Next: Conclusions and Speculations
Up: The Shepard Tone
Previous: The Shepard Tone
Contents
One could argue that if we start with , we are actually creating
partials which are lower than the audibility range (less than 20 Hz). And
by rescaling the signal (playing it faster) we are only changing the
audible high
frequency spectrum. For that reason we start the partials from 32 Hz.
Audio example 1 is an example of a Weierstrass function
with and
, therefore, the sound is composed of geometrically
related partials from 32 to 4096.
The example was created ^{4.1}at a sampling rate of 22050 Hz. It is first played at a sampling
rate of (22050 Hz)
and then at double that rate (44100 Hz). Even though one notices
that the center of mass of the energy has increased in the frequency
spectrum, one does not get the feeling that the pitch has moved one
octave higher. The selfsimilarity of this signal can be seen in
figure 41. In figure 41a the first
346 samples (which at a sampling rate of 22050 Hz is about 16 milliseconds),
is plotted against exactly half () of those samples in
figure 41b.
Figure 41:
A simple Weierstrass function with
and
.
(a) shows the starting 346 samples at sampling rate of 22050 Hz and
(b) shows exactly half of that
signal (the starting 173 samples). Notice that the two shapes differ only
in high frequency details. This similarity can be seen in higher
or lower time scales as well.

Figure 42:
A Weierstrass function with
and
.
(a) and (b) are the first 205 and 97 ( ) samples.

By carefully choosing , we can create a signal whose
perceived pitch will
descend by a semitone when the signal is played at twice the speed.
Therefore the relationship between the new and
scaling value (which is
2 since we are playing at twice the sampling rate), should be the twelfth
root of two (the frequency multiplier for a semitone). Therefore:



(4.4) 
Audio example 2 is an example of a Weierstrass function
with and
. Again, the sound is played at the original sampling
rate (22050 Hz) and then played at twice the sampling rate (44100 Hz).
The selfsimilarity of this signal in the time domain can be seen in
figure 42.
Listen to this and the previous example at first with no attempt to find
the pitch, and you will simply hear the movement of the mass of
frequencies. Then,
listen to the examples while concentrating on finding a pitch, and notice
that the paradox effects gets stronger.
Next: Conclusions and Speculations
Up: The Shepard Tone
Previous: The Shepard Tone
Contents
Shahrokh Yadegari
20010301