Music is composed algorithmically from a varying second order recurrence equation which defines sound frequency, x, at each step n knowing sound frequency and amplitude, y, of the previous step, n-1, as: x_n = c — ax_(n-1)~2 + by_(n-1), where the amplitude is given as y_(n-1)= f(x_(n-1),x_(n-2) with f an arbitrary function, and a, b, and c are also arbitrary ,constants to be chosen by the composer. This compositional model includes as special cases well-known recurrence equations as the logistic map, the Henon map, and the delayed Julia set. Another feature of our model is that the sound amplitude does not necessarily have the same dependence on previous frequencies in each step. In fact, by the use of discrete form boxcar functions it can change either in consecutive steps or after a finite number of steps. Compositional patterns are also incorporated by use of specific equations for the change of frequency. The effect of initial frequencies chosen is examined as well. Numerical examples are presented to show the chaotic behavior of our model. A final score is presented in musical notation.
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