The mathematics of piano tuning
Albert E. Sanderson · The Journal of the Acoustical Society of America · 1989
It is a well-known fact that piano strings vibrate inharmonically—the partials actually run sharp by amounts that vary from piano to piano and from note to note on any one piano. This affects the tuning significantly, and it is correct to say that no two pianos should be tuned exactly alike. This paper presents a mathematical solution to the problem of tuning such instruments for typical classes of inharmonicity versus note-number curves, the most important case being a straight-line logarithmic increase of about 3:1 per octave. Equal temperament is literally impossible in this case, and if one forces it with an electronic tuning, the instrument sounds very unmusical. The octave is no longer exactly 2:1, and must be redefined to allow for different types of octave partial matching. More general definitions of equal temperature and octaves lead to a tuning solution in the form of a difference equation. Although no two octaves or even semitones are exactly the same width, they are consistently wide by an amount that grows wider as the inharmonicity rises. Comparison with pianos tuned aurally shows that fine piano tuners have been solving this equation subconsciously for centuries.