Inharmonicity in other instruments

[Robert Scott]

Bill,

I was so fascinated by your idea:

  >if one could take a rosined bow and draw it across the sharp
  >edge of a bell, you would expect the sound to be quite different
  >and a more identifiable tone to be heard.

that I ran downstairs and got a bell and a violin bow and tried it.
The results depended a lot on how the bow met the bell.  Under
certain conditions I got a nice loud tone, and it was totally lacking
in that usual jumble of inharmonic partials that you associate with
bell sounds.  It almost sounded like a flute - hardly any harmonics.
And if I changed the angle of the bow, I could get a completely 
different tone, like when you play harmonics on a violin.  But
sometimes I got a very soft, thin sound that seemed to have a more
bell-like tone, containing many partials.  But I think that was
because the bow was not "grabbing" very well and the excitation
was more chaotic than periodic.  It sounded like white noise that
had been filtered.

As to your other question:

  >does not the rosin and the microscopic sized barbs of a bow
  >actually "pluck" the string too?  How rapidly does the 
  >"excitation" have to occur for it to be considered "continuous"?
  >If you give the piano key a rapid series of strokes such as when
  >playing a trill, would the inharmonicity of the strings involved
  >decrease or go away?

By "continuous" excitation I mean periodic excitation that is
as fast as the cycles of the fundamental.  When a bow is drawn
across a violin string, the friction between the bow and the string
alternates between grabbing and sliding.  When the string is going
with the bow, the friction is grabbing.  When the string is going
the opposite direction the friction is sliding.  Since grabbing
friction is greater than sliding friction, more energy is put
into the string on the "grabbing" half cycle than is taken away
on the "sliding" half cycle.  So there is a net positive transfer
of energy into the string, which keeps it going.  And this energy
transfer is from continuous pulsing that is synchronized with
the fundamental frequency.  And yes, if you could play A0 with a
repetition rate of about 27 hits per second, then the sound would
not exhibit any inharmonicity.

As to this:

 >if a reed has no inharmonicity, then how could my SAT pick up
 >the difference between the fundamental of a G2 reed and the
 >same reed as read on G3, G4, G5, and G6 and show an increasing
 >difference of 6¢ at the highest partial that was measured?

I would suggest the following double check of your measurement
method.  Find a tone source that is rich in harmonics and is
known to have no inharmonicity.  One such source would be a
cheap electronic keyboard.  Another source would be any
software that plays a continuous tone in your computer speaker.
Then measure the inharmonicity exactly the same way you did
for the reed and see if you get the same results.

 >When I play the G2 button and push the bellows normally, the
 >other 4 reeds sound in octaves that are apparently beatless.
 >When I push forcefully, they go out of tune with each other.
 >You can hear beats within the octaves.

This is a different experiment from the previous one, right?
Although the sound of a single physical reed has no
inharmonicity, the tracking of a number of nominally octave-
related reeds can certainly go out of whack as the air
pressure changes.  But that kind of inharmonicity comes from
the fact that there are separate tone generators for each octave.

While I'm at it, I would like to comment on Ron Nossaman's
suggestion:

 >I think what you are seeing here is the result of the higher
 >inertial mass of the lower pitched reeds. The reed excursion
 >will be comparitively wider under greater air pressure in a
 >heavier, more flexible (relative to mass) reed than in a
 >lighter, stiffer one. It should therefor take the heavy reed
 >a little longer to complete a cycle under more pressure.

If the excursion is greater, then so is the spring force acting
to return the reed to the center position.  In fact, in piano
strings, higher excursions are associated with an increase
in pitch, not a decrease.  The behaviour of a mass-spring
system under varying displacements is related to the shape
of the spring force vs. displacement function.  If the
spring force is exactly proportional to displacement, then
the period of the oscillations would be the same regardless of
the size of the excursions from the center.  The fact that
reeds go down in pitch with increased air pressure would lead
us to believe that the reeds, when looked at as springs,
have a spring force that is less than proportional as the
excursion is increased.  Now if they made reeds like they make
leaf springs in cars, you would have the opposite situation.
As the displacement increases, the spring force goes up more
than proportional.  That would make reeds go up in pitch
as the air pressure increases.

Robert Scott
Detroit-Windsor Chapter PTG