source file: m1405.txt Date: Tue, 5 May 1998 10:58:06 -0400 Subject: Chaos, octave equivalence, and subharmonics From: "Paul H. Erlich" 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.