Roger wrote:

>But it is stiffness that is the fundamental problem. Because it is
>proportional
>to the width, diameter or whatever the lateral dimension is CUBED of the
>vibrating body a small increase in d/l will produce three times the effect
>'inharmonicity'-wise (don't let's get into the definitions of that again).

We have to distinguish between the (micro) material property and (macro)
object property. Unfortunately there is no differentitation of terminology
for stiffness, as, for instance, in electrical resistance of a conductor vs
resistivity of the material. We can speak of material stiffness, a property
of the metallic structure of the wire (Young's modulus) - and that is the
only way I would use the term, because it is precise. There is also the
"stiffness" of the vibrating string itself (a more colloquial usage), which
depends on material stiffness and density, as well as the configuration -
frequency, diameter, and length. This usual formula for inharmonicity
factor (B) is one way to quantify the combined effect of these
parameters,  i.e. the degree to which the wire departs from an "ideal" wire
and behaves more like a rod.

We have to be very careful or we'll end up with propagating the myth that
Young's modulus of the wire is relevant. As I said last time, the only
relevant, audible effect related to _material_ stiffness can be attributed
to a wound string vs a (hypothetical) ultra-thick equivalent solid brass
string. Levaing aside wound strings (one of the major acoustic factors that
define a "modern" piano tone) sound....Any differences in stiffness
(Young's modulus) of yellow brass vs red brass vs iron vs steel are totally
insignificant compared to the other scaling factors that vary on an
instrument (diameter to length ratio). The myth that the effect of material
stiffness on inharmonicity is relevant to string scale design is just that
- a myth.

Stephen

Stephen Birkett Fortepianos
Authentic Reproductions of 18th and 19th Century Pianos
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Waterloo, Ontario
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