Membranes of any shape lack harmonic spectra, unlike strings

Evidence

Once again, for N beads there are N modes, each with a different frequency. As the number of beads grows, the new membrane modes recapitulate all the old ones and add new ones, just as in the case of a string. Figure 8.10 shows a rectangular array, but the array can be made circular or indeed of any shape. None of these shapes has an equally spaced, or harmonic, spectrum. This corresponds with the fact that drumheads when struck don’t give musical tones in the sense of a plucked string, although the degree of dissonance depends strongly on where the membrane is hit, which affects the amount each of the modes is excited. — Eric J. Heller, Why You Hear What You Hear- An…, 861

Categorization

• XREF: Explains why drums don’t produce definite pitch — a question that follows directly from hearing the difference between a drum and a stringed instrument.
• QUESTION: What exactly determines ‘degree of dissonance depends strongly on where the membrane is hit’ — is there a spot that minimizes excitation of the most problematic modes?