The Increasing Beat
Introduction
The mind grasps any
pattern of sounds that is repeated and considers this pattern a rhythm. An inherent property of any rhythm is tempo,
or the speed of the rhythm. An auditory
illusion designed by Kenneth Knowlton and Jean-Claude Risset
(Risset, 151) created the illusion of making a rhythm
sound like it was increasing when, in reality, it did not.
This illusion is
created by superimposing drum beats that have geometric relationships to each
other (whole note to a half note, half note to a quarter note, etc.). The illusion of the beat increasing in tempo is
created by slowly fading out the slower beat while fading in the successive
faster beat. By superimposing these drum
patterns, the mind latches onto self-similar patterns of drum beats as they
become dominant in amplitude without necessarily realizing that it does so.
Simple Case
The simple case
was shown in J. R. Pierce’s book: The Science of
Musical Sound. It starts with a
fundamental frequency 
[Hz].
The frequency describes the number of beats per second. Each of the beats is constructed by a sine
wave with a tone frequency 
[Hz].
The beats are then multiplied by a percussive envelope which makes the
tone sound more like a drum beat.
Figure 1. This is the percussive filter used in the
demonstrations. The characteristic features
of a percussion instrument, such as a drum, are the fast initial attack and a
sharp initial decay.
The individual
drum beat is then repeated at a rate of Hz over a
specific time period. A number of
patterns in geometric relation are then created in the same manner (with each
series having twice as many individual beats as the previous). Each of these patterns has a different
envelope for the amplitude, which will be multiplied to the pattern, then
summed to create a single drum beat.
Figure 2. Simple, linear envelopes used to determine
the amplitude of the
When these
patterns are combined, they create the illusion of the tempo increasing, when
in reality, the tempo remains the same.
Figure
3. This is a simple
beat pattern created with the linear envelopes from Figure 2. The fundamental frequency is 1.25 Hz and this
is a total of five superimposed patterns.
Figure 4. The simple case with bell curve shaped
envelopes.
Complex Case
Jean Claude Risset creates a complex version of the illusion (Risset, 152). The
concept for the complex case is the same: the illusion is achieved by having
the listener focus on the beat that has the loudest magnitude.
In Risset’s example, his initial frequency of beats is 
and the number
of geometric beats is 5. In order to
create a smoother effect of increasing tempo, Risset reduces . The function used to describe the decreasing
of beats is an exponential function ranging from 
to 
with a period of 66 seconds described as 
, where 
is the time in seconds and 
. Risset focuses the
bell shaped curves of the amplitude to the middle of time it takes to decrease by a half. For example, 
in 13 seconds
so the center of the bell shaped curve for the second beat is at 6.5
seconds. In Risset’s
example, the sample is about 40 seconds long, which results in a 
decrease of , but by this
time, the listener is focused on the amplitude of the fifth beat (which is 
), so the
listener is hearing a beat which is 
, where is the initial frequency (1.25 Hz).
Figure 5. The decrease of the
fundamental frequency (r--beats per second) over time. The function ranges from 1 to 1/32 with a
period of 66 seconds.
The complex
example is more convincing than the initial, simple example due to the use of
the decreasing function for . This leads to a smoother transition between
each of the beats and the changing of beats is not as noticeable (a total shift
in as opposed to a shift in 
).
Figure 6. This is the wave
form of the pattern created by Risset and
Knowlton. The fundamental frequency is
decreasing while the amplitude of the faster beats are
increasing.
References
Risset, Jean Claude (1989). “Paradoxical
Sounds.” Current Directions in Computer Music
Research. 149-158. MIT Press:
Pierce, J. R. (1983). The Science of Musical Sound.