Klarinet Archive - Posting 000180.txt from 1995/02

From: Jonathan Cohler <cohler@-----.NET>
Subj: Re: Caution More Physics!
Date: Wed, 8 Feb 1995 18:32:37 -0500

Jon Delorey <Jon_Delorey@-----.COM> said:

>Regarding Non-Harmonics.
>>6. Now to answer the seeming paradox about
>> how you can get inharmonic (or non-integer-multiple
>> harmonics) from a driven system.
>> The answer is that the driving force coming into
>> the clarinet is not a simple sinusoidal driving force.
>> If it were, then as I described in 4 above, the air
>> column would vibrate ONLY at that sinusoidal driving
>> frequency at an amplitude determined
>> by the input force and the resonance curve.
>
> I don't understand this statement. A non-linearity is needed in order to
> produce non-harmonic partials.

No. A vibrating air column is about as linear a system as you can get.
The "natural modes" or "resonances" of the tube are determined by the shape
of the tube. An "ideal" cylindrical tube closed at one end will have
integer multiple modes for wavelengths long compared to the width of the
tube. Any deviation from perfect cylindrical shape will modify the modal
frequencies. One can measure the input impedance curve (the resonance
curve) of the clarinet by measuring resultant pressure (using a microphone,
for example) divided by input air volume using a speaker as the resonator.
Varying the frequency of the produced by the speaker over the whole range,
you will find that a clarinet has modes (resonances) that are slightly
flatter than the odd integer multiples (and they get flatter as you go up).

A non-linear system produces rectification (shift of average value),
harmonic generation (integer multiples) when the input is a single
sinusoid. If there are multiple sinusoidal inputs, nonlinearity also
produces modulation (sum and difference frequencies).

Although there are certainly non-linearities involved in the reed (this
physics is much more complicated), I believe these are all second-order
effects. The tube is the dominating influence, because the reed can
vibrate comfortably at any of the frequencies in the range of the clarinet
(at least, a good one can!). I think that, like a piano sounding board,
the reed has a very broad resonance curve.

>As far as the reed being a white noise generator this makes sense, but since
>the
> tube is not ideal, the reed in combination with the tube is not a white noise
> generator in the strictest sense. The shape of the noise must vary with the
> keying used for the particular note. This would affect the harmonic content
>of
> the resulting waveform.

Yes and no. It is definitely not a strict white noise generator. But for
the purposes we are discussing I believe it has roughly equal amounts of
energy across the spectrum of interest. Try blowing into the instrument
and making only air sound while moving your fingers around. It changes the
quality of the white noise only slightly. (I know this, because I just
played Waves by Takemitsu, which ends with one minute of air noise! You
get more effect at the end by moving the clarinet around and pulsing your
breath in different ways. The keys don't help too much.)

>
>The non-harmonic partials has to also make the manufacturing of a clarinet that
> is *in tune* for nearly four octaves very difficult (never mind having to
> accomadate a non-harmonically related even-tempered scale). It's the kind of
> problem that makes engineering interesting and a good day job.

That's absolutely true. All clarinets are quite out of tune. It requires
the skill of the player to take advantage of the fact that the resonances
have some width, and are therefore flexible to a certain degree. The
narrowness of the resonances vary with the fingerings, and those resonances
that are the quite narrow are very hard to adjust. Conversely, those with
wide resonances are easier to adjust, but are also deader (less resonant)
sounding notes. This area of adjusting intonation also involves the reed
(and the oral cavity) a great deal.

There are tradeoffs in clarinet design as to which notes the designers
choose to make out of tune. The Buffet, which is among the best, has many
out of tune notes on all of its instruments. The instruments that I have
found to be the most resonant and most in tune across the range, are those
made by Luis Rossi, a Chilean clarinet maker (and excellent clarinetist
too). I played on Buffets for more than 20 years until I heard the Rossi,
and there is really quite a difference (this is not an ad). One reason,
the Rossi's are more in tune is that the traditional two middle joints are
combined into one piece. This allows for proper, optimal hole placement.
Instruments that have two middle pieces suffer from the fact that the holes
must be moved around to accomodate the joint, which has a dramatic affect
on the intonation of several notes.

The intonation problem is a major reason why playing with hard (this is a
relative term) reeds is not a good practice. Some people have the
misconception that a hard reed stabilizes the sound. What it actually does
is it reduces the flexibility that one has in adjusting out-of-tune
pitches. It also reduces the amplitude of higher partials making the sound
duller and less brilliant (some people call this "darker"). The amount of
effort required to produce sound is also increased, and therefore you can't
play phrases as long.

With a lighter (not too light of course) reed, you can control pitch much
better and you have access to the brighter sound. You can also make the
sound darker with lip, throat, tongue and mouth techniques. With the hard
reed, you're stuck sounding "dark"; the only thing that makes it brighter
is playing louder. Lighter reeds allow you to expend less energy in
producing the sound, allowing for more relaxed performance and greater
stamina.

>
>Speaking of boundary conditions, there was a posting a while back that stated
> the harmonics in an organ pipe were dependent on position. Would this be due
> to the proximity of the floor or ceiling, and is this a factor in the harmonic
> output of a clarinet?
>

I don't know a lot about organs, but it is not a factor with the clarinet.
Nothing substantially outside the tube effects the sound production. Of
course, if you put the clarinet one half an inch from the floor while
playing a third-line B, you'll hear a big difference, but that's an extreme
example.

>Do you have a bibliography for some of these results? That may be an easier
>way
> to answer these questions since the physics can get long winded since
>equations
> and pictures of spectrums are hard to reproduce on a text based posting.
>

A couple of good basic books are:

"Music, Speech, Audio" by William J. Strong and George R. Plitnik,
published 1992 by Soundprint, 2250 North 800 East, Provo, UT 84604

"Fundamentals of Musical Acoustics" by A. H. Benade, published 1976
by Oxford University Press. (Also a 1991 Dover edition)

"The Physics of Sound" by R.E. Berg and D.G. Stork, published 1982
by Prentice Hall.

and of course one of the bibles of Physics in general:

"The Feynman Lectures on Physics" by Richard P. Feynman, Robert B.
Leighton, and Matthew Sands, published 1963 by Addison-Wesley Publishing
Company, Reading, Mass.

-------------

Jonathan Cohler
cohler@-----.net