Inharmonicity

[Israel Stein]

At 07:49 AM 2/4/2007, Robert Scott wrote:
>>RicB said: "Inharmonicity is a distinctive characteristic of pianos."
>>Ric,
>>I competely disagree. All instruments have inharmonicity; just less 
>>than the piano.
>>Joe
>Well, not exactly.  A pipe organ does not have any inharmonicity. 
>Neither does a violin (when it is being bowed).  These instruments 
>produce true harmonics that are exactly multiples of their 
>fundamentals.  And some instruments have more inharmonicity than a 
>piano - like for instance chimes.
>
>Robert Scott
>Ypsilanti, Michigan

Sorry, Robert.

You are correct about the organ, but incorrect about the violin. 
Inharmonicity is a property of vibrating strings, caused by stiffness 
of the material of which the strings are made. Stiffness causes a 
distortion of the string's vibration at the termination points - 
there is a part of the string that is more or less straight, before 
the "waves" can start. The result is that those sections of the 
vibrating string (1/2, 1/3, 1/4) etc. that give rise to the overtones 
(or partials above the first) are shortened to a greater proportion 
of their length by that distortion as one goes up the overtone 
series. If the distorted section of string is, say, 2 mm - that is a 
greater proportion of 1/4 of the string than of the whole string. 
This is what gives rise to inharmonicity - a shorter section produces 
a higher frequency, and so the higher you go up the overtone series, 
the sharper is the partial relative to the previous ones.

The stiffer the string, the greater the distortion - and therefore, 
the inharmonicity.

Stiffness is affected by three factors: density of the string 
material, thickness of the string and tension of the string. To have 
a string with zero inharmonicity, you must have a string with zero 
stiffness which is impossible, since that would require a string made 
of something with zero density, of zero thickness under no tension...

Since the modern piano uses very thick steel strings (some with 
copper windings) under a great deal of tension, its inharmonicity is 
through the roof and requires special treatment in tuning. In 
violins, on the other hand, the inharmonicity is negligible - thin 
strings under low tension...

Israel Stein