Digital Signal Processing Reference
In-Depth Information
Chapter 3
The Uncertainty Principle
Musical notes have something to do with the simultaneous presentation of the time and
frequency domains as they are to be found in the three-dimensional Illustration 28 ff
(Chapter 2) of periodic signals. The height of the notes on the lines of the score gives the
pitch of the tones; that is in the final analysis the frequency. The form of the notes gives
their duration in time. Notes are written by composers as if the pitch and length could be
determined quite independently of each other. Experienced composers have, however,
long been aware of the fact that, for example, the low notes of an organ or a tuba must last
a certain time in order to be felt to be sonorous. Sequences of such low notes can therefore
only be played at reduced speed.
A strange relationship between frequency and time
and its practical consequences.
It is one of the most important insights of oscillation, wave and modern quantum physics
that certain quantities - such as frequency and time - cannot be measured independently
of each other. Such quantities are termed complementary.
Illustration 44:
Simultaneous representation of the time and frequency domain in musical scores.
Norbert Wiener, the world famous mathematician and founder of cybernetics, writes in his autobiography
(Econ-Verlag - publishers): "Now let us look at what a musical score actually denotes. The vertical
position of a note in the line system gives the pitch or frequency of a tone, while the horizontal position
allocates the pitch to time. ..."Thus musical notation appears at first sight to be a system with which
signals can be described in two independent ways, i.e according to frequency and duration". However,
"things are not quite so straightforward. The number of oscillations per second which a note comprises is
a piece of information which refers not only to the frequency but also to something which is distributed in
time" ..."Beginning and ending a note involves a change in its frequency combination, which may be very
small but which is very real. A note which lasts for only a limited period of time must be seen as a band of
simple harmonic movements none of which can be regarded as the sole simple harmonic movement pres-
ent. Precision in time implies a certain indefiniteness in pitch, just as precision of pitch involves vagueness
in time".
