Harmonic Vibrational Modes Research Articles

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 in cents (hundredths of a semitone) is given by 3.4 × 1013n2d2/ν02l4, where n is the mode number, d is the diameter of the wire in cm, l is the vibrating length in cm, and ν0 is the fundamental frequency. A value of Q/ρ = 25.5 × 10 10 (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 roughly 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.

An Electromagnetic Monochord for the Measurement of Audio Frequencies

The desirable properties for an instrument for measuring frequencies over the audible range are briefly mentioned and a suggested electrically excited monochord is considered. In practice certain difficulties are found and lead to a proposal for employing the harmonic modes of vibration of the stretched wire. The theory of this is given, as well as the relation between the frequency response and the position of a movable bridge. The constructional details and the method of calibration from a single tuning fork are described in turn. Miscellaneous considerations concerning the practical operation conclude the paper.