Chaos, octave equivalence, and subharmonics

One of the major results in chaos theory is the universality of certain
features of non-linear dynamical systems. In virtually all dynamical
systems which can behave chaotically, there is a phenomenon called
period-doubling which describes the transition from stable, periodic
behavior to chaotic behavior. As a relevant parameter of the system is
increased from a stable to a chaotic value, the system repeats itself
every 2, 4, 8, 16, . . . periods of the stable period. The transitions
from one power of two to the next occur closer and closer together; the
changes in parameter values required to produce each successive period
doubling approach a decreasing geometric sequence with scaling parameter
equal to the Feigenbaum constant (4.6692016091029 . . . ). This means
that at a finite value of the parameter, the period will be infinite,
i.e., we have chaos.

There may be many stages in the hearing process in which non-linear
dynamics come into play. It would probably be counterproductive to allow
this non-linearity to be enough to lead to chaos, while a parallel
structure of processors with different, lesser degrees of non-linearity
might actually aid in the recognition of pitch. It is known that when a
(not too high) pitch is heard, there are neurons that fire at the same
rate as the vibration rate of the pitch itself. Other neurons in the
brain are known to have a non-linear response to their input from other
neurons. Since a response non-linear enough to lead to chaos would
essentially be destroying all frequency information, most of the neurons
would oscillate at the input frequency or at octave equivalents below
that frequency. Perhaps a certain, low octave range is where pitch
judgments are actually made. Notice how very high tones seem ambiguous
in pitch.

Whether this or the winding of the cochlea explains octave equivalence,
there may have been evolutionary advantages conferred by the ability to
reduce unimportant information and potential confusion from overtones by
compressing pitch information to within one octave, which led to the
brain or ear being designed the way they are.

As for the apparantly irregular "subharmonic" which Gary observed in the
bassoon waveform, this can easily be explained by assuming some
parameter of non-linearity (perhaps lip pressure) was hovering around a
value at which an initial period doubling occurs. So the amplitude of
this period-2 subharmonic could have been changing, and it could cease
to exist for a while, returning again just as easily after either an odd
or even number of period-1 oscillations.

Here's an observation about instrument or vocal "subharmonics": Beyond
the onset of chaos, chaotic regions alternate (in a fractal pattern)
with regions whose periods are non-power-of-two multiples of the stable
period. The last of these subharmonic periods to occur, but the broadest
in allowed parameter values, is period 3. So within a wide enough range
of highly chaotic parameter values, one is likely to stumble upon
period-3 behavior. Increasing the parameter value further leads to the
period doublings, which in this case means period-6, period-12,
period-24, . . . with the same Feigenbaum constant, and back to chaos.
But decreasing the parameter leads directly back to chaos, via a
phenomemon known as intermittency, where very nearly period-3 behavior
persists for stretches of time, unpredictably alternating with stretches
of chaotic behavior. (The same thing is true for every odd number above
3, although the smallest parameter value needed to achieve a given odd
subharmonic, and the range of parameter values in which it persists, are
decreasing functions of that odd number). Therefore, assuming the
parameter value varies smoothly with time, and at some times takes on
values corresponding to simple period-1 vibration, the only subharmonics
which can exist without chaos ever occuring are the subharmonics
corresponding to powers of 2. Period-3 oscillation (or, to a lesser
extent, periods of higher odd numbers) can be relatively common but
cannot smoothly connect with simpler behavior.