Appendix II - Creating sounds with MATLAB

WAv files have sampling frequency 0f 22050. If we read a WAV-file into a MATLAB vector Y via

Y=wavread('filename');

we can replay the sound with the command

sound(Y,22050);

provided the computer has a sound card.

This now allows us to create our own sounds. For example the sequence of commands

t=linspace(0,4,40000);
y=sin(400*2*pi*t);
sound(y,10000);

will create a sound of 400 Hz that lasts 4 seconds. We used a sampling rate of 10000 here, since this seems sufficient for a sound of 400 cycles per second, i.e. in each cycle the function $y=sin(800\pi t)$ is sampled 25 times.

As a little advanced exercise we will write a program that plays a C-major scale. To start we need to find the frequencies of the individual sounds. A middle A is usually used to tune instruments. That note has a frequency of 440 Hz. The base A has half the frequency, and there are twelve notes in the octave between a base A and a middle A. Similar the octave between a low C (the start of the C-major scale) and the high c its end contains twelve notes (including the so called half-tones).

$${\bf C}, {\bf C\char93 }, {\bf D}, {\bf D\char93 }, {\bf E}, {\bf... ...har93 }, {\bf G}, {\bf G\char93 }, {\bf A}, {\bf B}, {\bf B\char93 }, {\bf c}.$$

The frequencies $f_n$ and $f_{n+1}$ of two consecutive notes satisfy

$$f_{n+1}=C f_n$$

for some constant $C$. In other words, the frequencies form a geometric sequence not an algebraic (or linear) sequence. To find the value of $C$ we observe that

$$f_{n+12}=2f_n,$$

and therefore

$$C^{12}=2,$$

or

$$C=2^{1/12}=1.05946$$

This now allows us to compute all the frequencies for the notes from C to c. They are listed below.

Table 1: Approximate Frequencies of the notes between C and c in Hz

C	C#	D,	D#	E	F	F#	G	G#	A	B	B#	c.
262	277	294	311	330	349	370	392	415	440	466	493	523

In order to sound a C for 2 seconds, we write the following program:

t=linspace(0,2,20000);
y=sin(262*2*pi*t);
sound(y,10000);

and call it ``cee.m''. We can write similar programs for all other notes and then the C-major scale

pause;
cee;
pause;
dee;
pause;
ee;
pause;
eff;
pause;
gee;
pause;
aa;
pause;
bee;
pause;
hcee;

call it ``scale .m''. The pause commands are requests for key board input. Pressing any key will move the program to the next line. Typing

scale

will allow us to play a C-major scale by pressing any key to move to the next note. An obvious improvement would be to allow for different length of the notes.