Version 14 of Making mathematical waves
Updated 2007-07-03 11:54:39 by theoverWhen making sounds, the most fundamental way to create samples which represent pure and well defined waves is to use mathematically defined waves, for instance by explicit formulea as a function of time or sample number, assuming the samples are from a function of time with frequency spectrum limited to the Niquist rate (half the sampling frequency). The latter is not completely true for the below example, but the spectrum is limited enough to give usefull results.
This is where Tcl comes in to make a usefull script which executes exactly the right commands to let us do that. Not a very big script (not counting bwise) but one which calls two major applications (maxima and gcc), and which does the essential packing of a fortan function in a suitable file, and finally automatically executes the compiles program, which writes a standard wav file (see snack) to play with some media player.
Maxima can be integrated with tcl in two main ways, see and . Here I use the easier programmable and memory friendly way to start up the executable from the tcl (bwise) script.
This is the main tcl script, which is combined with (sourced) bwise:
proc domaxima { {m} } {
set t "display2d:false;\n$m;"
return [string range [exec maxima << $t | tail -2 ] 6 end-7]
}
proc domaximafor { {m} } {
set t "display2d:false;\nlinel:3000;\nfortran($m);\n"
return [string range [exec maxima -q << $t ] 42 end-18]
}
proc formake { {e} } {
global f
set t [subst -nocommand {
subroutine sayhello(x,r)
real x,r
r = $e
return
end
}]
set f [open sub.f w]
puts $f [string trim $t \n]
close $f
exec gcc -ffixed-line-length-none -o fm sub.f mw.c wav.o -lm
#exec gfortran -ffixed-line-length-none -c sub.f
#exec gcc -o fm sub.o main.c -lm
return [exec ./fm]
} Of course maxima must be present on the system and reachable according to the PATH shell variable (like wish and tclsh). I've done this setup on Linux, which when set up right gives excellent maxima and compile times, with complicated formulas (see below) a sceonds long wav file is created in under a second (!). I guess this tcl/maxima/fortran/C setup is therefore usefull for general application, too.
You need to have these files in the current directory which contains the wav file and C loop part of the target program:
/* wav.c */
#include <stdio.h>
#include <fcntl.h>
#include <sys/stat.h>
#include <string.h>
#include <sys/types.h>
#include <stdlib.h>
/* #define CYGWIN */
int fd;
FILE *fp;
#define MSEC(t) (int)(t*44.1)
iwrite(fd,n,l)
int fd,l;
unsigned int n;
{
write(fd,&n,l);
}
int initwav(s,l) /* wav header, s=filename, l=#samples */
char *s;
int l;
{
#ifdef CYGWIN
fd = open(s,O_WRONLY|O_CREAT|O_TRUNC|O_BINARY,S_IRUSR|S_IWUSR|S_IRGRP);
#else
fd = open(s,O_WRONLY|O_CREAT|O_TRUNC,S_IRWXU);
#endif
if (fd < 0) return(-1);
write(fd,"RIFF",4);
iwrite(fd,(2*l+36),4);
write(fd,"WAVE",4);
write(fd,"fmt ",4);
iwrite(fd,(0x10),4);
iwrite(fd,((short) 0x01),2);
iwrite(fd,((short) 1),2); /* Mono */
iwrite(fd,(44100/1),4); /* Sample rate */
iwrite(fd,(2*44100/1),4);
iwrite(fd,((short) 2),2);
iwrite(fd,((short) 16),2);
write(fd,"data",4);
iwrite(fd,(2*l),4);
return(0);
}
void writewav(p,n)
short *p; /* Sample values */
int n; /* #samples */
{
int i;
for (i=0; i<n; i++)
write(fd,&p[i],2);
}
void closewav()
{
close(fd);
} /* mw.c */
/* This is file: main.c */
#include<stdio.h>
#include<math.h>
extern void sayhello_(float *, float *);
extern int initwav(char *,int);
extern int writewav(short *, int);
extern void closewav();
int main(argc, argv)
int argc;
char *argv[];
{
float in, out;
float x, start, stop, incr;
short s;
if (argc == 1) {
start = 0.0;
stop = 3.0;
incr = 1.0/44100.0;
if (initwav("math.wav",3*44100) != 0) return((int) -1);
for (x=start; x<(stop-incr/2); x+=incr) {
in = x;
sayhello_(&in,&out);
/* printf("%f %f\n", x, (float) out); */
s = (short) (32000*out);
writewav(&s,1);
}
closewav();
}
return((int) 0);
}Running the tcl script with a complicated formula (using commands, not BWise blocks):
formake [domaximafor {(sin(6.2831*110*x)*exp(-2*x)+(1/2)*sin(2*6.2831*110*x)*exp(-4*x)+(1/3)*sin(3*6.2831*110*x)*exp(-6*x)+(1/4)*sin(4*6.2831*110*x)*exp(-8*x)+(1/5)*sin(5*6.2831*110*x)*exp(-10*x)+(1/6)*sin(6*6.2831*110*x)*exp(-12*x)+(1/7)*sin(7*6.2831*110*x)*exp(-14*x)+(1/8)*sin(8*6.2831*110*x)*exp(-16*x)+(1/9)*sin(9*6.2831*110*x)*exp(-18*x))/(1+(1/2)+(1/3)+(1/4)+(1/5)+(1/6)+(1/7)+(1/8)+(1/9))}]generates this fortran file:
subroutine sayhello(x,r)
real x,r
r = 2.52E+3*(exp(-18*x)*sin(6.2202690000000002E+3*x)/9.0E+0+exp(-16*x)
1 *sin(5.5291279999999997E+3*x)/8.0E+0+exp(-14*x)*sin(4.837987000
2 0000001E+3*x)/7.0E+0+exp(-12*x)*sin(4.1468459999999995E+3*x)/6.
3 0E+0+exp(-10*x)*sin(3.4557050000000004E+3*x)/5.0E+0+exp(-8*x)*s
4 in(2.7645639999999999E+3*x)/4.0E+0+exp(-6*x)*sin(2.073422999999
5 9998E+3*x)/3.0E+0+exp(-4*x)*sin(1.3822819999999999E+3*x)/2.0E+0
6 +exp(-2*x)*sin(6.9114099999999996E+2*x))/7.129E+3
return
end
And this [L1 ] is the resulting wav file, its a 260 kilobyte 16 bit wav file.
Part of he formula in neater form: 
...
Finally, it is also possible to read the resulting wav file in a sampler or sample software and 'play' it with different keys on a (usb or midi) keyboard, which is great fun.
As a side remark: in Electrical Engineering and various other disciplines the above type of combined exponential / sine terms are very important and fundamental types of formulas which arise as the solution to important network theoretical linear but reactive circuits or systems.
Needless to say, because of the block facilities or bwise and general tcl/tk possibilities, a powerfull environment can be made.