Lorenz7 ============================================================================================================== ============================================================================================================== LORENZ TIME SERIES Ignasi ?lvarez Garriga 1998 rev. 2005 ============================================================================================================== NOTE: this instrument's sound is heavilly influenced by the project's krate (ksmps). Should be set to 1, but feel free to experiment. ============================================================================================================== Uncomment the score i statements or use the score events on i/o !! ============================================================================================================== The original Lorenz system of equations is: Dx=A*(x-y) Dy=B*x-y-z*x Dz=x*y-C*z To build an audio instrument with it, we will use the Heun's integration method. A, B and C parameters will have realtime control using Csound i/o features. The Dt, or integration step, will have realtime control too. Due to sensibility to initial conditions, dense periodic points, and mixing properties, the Lorenz system reacts differently to N iterations with M size Dt, than to n*N iterations at size Dt/n (non-linearity). That is why we may need the "zoom" trick in adittion to variable length Dt steps. For convenience, the zoom parameter runs as an init P-parameter on .sco, as the loop algorithm, with its 'if' operator, would otherwise not accept it. I am listening to any suggestions in this point ... As enough processing power is available, we are supposed to be able to run more than one instance of this instrument using score statements: for the case in wich we run it simultaneously, the p7, p8, and p9 score parameters will allow to introduce a "chorus" effect naturally derivating from sensibility to initial conditions property of the system. SCORE PARAMETERS p4 = zoom (time we run the integration loop for each audio sample / las veces que corremos el bucle de integraci?n para cada muestra de audio ) p5=pan L ; Simple stereo image weighting / Sencillo balanceado de la im?gen est?reo p6=pan R p7 = diferential to kbparamA p8 = diferential to KbparamB p9 = diferential to KnparamC ____________________________________________________________________________________________________ El sistema de ecuaciones de Lorenz original es: Dx=A*(x-y) Dy=B*x-y-z*x Dz=x*y-C*z Para construir un instrumento de audio con ellas, utilizaremos la integraci?n por el m?todo de Heun. Los par?metros A, B y C tendr?n control a tiempo real utilizando las prestaciones de i/o de Csound. Tambi?n tendr? control a tiempo real el paso de integraci?n Dt. Debido sus propiedades de sensibilidad a las condiciones iniciales, densidad de puntos peri?dicos y mezcolanza, el sistema de Lorenz reacciona de modo distinto a N iteraciones con paso Dt de tama?o M, que a N*2 iteraciones con tama?o de paso Dt/2 (no-linealidad). Por esa raz?n se necesita el truco de "zoom" adem?s de la longitud variable en los pasos de Dt. Por conveniencia, el par?metro zoom es un P-par?metro incial en .sco, ya que el algoritmo del bucle, con su operador 'if', no lo aceptar?a de otro modo. Agradecer? sugerencias sobre este punto... A medida que dispongamos de suficiente capacidad de proceso, se supone que podremos ejecutar m?s de una instancia de este instrumento mediante las entradas en el score: para el caso en que se ejecuten simult?nneamente, los par?metros p7, p8 y p9 del score permiten introducir un efecto de "chorus" natural derivado de la propiedad de sensibilidad a las condiciones iniciales del sistema. PAR?METROS DE SCORE p4 = zoom ( numero de veces que corremos el bucle de integraci?n para cada muestra de audio ) p5=pan L ; Sencillo balanceado de la im?gen est?reo p6=pan R p7 = diferential to kbparamA p8 = diferential to KbparamB p9 = diferential to KnparamC ============================================================================================================== ============================================================================================================== gitbdeclip_<INSTR_ID> ftgen 0, 0, 16384, 7, 0, 16, 1, 16352, 1, 16, 0 idur = abs(p3) izoom = p4 iamp = p5 ipanL = sqrt(p6) ipanR = sqrt(1-p6) iparamA = p7 iparamB = p8 iparamC = p9 ka = <paramA> kb = <paramB> kc = <paramC> kdt = <dt> kdt2 = <dt>/2 kgainmaster = <gain> kgainx = <gainX> kgainy = <gainY> kgainz = <gainZ> ka = ka + iparamA kb = kb + iparamB kc = kc + iparamC ky init 1 kz init 1 kx init 1 kxo init 0 kyo init 0 kzo init 0 kynew init 0 kxnew init 0 kznew init 0 kynewo init 0 kxnewo init 0 kznewo init 0 kcont init 0 loop: ;sistema d'equacions kxnew=ka*(ky-kx) kynew=kb*kx-ky-kz*kx kznew=kx*ky-kc*kz kxo=kx kyo=ky kzo=kz kxnewo=kxnew kynewo=kynew kznewo=kznew kx=kxo+kxnewo*kdt ky=kyo+kynewo*kdt kz=kzo+kznewo*kdt ;sistema d'equacions kxnew=ka*(ky-kx) kynew=kb*kx-ky-kz*kx kznew=kx*ky-kc*kz kx=kxo+kdt2*(kxnew+kxnewo) ky=kyo+kdt2*(kynew+kynewo) kz=kzo+kdt2*(kznew+kznewo) kcont=kcont+1 if kcont!=izoom kgoto loop kcont=0 araw = kgainmaster*(kx*kgainx+ky*kgainy+kz*kgainz) ; declip aclip oscil iamp, 1/idur, gitbdeclip_<INSTR_ID> asig dcblock araw * aclip ; Optionally output to a Display object: ;outvalue "kdisplay", 0.5+kx/25 ;outvalue "kdisplax", 0.5-ky/25 blueMixerOut asig*ipanL,asig*ipanR paramA 26 16 true 1.0 10.0 8.1415 60 true 33 94 paramB 99 17 true 14.0 28.0 26.6504 60 true 106 95 paramC 171 16 true 0.0 1.0 0.8165 60 true 178 94 dt 305 24 true 2.0E-4 0.05 0.028063 60 true 328 102 gain 303 136 true 0.0 5000.0 888.5 60 false 305 215 gainX 30 143 true -1.0 1.0 -0.5148 60 true 42 223 gainY 102 143 true -1.0 1.0 0.4756 60 true 114 223 gainZ 172 144 true -1.0 1.0 0.9028 60 true 184 224