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Calculating Inharmonicity

As mentioned in "Tension" piano strings are not completely flexible. They behave like bars to. In a bar the high frequency parts of a wave spread out faster than the rest. A wire with stiffness acts like a flexible string for low frequencies and like a stiff bar for high ones. The more the tension is the restoring force of a string, the less the stiffness influences the sound. It is therefore that string diameter at a given string tension is chosen close to the maximum load.

Fletcher and Rossing state that Morse shows that the allowed frequencies of a string with small stiffness can be written:


where:
fn = frequency of the n-th partial
n = partial number
f0 = fundamental frequency of the same string without stiffness

π = 3,1416
E = E-modulus
d = string diameter
l = speaking length
T = string tension

They further state:

The inharmonicity In, wich expresses the ratio fn/nf1, is given by:
,
where b = 866B is the inharmonicity coefficient in cents.

In fact this is a simplification of the formulas of Morse, who attached conditions to the validity of his formulas. It is clear that not all strings of a piano can fulfil these conditions. As the simplification of Fletcher and Rossing is most widespread I have used this in the spreadsheets to calculate the inharmonicity coefficient, I called it δ.

With this the frequencies of the partials could be calculated like explained in “Frequency”, but the results of these calculations prove not always to correspond with measurements. There are several possible reasons for this among which the formulas cover not the behavior of the strings in all circumstances. Like strings are influenced by the behavior of bridge and soundboard. One of the conditions for the validity of the formulas of Morse is that the endings of the strings are motionless fixed.
Nevertheless, with this calculation of inharmonicity possible unwanted differences in the sound of unisons can be revealed.


 
 
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