[CAUT] Inharmonicity, "Link to Young Paper"

[Fred Sturm]

Hi Jim,
    Tuning stretch is maybe acting like a red herring in what I wrote. The
notion I was trying to convey is that the resonances that provide feedback
to the string should include the whole assembly, strings included, not just
soundboard resonance. The tensions and pitches of those strings might react
as well with various partials of a string being measured. In other words,
there may be a complex mix of resonances including those created by the
strings (at tension) in reaction to the soundboard bridge structure. And
they may be powerful enough to couple with individual partials and affect
the measurement.
    Getting back to stretch, I was suggesting that one might test this
theory by measuring partials of individual strings with somewhat varied
tuning on the whole piano, to see if that changed the "para-inharmonicity."
And I suggested that varied stretches might be the varied tunings used for
such a test.
    Again, what I was getting at is the fairly consistent tendency to get
lower inharmonicity numbers when measuring a piano before and after tuning
it. There must be a physical reason. The one that comes to my mind is some
sort of coupling of "all the strings" (and their various partials) with the
string being measured. The only thing that has been changed in this case is
the tuning of all the strings.
Fred
On 1/20/07 1:15 PM, "James Ellis" <claviers at nxs.net> wrote:

> 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 
> 
>