Schuck, O.H., Young, R.W. (1943). Observations on the vibrations of piano strings. J. Acoust. Soc. Am. 15, 1-11

The behavior of single piano strings struck in their normal positions was investigated through measurement of the frequencies of the partials and their amplitudes as functions of time after striking. Three-dimensional figures showing partial amplitudes versus time are presented. The frequencies of the partials of the tone were found to be those of the normal modes of vibration and are therefore referred to as the modal frequencies of the string. The higher modal frequencies aare progressively sharp with respect to true harmonics of the fundamental; the sharpening follows approximately a square law with respect to mode number, to the extent that in one case the 15th partial had just about the frequency of a 16th harmonic. The resulting inharmonicity in piano strings is least in the two octaves below middle C; it rises sharply at the high end and more or less sharply at the low end, de- pending upon the type of piano. A correlation between inharmonicity and subjective tone quality rating is suggested. It is shown that the commonly observed "stretched" tuning (sharpening of the treble, flattening the bass) is a natural consequence of the inharmonicity.