We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.
The strings of a piano have some degree of stiffness; this can be modelled using the Euler-Bernouilli beam model. For a pinned string we compute the corresponding spectrum, which turns out to be slightly inharmonic. Therefore, equal temperament (12 equal divisions of the just octave) is not the best choice to tune a piano. We study several possible tunings aiming to improve its consonance. A good solution is obtained if one tunes a note and its fifth by forcing the beats among their third and second partials, respectively, to disappear.