The movement of the membrane that produces a sound emitted by an ideal drum can be decomposed in a countable quantity of vibrations. The profiles of these vibrations can be described by means of the Dirichlet eigenfunctions corresponding to the shape of the membrane. To each eigenfunction it is associated a positive number called eigenvalue. The square root of this value represents, up to a multiplicative constant depending on physical parameters, the frequency of the sound produced by the vibration described by the corresponding eigenfunction. In this seminar, after a revision of the classical theory, we introduce a new type of eigenfunctions that satisfy orthogonality constraints with respect to a given family of functions. Finally, time permitting, we treat some shape optimization problems regarding the corresponding eigenvalues. This talk contains a part of the results achieved working with Dorin Bucur and Davide Zucco. The talk will be part of the ASK seminar series.