Yahoo Tuning Groups Ultimate Backup mills-tuning-list Replies to Graham Breed

>> I maintain that any systematic inharmonicity in brass instruments, which
>> is either nonexistent or extremely small, has little or nothing to do
>> with the failure of the resonant modes of the instrument to form an
>> exact harmonic series.

>Now you're getting suitably equivocal. Before you were saying that no
>inharmonicity could possibly occur, which places too much faith in the
>simplified models. Ideally, we should develop more complicated models to
>give a quantitative estimate of the (upper limit of the) inharmonicity.

I'm not getting equivocal at all. The "simplified models", as Gary
observed, seem to handle the vast majority of cases. Dave Hill did find
specific values for the inharmonicities, which I would ascribe to
methodological errors, etc. But even if you don't believe me, and take
Dave's values for the inharmonicities, they are clearly of a different
order of magnitude (typically < 1 cent) than the departures of the
series of resonant frequencies from a harmonic series (typically dozens
of cents). That's what I've been trying to say.

>Any noise in the input will cause peaks at the resonant modes.

They are not really "peaks" so much as "hills" since there are no
constructively interfering standing waves to sharpen them. While the
noise energy will be spread throughout the spectrum, the energy from the
driver (reed, lips, bow) will manifest in Dirac-delta-function-like
peaks at harmonic overtones in the spectrum.

>Inharmonicity will cause problems with JI whether or not it's systematic.
>Any noise in the input will cause peaks at the resonant modes. The result
>will be inharmonicity.

Listen, Gary and I have been defining systematic inharmonicity as cases
where, after the noise is removed, the partials deviate from a harmonic
series. Since noise doesn't exhibit interference effects, the only
inharmonicity relevant to JI is systematic inharmonicity.

>If the overtones are not centered at the resonant peaks, the amplitude of
>each overtone on output will depend on it's pitch.

Good point. I don't know about the rest, but you said it isn't
important, so . . .

>Examples of recorded resonances were
>given for a range of wind instruments, and they're all significantly off
>from the integer ratios.

Yes indeed!

>The resonances of flutes were also given relative to 12-equal. They were
>up to 20 cents out! It's only the flautist's skill that adjusts them to
>the desired scale.

The standard fingerings take this into account, in case you didn't know
that already. So, if knowing the standard fingerings is considered part
of "skill", then you are right.

>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.

While most physical nonlinearities are characterized by a quadratic
fixed point, this appears to be one with a zero-slope cubic nonlinearity
(inflection point). Instead of a period-doubling cascade, changes in the
parameter value for this type of system lead to a Fibonacci sequence of
period lengths. So changing the parameter in this case could decrease
the period to 2 or increase it to 5. I don't see how this is more
"stable" that a typical order 1/n^2 subharmonic for a
quadratic-fixed-point nonlinearity.

>> 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?

I don't know, but this would be at a higher processing level than the
neurons forming the auditory nerve, which is what is typically probed.
In any case, Gary was asking for a mechanism by which powers of two
would be distinguished from powers of any other number. Chaos theory
presents a very clear reason to think that such distinctions are often
made in nature.

>> 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.

What are "such high frequencies"? I did say "not too high" above!

>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 actual frequency at which periodicity pitch gives up is about 5-6
kHz. Really high!

>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.

Are white rats musically literate?

>More specifically, literate in Western
>Classical Music, where the notation is octave based.

Many other cultures use octave-based notation.

>> 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.

Not unlikely at all! Look again at a bifurcation diagram -- there is
quite a large range of values where the period-2 behavior has a
relatively constant amplitude, but at one end of this range the
amplitude suddenly decreases and period-1 behavior ensues. If the
bassoonist is trying to keep within this range, say because he is trying
to play as loud as he can and octave subharmonics don't bother him but
chaos does, the parameter will likely take on just such a range of
values.

>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

>A period doubling cascade can be
>linked to amplitude, though.

These don't sound like examples of real-world dynamics.

>It's unlikely that a period 3 cascade could be picked out from a chaotic
>region with one parameter.

Not unlikely at all, I've done it with my vocal cords.

>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.

Actually, a differential equation needs to have at least 3 parameters
for chaos to occur. However, the picture along whichever of the
parameters is responsible for chaos will be the good old bifurcation
diagram. The Mandelbrot set represents a discrete (not continuous)
dynamical system, which only needs one parameter to exhibit chaos. If
you restrict yourself to the real line, you see the usual bifurcation
diagram again (proceeding from right to left).

>I think there
>might even be a period 3, 9, 27, ... cascade in the Mandelbrot set.

Naah, the Mandelbrot set is clearly all about period doubling; the
iteration of a point on the set's boundary doubles the angle the point
makes on the unit circle (to which the boundary can be conformally
mapped). Although you could probably find some contorted path in the
complex plane to support your guess.

Replies to Graham Breed

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