Inharmonicity

Every note we play on a piano exists of a fundamental and a few partial sounds.
In short the inharmonicity of a string is the deviation of the different partial sounds according to each other and according to the fundamental of the string..

If for instance the fundamental is 440Hz, the first partial is 880Hz.
When that first partial is not exactly 880Hz but for example 880.4Hz we call this difference the inharmonicity.

The inharmonicity is determined by the stiffness of the string.

The stiffer the string, the higher the inharmonicity.
A short thick string is much stiffer than a long thin one producing the same sound.
The inharmonicity of the bass strings of a concert grand piano is much lower, because the strings are much longer and thinner, sounding much lower than those of a baby grand.

Usually one cannot hear the fundamentals of the lowest bass strings. Partially due to the tones sounding on the limit of the human sense of hearing, but mainly because the soundboard is not capable to transmit these tones well.

Even listening to a small transistor radio, that does not give low frequencies at all, we can tell the difference between a small and a big instrument.
The human ear is used to construct a fundamental from the different partials.
The inharmonicity is perceptible with great ease.
The level of inharmonicity can tell us if we are listening to a grand piano or a small upright.

What do we perceive as a sound being  'beautiful'?

That, of course, is a matter of taste, but maybe habituation takes a part too.
Very small instruments (high inharmonicity) have a sound we usually don't like so much.
But do we like the sound of an enormous instrument like the Klavins 370?
This instrument has an extreme low inharmonicity and a enormous big soundboard too.
Thanks to that colossal soundboard it should be able to radiate the lower sounds.
Still we have to get used to this sound.

Can we change the inharmonicity of an instrument?

Most piano manufacturers have started to determine the scale by trial-and-error.
(the scale comprises the length and thickness of the strings)
Renowned manufacturers have had more possibilities to experiment themselves, but the small ones often copied the scale from others.

It was only in the fifties of the last century people have developed a formula which can be used to calculate the inharmonicity.



In a restoration project it is not possible to change the length of the strings, but by adapting the thickness of the steel and the copper of the windings, it is possible to greatly improve the sound and tuning of the instrument.