Inharmonicity of Plain Wire Piano Strings

The inharmonicity of plain wire strings in situ has been measured in six pianos of various styles and makes. By inharmonicity is meant the departure in frequency from the harmonic modes of vibration expected of an ideal flexible string. It is shown from the theory of stiff strings that the basic inharmonicity cents (hundredths of a semitone) is given by $3.4×1013n2d2/v02l4$⁠, where n is the mode number, d is the diameter of the wire in cm, l is the vibrating length in cm, and v₀ is the fundamental frequency. A value of $Q/ρ = 25.5 × 1010 (cm/sec)2$ was assumed for the steel wire, where Q is Young's modulus and ρ is the density. The observations are entirely compatible with the relationship given. In general terms the inharmonicity of the plain steel strings is about the same in all the pianos tested, being about 1.2 cents for the second mode of vibration of the middle C string. Above this point every eight semitones it is doubled. Below middle C the inharmonicity is consistently less in large pianos than in small ones.