118

D. Rocchesso: Sound Processing

multivariable function

contour plot

polynomials

coefficients

xlabel('l [m]');

ylabel('t [N]');

zlabel('h [Hz]');

% replot; % Octave only

Of a multivariable function we can also give the contour plot, i.e., the plot of

curves obtained for constant values of the dependent variable. For example, in

the function (5), if we let the dependent variable to take only seven prescribed

values, the cartesian plane of length and tension displays seven curves (see fig. 4).

Each curve corresponds to an horizontal cut of the surface of fig. 3.
1

2

3

4

800

900

1000

1100

1200

38.8

59

79.3

99.6

120

140

161

Pitch of note as a function of string length and tension

l [m]

t [N]

Figure 4: Contour plot of pitch as a function of string length and tension

The Octave/Matlab script producing fig. 4 is the following:
r=0.0367;

% definition of density

l=[0.5:0.1:4.0]; % domain for the string length

t=[800:10:1200]; % domain for the string tension

h=(1./(2*l')*sqrt(t./r))'; % expression for pitch

% contour(h', 7, l, t); % Octave only

co=contour(l, t, h, 7); % Matlab only

clabel(co);

% Matlab only

title('Pitch of note as a function of string length and tension');

xlabel('l [m]');

ylabel('t [N]');

zlabel('h [Hz]');

A.3

Polynomials

An important class of one-variable functions is the class of polynomials, which

are weighted sums of non-negative powers of the independent variable. Each

power with its coefficient is called a monomial. A polynomial has the form

y = f (x) = a

0

+ a

1

x + a

2

x

2

+ . . . + a

n

x

n

,

(6)

where the numbers a

i

are called coefficients and, for the moment, they can be

considered as real numbers. The highest power that appears in (6) is called the

order of the polynomial.