RE: More on instrument aperiodicity

>> Hence the amplitude of
>> the partials can vary, sometimes wildly, rendering the waveform
>> aperiodic. However, for the time domains relevant for the ear's
>> analysis, this does not lead to any relevant alterations in the
>> frequencies of the partials.

>So, you're presumably still with exact integer ratios. Maybe, though,
>you mean that pitch fluctuations place a lower limit on the pitch
>precision that is meaningful, and that inharmonicity is below this
>threshold. It isn't clear.

Put it this way: although analysis of a signal into frequency components
with constant amplitude has a unique solution, analysis of a signal into
frequency components with varying amplitude doesn't. In the typical case
the variations in amplitude will be slow enough so that any inharmonic
frequencies that might come up in the latter, non-unique mode of
analysis would be too close together to be resolved by the ear. What I'm
suggesting is that the ear in fact does the latter and hears a harmonic
series with changing amplitudes, rather than inharmonicity.

>If you're now saying that inharmonicities of a fraction of a cent are
>not physically impossible, I'm happy to concur with that. My guess
>would be less than 0.1 cents, although I have no proof of that, and
>would like some. Hopefully, we can stop theorising and wait for some
>more data.

Again, with the amplitude fluctuations, you can't pin the frequencies
down in a well-defined way. However, I'm saying that for wind, brass,
and bowed string instruments, it would be consistent to assume that you
have exact integer overtones with a particular amplitude spectrum for
each one (plus noise). Such an assumption would not differ from the
ear's own interpretation in any meaningful way.

>>>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.

>Why couldn't they be reinforced by standing waves?

The driving force has to be in phase with the reflected wave for a
resonant standing pattern to develop. In noise, there is no phase
regularity whatsoever.

>If the noise
>provides a constant impulse at all frequencies, that should lead to
>peaks at the resonances

Hills, yes.

>like with a flute, maybe still corrected by an
>order of magnitude towards harmonicity.

No, not like a flute. A flute tone is not just selectively amplified
noise. The turbulent airflow that is responsible for the driving force
of a flute is not fully understood, but is definitely more regular than
just noise.

>Whether this means two peaks
>occur close together, or the original peak is shifted slightly, I don't
>know.

In the ideal of an infinitely long sampling time, the original peaks are
Dirac delta functions, while the hills of noise remain hills, so the
former are not shifted at all.

>I'm not deliberately causing trouble here, only I usually agree with you
>and it distresses me when I don't. Hopefully, this was just a problem
>of expression, and we can put it all behind us, marching defiantly
>towards a better future.

Yes!

>> 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.

>I can't find this definition anywhere. I was assuming "systematic"
>meant the deviation from integers had to be constant from one cycle to
>the next.

Well, we both mentioned noise as an extraneous factor at various points
in the discussion. So the definition was implicit.

>The inharmonicity in the resonances is caused by the holes being there

As well as end effects even when all the holes are closed.