Biomedical Engineering Reference
In-Depth Information
9 Complex Rhythms
There is more to birdsong than just a beautiful melody. So far, we have
found “melodies”, or sequences of more or less complex sounds. We have
focused on the acoustic properties of the individual elements. Now we turn
to another aspect of birdsong production: its
rhythm
. Here we use “rhythm”
in a very musical sense: we mean “the pattern of musical movement through
time” [American Heritage Dictionary 2000], or the timing of sounds within
the melody. As we shall see, complex rhythms in birdsong may be found in
the song of a single bird or a duetting couple. How are they generated?
9.1 Linear vs. Nonlinear
Forced
Oscillators
We can obtain a precise image of what we mean by
complex
rhythms by
simply hitting the table periodically with our left hand (which will behave
as a clock in this discussion) while also hitting the table, at a variety of
frequencies, with our right hand. The variety of possible timings between
impacts of the right hand constitute the various rhythms of the problem. If
both hands hit the table the same number of times in a given time interval,
we say that they are locked into a “period one” solution (a particular way
of achieving “period one” is by always hitting the table with both hands at
the same time). Let us now hit the table with our right hand only once per
two impacts of the left hand. This is a “period two” solution-the complete
pattern does not repeat itself until the left hand has hit the table twice.
Now, a more complex excercise. Let us generate a “period three” solution.
At this point we have a choice: the right hand can hit the table only once
in the time interval that it takes our left hand to hit the table three times,
quite analogously to the previous exercise for “period two”. But we could
also generate a “period three” solution in which the right hand hits the table
twice in the time it takes the left hand to hit it three times. These are two
different “period three” solutions. In order to distinguish them, one defines
the
rotation number r
=
p/q
,where
q
is the period of the solution, and
p
the number of recurrences performed by the driven system. In our two
examples of “period three” solutions, we would have
r
=1
/
3and
r
=2
/
3,
respectively. Can we generate these different rhythms with simple coupled
physical systems?
