[pianotech] inharmonicity in piano wire

[Mark Schecter]

Robert,

Thanks for this explanation. It confirms my general sense of the matter.

-Mark Schecter

Robert Scott wrote:
> Mark Schecter wrote:
> 
>> ...inharmonicity is due to the material's stiffness at the
>> attachment points only?...
> 
> No, inharmonicity is affected by the material's stiffness anywhere
> that it bends - which is mostly at the termination points and at the
> nodes.  That is why the effect is more pronounced for higher
> partials.  They bend in more places than the fundamental or lower
> partials.
> 
> The model of a vibrating string that produces pure harmonics (no
> inharmonicity) is one where the string has some mass and some
> longitudinal spring constant, but no transverse stiffness - i.e. no
> resistance to bending.  The frequency of each resonance is determined
> by how a displaced mass responds to a restoring force.  In the case
> of the ideal string, the restoring force is entirely made up of the
> transverse component of the string tension when the string is
> displaced.  But when you add stiffness (resistance to bending) into
> the picture, the restoring force for a displaced mass is partly due
> to that stiffness, which is in addition to the restoring force due to
> string tension.  That raises the pitch of those resonances that
> engage in more bending.
> 
> Robert Scott Ypsianti, Michigan
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