[pianotech] inharmonicity in piano wire

[Robert Scott]

Bruce Dornfeld writes:

>...The inharmonicity does not come from the piano wire alone.
> If you bow (if your violin bow doesn't fit, rosin up a scrap of old felt)
> a piano string, the inharmonicity will be less than striking the string
>  with a piano hammer!  The formulas we have for inharmonicity in a 
> piano do not show these things; they assume the piano will have felt
> hammers, bridges and a soundboard fixed on all sides.  How much these
> other things influence inharmonicity, I don't know, but the image we
> like to hold in our mind to understand it normally simplifies things too much.    

No, it really is that simple.  The resonances of a string in a piano are a function of the string and how it is held at the termination points.  It does not depend on the condition of the hammers.  It might depend very slightly on the condition of the bridge or soundboard if either of these items happen to have strong resonances of their own, and if those resonances are very close to the pitch of some partials.  But typically the resonances in the string itself are much more pronounced and will not be pulled much from their natural frequencies.  Or it might depend very slightly on how hard the string is struck, since higher string excursions produce higher tension and higher pitch.  But for anything in the "mf" to "pp" range, this effect will be unmeasureable.

The phenomenon you mention about bowing a string is a different case.  It is not like hitting a string with a hammer and letting it ring.  When you bow a string, you force all the partials to be coupled continuously through the contact of the bow.  The inharmonicity is not merely lower.  It is actually zero, because the partials are locked as true harmonics.  But the coupling will persist only so long as the bow remains in contact with the string.  As soon as you lift the bow from the string and allow it to ring free, the partials will immediately switch to their natural (inharmonic) frequencies.

There was an excellent class given at an Annual Convention by James Arledge and another James who I can't remember.  The class was about inharmonicity studies on bass strings.  But they did not bother with bass strings in a piano.  They constructed a special jig that held just one string, and they measured all the partials up to some very high number with greater accuracy than you could in the piano.  If inharmonicity depended so much on other factors, then this whole class would have been a waste of time.  But it doesn't, so it wasn't.

Robert Scott
Ypsilanti, Michigan