Klarinet Archive - Posting 000034.txt from 1999/01
From: Martin Pergler <pergler@-----.edu>
Subj: [kl] Inharmonic partials, was RE: [kl] re:Intonation training
Date: Sat, 2 Jan 1999 10:56:45 -0500
On Fri, 1 Jan 1999, Edwin V. Lacy wrote:
> On Thu, 31 Dec 1998, Martin Pergler wrote:
> > "Out-of-tune-ness" between 2 instruments can be caused not only by
> > mismatched fundamental frequencies or percieved pitch whan playing
> > alone, but even between higher peaks. For instance, an overly
[...]
> According to all I have ever heard and read about the field of acoustics,
> the opinion of acousticians is that in the case of wind instruments, the
> above would be impossible. I have questioned many of them about precisely
> this question, and their answer has always been that the partials *must*
> be whole number multiples of the frequency of the fundamental.
>
> I would appreciate being made aware of any references which would support
> your position and refute the one which acousticians have given me.
>
Dear Ed: (and others interested)
I am home for Xmas and do not have books or a Univ library to help
me, so I cannot provide you a reference right now, but I suspect it
can be found in Benade's Acoustics of Musical Instruments (not sure
of exact title). It's been a while since I read it.
Here is a brief explanation---hopefully correct, since as you have
politely refrained from explicitly pointing out, I am not actually
an acoustician or even a physicist---and example you can try at
home.
A bowed string will vibrate in frequencies *exactly* in integer
multiples of a base frequency. No problem.
A struck membrane will vibrate with frequencies which are related
in far from small whole number ratios. I seem to recall the first
two standing wave patterns are at f and about 1.6 and 2.2 f.
Because of this, we do not perceive it as having pitch.
Now consider a perfect smooth cylinder, open at one or two ends, or
a cone. Call this our pipe. By mechanism of a reed, or turbulent air
flow over an edge in a flute or recorder, we introduce air vibrating
in a very inharmonic frequency pattern. If we "heard" this, there
would be no recognizable pitch; fortunately we don't. However, the
vibrations in this air create vibrations in the air inside the whole
pipe. The geometry of the pipe determines that vibrations at almost
all frequencies are attenuated. The ones which are not are the ones
which set up so called "standing waves" at resonant frequencies of
pipe. For the perfect pipes mentioned above, these resonant
frequencies are at integer multiples or even odd integer multiples
of a base frequency which is determined by the length of the pipe.
The strong waves at these resonant frequencies in the air inside
this pipe induces vibrations at these frequencies in the air
outside. This propagates to our ears and we hear a pitch at the base
frequency, since the brain/ear is "programmed" to generally
"identify" the base frequency if frequencies only of a integer
multiple of it are present; the rest of the freq spectrum is
perceived as timbre. Your friends and I are in full accord up to
this point.
The important point here is that the geometry of the pipe is a
selector of which small number of frequencies of the complex
nonharmonic spectrum present in the driving air column is
propagated.
Now consider a complex musical instrument like a clarinet. The bore
shape is complex, there are tone holes, which we cover in some
pattern with our fingers. As an *approximation*, via a fudge for the
nature of the driven oscillations by a single reed, we can
approximate it as a perfect pipe of one of the geometries above (I
forget right now, someone will jump in with the answer) of the
length of the distance to the first uncovered tonehole, so that
resonant frequencies are at odd-integer multiples of the base
frequency.
This approximation, due to great ingenuity by clarinet makers for
compensatory adjustments, is generally very good and thus we hear a
beautifully in-tune note at a given pitch. However, it is only an
approximation, as we can see via forked fingerings (the geometry
beyond the first hole does matter) and overblown fingerings (opening
vents or holes encourages certain nodes to form and changes the
pattern of resonant fingerings). The sheer presence of (even closed)
tone holes actually changes the geometry. The resonance freq pattern
is perhaps best described as "usually very close to harmonically
related". Where it is not, we here diffuse pitch or worse.
I said generally the approximation works well. But we can see
exceptions, I'll detail those, with an at-home demo for those with
IBM-type machines, in the next message.
Martin
--
Martin Pergler pergler@-----.edu
Grad student, Mathematics http://www.math.uchicago.edu/~pergler
Univ. of Chicago
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