lorenz — Implements the Lorenz system of equations.
Implements the Lorenz system of equations. The Lorenz system is a chaotic-dynamic system which was originally used to simulate the motion of a particle in convection currents and simplified weather systems. Small differences in initial conditions rapidly lead to diverging values. This is sometimes expressed as the butterfly effect. If a butterfly flaps its wings in Australia, it will have an effect on the weather in Alaska. This system is one of the milestones in the development of chaos theory. It is useful as a chaotic audio source or as a low frequency modulation source.
ix, iy, iz -- the initial coordinates of the particle.
iskip -- used to skip generated values. If iskip is set to 5, only every fifth value generated is output. This is useful in generating higher pitched tones.
iskipinit (optional, default=0) -- if non zero skip the initialisation of the filter. (New in Csound version 4.23f13 and 5.0)
ksv -- the Prandtl number or sigma
krv -- the Rayleigh number
kbv -- the ratio of the length and width of the box in which the convection currents are generated
kh -- the step size used in approximating the differential equation. This can be used to control the pitch of the systems. Values of .1-.001 are typical.
The equations are approximated as follows:
x = x + h*(s*(y - x))
y = y + h*(-x*z + r*x - y)
z = z + h*(x*y - b*z)
The historical values of these parameters are:
ks = 10
kr = 28
kb = 8/3
Here is an example of the lorenz opcode. It uses the file lorenz.csd.
Example 236. Example of the lorenz opcode.
See the sections Real-time Audio and Command Line Flags for more information on using command line flags.
<CsoundSynthesizer> <CsOptions> ; Select audio/midi flags here according to platform ; Audio out Audio in -odac -iadc ;;;RT audio I/O ; For Non-realtime ouput leave only the line below: ; -o lorenz.wav -W ;;; for file output any platform </CsOptions> <CsInstruments> ; Initialize the global variables. sr = 44100 kr = 44100 ksmps = 1 nchnls = 2 ; Instrument #1 - a lorenz system in 3D space. instr 1 ; Create a basic tone. kamp init 25000 kcps init 220 ifn = 1 asnd oscil kamp, kcps, ifn ; Figure out its X, Y, Z coordinates. ksv init 10 krv init 28 kbv init 2.667 kh init 0.0003 ix = 0.6 iy = 0.6 iz = 0.6 iskip = 1 ax1, ay1, az1 lorenz ksv, krv, kbv, kh, ix, iy, iz, iskip ; Place the basic tone within 3D space. kx downsamp ax1 ky downsamp ay1 kz downsamp az1 idist = 1 ift = 0 imode = 1 imdel = 1.018853416 iovr = 2 aw2, ax2, ay2, az2 spat3d asnd, kx, ky, kz, idist, \ ift, imode, imdel, iovr ; Convert the 3D sound to stereo. aleft = aw2 + ay2 aright = aw2 - ay2 outs aleft, aright endin </CsInstruments> <CsScore> ; Table #1 a sine wave. f 1 0 16384 10 1 ; Play Instrument #1 for 5 seconds. i 1 0 5 e </CsScore> </CsoundSynthesizer>