More on instrument aperiodicity

Paul Erlich wrote:

> 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.

Hang on. In TD 1392, Gary Morrison started the discussion with:

> Another consideration that deserves some thought along these lines
is
> that very few acoustic instruments have their overtones within 1 cent
of
> exact harmonics.

Then you said

> Last time we had this discussion, we agreed (I thought) that bowed
> strings, winds, and brass instruments had exact integer harmonics for
as
> long as they sustain a consistent tone. That is a physical fact, true
of
> any system with a 1-component driver, which oscillates periodically
and

Firstly, I took "exact" to mean an integer to an infinite number of
decimal places. "How can anyone assert such accuracy?" I thought.
Also, the implication is obviously of far _higher_ accuracy than the <1
cent previously suggested. I also queried how you could be so sure that
a reed or whatever is exactly a 1-component driver, but Gary made the
point for me.

You replied with a model for the operation of reeds/lips that assumed
laminar airflow, and concluded with:

> 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.

I presumed, with all the uncertainties in the construction of the reed,
that resonance with the air column was the main factor governing
inharmonicity, and was corrected on that point.

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.

This is interesting, though:

>>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? If the noise
provides a constant impulse at all frequencies, that should lead to
peaks at the resonances like with a flute, maybe still corrected by an
order of magnitude towards harmonicity. Whether this means two peaks
occur close together, or the original peak is shifted slightly, I don't
know. Definitely smaller than the 0.1 cents for most instruments,
though, because of the low noise level. Maybe not even the only source
of inharmonicity, but the easiest one to quantify.

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.


> 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.

>>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.

I can't go back and check now, but I think both cases were covered.
Harmonics played by the performer were out by less than however else
they were measured. There is still a deviation with the usual
fingerings, though, of quite a few cents. Enough to scupper the
difference between 12-equal and meantone. Instrument manufacture may
have improved since then, of course.

The inharmonicity in the resonances is caused by the holes being there,
and there's some complex relationship between this and the holes
producing slightly the wrong pitches. I didn't read it for long enough
to be sure. I usually ignore acoustic instruments, but thought I'd
better clue up for this discussion.