Music Composition from the Cosine Law of a Frequency-Amplitude Triangle

We propose a frequency-amplitude triangle whose cosine law is used as a recurrence equation to algorithmically compose music The Mangle is defined by two sides and the angle between them, one of the sides is the sound frequency. x, at a step. the other side is the sound amplitude. y, at the same step. while the angle between them is a free parameter to be chosen by the composer The result me third side of the triangle is defined as the square root of the sound frequency. x, of the next step The sound amplitude. y, at each step depends. in general. on the amplitude and frequency of the same or previous step(s) in any way the composer chooses To test our proposition for bifurcation and chaos against known solutions, we consider in detail the case where the,amplitude depends only on the frequency of the same or some previous step as y(n) = b(x(n) (m))(lambda), with b being a free parameter, m either zero or a positive integer, and lambda a real number greater than or equal to zero If lambda is taken as zero. the resulting equation after a change of variables becomes the one parameter logistic recurrence equation From the known range of the parameter for bifurcation and chaos, we obtain the range of sal ties for sound amplitude and angle 0 for periodic and chaotic music For specific values. the frequencies corresponding to musical notes are obtained from the proposed triangular recurrence equation and musical chaotic scores ale composed and presented Other lambda's are also considered and their effect on the periodicity of the frequencies is examined