Influence of inharmonicity on the tuning of a piano—Measurements and mathematical simulationa)

Source abstract

It is well known that inharmonicities in piano string vibrations and especially any irregularities in their progression, are partially responsible for the difficulties encountered in tuning the instrument. This paper studies the quantitative influence of inharmonicities by including their actions in the mathematical relations governing musical intervals. The measurement of fundamental and rank 2, 3, 4, 5 partial frequencies of the piano strings is discussed, and a method using a relatively simple electronic apparatus is described. Results obtained on a grand piano are given. These values are entered in a mathematical simulation of tuning. The first-hand calculation procedure used successive steps inspired from the classical tuning process: proposed beat rates, verification on other intervals, corrections. A much easier simulation is now obtained with a personal computer using either a commercially available spreadsheet software, or a specifically developed small program written in basic. Results of the simulation are precisely determined fundamental frequencies. Test of tuning the measured piano by zero beating it with calculated frequencies was successful. Several similar experiments were made on other pianos, leading to the idea of a ‘‘personalized tuning’’ for each instrument, provided an adequate electronic generator is available.