Mathematical Fundamentals

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Pitch of note as a function of string length

l [m]

h [Hz]

Figure 2: Pitch of a note as a function of string length

h=1./(2*l)*sqrt(t/r); % expression for pitch

plot(l,h);

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

xlabel('l [m]');

ylabel('h [Hz]');

% replot; % Octave only

In order to visualize functions of two variables, we can also use three-

dimensional representations. For example, the function (5) can be visualized

as in fig. 3 if the variables length and tension are defined over intervals and the

density is set to a constant. In such a representation, the function of two depen-

dent variables becomes a surface in 3D. The Octave/Matlab script for fig. 3 is
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l [m]

Pitch of note as a function of string length and tension

t [N]

h [Hz]

Figure 3: Pitch of a note as a function of string length and tension

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

mesh(l,t,h);

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