Csound

I think I fixed the bug with rounding for cpsxpch but I will check.
==John

I always tell people that the mathematics involved in tuning theory is
fairly simple, and now you bring up nonlinear dynamics. Great. Musicians
take note that you do not need to understand any of this!

> 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.

Yes, chaos would be counterproductive. There are plenty of nonlinear
equations that produce stable subharmonics without being chaotic, though.
These would probably be more efficient than one that could suddenly lapse
into chaotic behaviour.

Jordan & Smith give the following example of a stable order 1/3
subharmonic:

x'' + x - x^3 / 6 = 1.5*cos(2.85*t)

Where the '' means differentiate twice with respect to t.

> 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.

Your theory suggests that neurons should fire at a 1/2^n subharmonic of
higher pitches. Has this been observed?

> 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.

I would question whether neurons can react at such high frequencies at
all. I don't have experimental data to hand, but I remember evidence that
low frequencies are perceived in a different way to high ones, probably
because of this.

The usual mechanism suggested for high frequency perception is that each
hair in the inner ear responds to vibrations of a particular frequency,
plus their harmonics. The hair is connected to a nerve cell, and that
triggers when the hair vibrates. For high frequencies, I don't think the
periodicity of that nerve impulse is used by the brain. This mechanism
does show how the ear could have a predisposition towards harmonic
overtones. I think there is one hair for every few cents. More precise
measurements of pitch can be performed by averaging out the stimulus from
two or three hairs. I think octave reduction would have to be performed
at a later stage.

> Whether this [period doubling] 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.

I don't disagree with this. I'm still awaiting confirmation of this
cochlea winding theory, though. Also, I want evidence of an innate
predisposition to octave invariance. The latest book's I've read --
which, as they came from libraries, are a decade or three old -- say that
the original experiments showing octave invariance were performed on
musically literate subjects. More specifically, literate in Western
Classical Music, where the notation is octave based. The effect in the
musically naive was much reduced. More experiments have no doubt been
performed since, so perhaps somebody could update me. Such experiments
should show a comparison between 4/1 and 3/1, not only 2/1 and smaller
intervals.

Octave invariance, and harmonic theory in general, lie outside the scope
of this list. I'll write up my ideas for my website sometime.

> 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.

This assumes the system is teetering in a bifurcation point, which is
actually very unlikely. I'd guess the amplitude of the subharmonic is
chaotic. Or, the subharmonic is entirely chaotic but appears to be order
1/2 at certain times. If the subharmonic comes from something like
Duffing's equation, it will probably arise in the tube, and so be isolated
from parameters like lip pressure. A period doubling cascade can be
linked to amplitude, though.

[lengthy explanation omitted to conserve bandwidth]
> 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.

It's unlikely that a period 3 cascade could be picked out from a chaotic
region with one parameter. In the Mandelbrot set, though, there is a
fairly large period 3 region. Similar things might occur with
differential equations of 2 parameters for all I'd know. I think there
might even be a period 3, 9, 27, ... cascade in the Mandelbrot set.