Parametersalpha_type = type of vector (to generate the Membership Matrix)memb_type = type of Membership Matrix

Walshing the Output

1. Walsh Functions are Retangular Functions2. They form a basis for Continuous Functions3. They can be represented by Hadamard Matrices H(n) 4. They can be used to sequencing grain streams

1111111110101010110011001001100111110000101001011100001110010110

H(8) =

Walshing Crickets

Click to listen

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