Klarinet Archive - Posting 000889.txt from 1997/05

From: Jonathan Cohler <cohler@-----.net>
Subj: Re: Pitch of tuning notes
Date: Sat, 31 May 1997 10:12:42 -0400

Ian Dilley wrote:

>The partials of a piano string are not in the perfect mathematical
>relationship that most people assume. They tend to run sharp.
>Presumably this is due to the strings not being infinitely thin and
>flexible. When you tune an octave you make it as beatless as
>possible. ie. match the first partial of the lower note to the
>fundamental of the upper, the third partial of the lower to the 1st of
>the upper etc.

Ian,

I believe that your statement here is basically correct. The physical
reason is that the vibrating element (the string) is not a totally linear
system. As I recall, the non-linearities are most noticeable in the very
high-tension strings of the highest notes of the piano (and not very
noticeable at all down lower). I believe it has to do with the properties
of the metal strings at very high tension that makes them non-linear (i.e.
the force of the tension is not always directly proportional to its
displacement from equilibrium).

>
>I'm sure the same thing happens with other instruments. A unison
>sounds in tune when most of the partial are in tune with each other.
>I believe a clarinet has partials which tend to run flat. So, when
>playing a unison with a piano that sounds in tune, the fundamental of
>the clarinet could be sharper than that of the piano.

Here's where you run amok. The vibrating element in the clarinet is an air
column, which is one of the most linear physical systems you can create.
Therefore, all of the notes it produces are harmonic and therefore made up
of harmonic partials (we've had this discussion on the list before). In
other words, its partials are exact integer multiples of the fundamental in
all cases. (It is true, as an aside, that the resonant modes of the
clarinet tube are not exactly aligned with the harmonic partials that it
produces, but this is an entire discussion unto itself.)

Therefore, the best intonation of a clarinet (or any instrument) when
playing a unison with a piano is that when the fundamental is precisely in
tune with the fundamental on the piano.

Now, when you start talking about thirds, sixths, sevenths, then it gets
more interesting...

-------------------
Jonathan Cohler
cohler@-----.net