[CAUT] Inharmonicity, "Link to Young Paper"

[James Ellis]

Fred,

I think we are talking about two different things.  What I'm talking about
has nothing to do with tuning "stretch".  I am saying that a resonance in
the bridge/soundboard system of a piano that happens to be near resonance
with a given partial of a given string will affect the frequency of that
particular partial.  Then if we plot all the partials of that particular
string, we will find that the one affected by the bridge/soundboard
resonance does not conform to the curve we plot for all the others, i.e.,
that the frequencies of the partials of a given string are proportional to
the square of the partial number, but one of them lies somewhat off the curve.

If the string is long and uniform, and the terminations nearly perfect,
when we plot the inharmonicity of all the partials they will fall on a
nearly perfect exponential curve.  If the string is very short, the
terminations will affect the shape of the curve.  If there is some
resonance in the system that falls near one of the measured partials, it
will "pull" that partial off frequency, and it won't be right on the curve
with the others.  

I suspect this sort of thing happens in real pianos, and if you change the
tuning, you may move the affected partial away from the resonance that is
pulling it off frequency.  This is bound to happen to some degree in the
duplex treble octaves, and I suspect it happens in other parts of the scale
too.

Jim Ellis