Soundboards and Inharmonicity

[Robert Scott]

Richard Moody writes:

>I am asking if soundboards DO have inharmonicity?   If so, 
>how do you find out how much? 
   ...snip...
>I say that if a vibrating string has inharmonicity due to stiffness 
>and diameter, then a vibrating sound board MIGHT have inharmonicity 
>in itself due to stiffness and diameter.(thickness).

Richard, let me see if I can clear up what the issue really is.
In trying to answer the question "Does the soundboard affect the
inharmonicity of the strings" it is a mistake to focus on the
soundboard as a separate item with its own "inharmonicity". 

If you tried to measure the inharmonicity of the soundboard all by 
itself, then sure, you could take off all the strings and then strike the
soundboard and explore its resonances.  The soundboard all by itself
does have resonances, but they are not strong resonances.  They are
highly damped and thus have very short decay times.  If a soundboard
did have strong resonances it would sound terrible - just like a
speaker cone with strong resonances.  And assuming you could find more
than one such resonance, there is no reason for two such resonances 
to be even approximately harmonically related.  For example, if you
found a soundboard resonance at 10 Hz and another at 28 Hz, what would
you conclude about inharmonicity?  Is the 28 Hz to be thought of as a
very sharp second partial or a quite flat third partial?  The answer is
neither.  They are just two resonances.  There is no physical mechanism
in a soundboard to make the partials approximately true harmonics.  In
the case of strings the physical mechanism is the close similarity to
the ideal case of infinitely thin strings which do have perfectly
harmonic partials.  To the degree that real strings deviate from this
ideal, they have inharmonicity.

So that is about the best you can do with the notion of a soundboard
having inharmonicity all on its own.  So how would these weak resonant
characteristics affect the soundboard-with-strings system?  The answer
is not very much.  Since the soundboard resonances are so very broad,
they are not going to be measureably different at two frequencies
that are only .1 cents apart.  But string resonances are so narrow
that they can respond differently with a .1 cent shift in frequency.
Therefore the string resonances will dominate over those of the 
soundboard.  Soundboard resonaces will have an immeasureable effect
on the overall inharmonicity of a sounding string.

In the search for a link between inharmonicity and something beyond the
string itself, there other candidates to consider.  Suppose humidity
makes the soundboard swell up.  You tune the piano with a swelled
soundboard by letting off some of the excess tension at the tuning
pin.  This causes a shift in the bearing point under the V-bar.
Since inharmonicity is highly sensitive to stiffness and kinks near
the bearing point, it is easy to see how random variations in
the inharmonicity could result from the string sliding to a new
position under the V-bar, or, for that matter, at the bridge.
And for those who claim that hammer voicing affects inharmonicity,
I will just repeat my claim that the only possible way voicing
could affect inharmonicity is indirectly though its affect on
volume.  The inharmonicity of a string sounding forte is going to
be slightly different than a string sounding pianissimo.

-Robert Scott
 Ypsilanti, Michigan