Musical notes are complex
tones consisting of a fundamental frequency,
and higher harmonics (or partials) that are integer multiples of the
fundamental frequency (2f,
The animation below shows the composition of a complex tone. The waveform
of the fundamental is shown (in blue) together with the waveform of
the first four partials (in black) and the resultant complex tone (in
green). The green waveform represents the fluctuation of the acoustic
pressure with time produced at some location near a musical instrument. The relative amplitudes of the components are typical of the
structure of the sound pressure fluctuation produced by a musical
instrument. The mixture
of tones determines the timbre of a musical instrument. In our example,
the resultant pressure fluctuation, PR, is the sum of the fundamental, P1,
and the first four harmonics, P2, P3,
P4, P5, with appropriate
amplitudes. In the case illustrated:
P1 = sin
P2 = (1/2) sin 2ωt ,
P3 = (1/3) sin 3ωt ,
P4 = (1/4) sin 4ωt
, P5 = (1/5) sin 5ωt
ω = 2 π f
is the angular frequency. If we add more harmonics, the resultant will be
a better approximation to a saw-tooth wave.
The pitch of a note is partly related to the fundamental frequency of the
complex tone. We can now listen to a complex tone
made up of a fundamental at 147 Hz and four higher harmonics. In musical
language, this corresponds to the note D played in the first octave.
Let's now listen how a
complex tone is modified after removing successively its lowest partials.
Let's take the example of the note D. In the audio file, you can listen to
the first tone composed of the fundamental plus four harmonics.
If you click on the blue button above, this should play the tone
in a separate window.