Drum membrane modes are not harmonically related

Evidence

Like a string, a thin membrane has little resistance of its own to bending and so derives its resistance to deformation from the tension that is applied at its extremities—for example, a rim in the case of a drum. Even if the rim is a simple shape (circles, ovals, squares, and the like, all lying in the same plane), the partials are not harmonically related. The frequencies of the particals depend on the tension and the shape of the rim. — Eric J. Heller, Why You Hear What You Hear- An…, 1357

The frequencies of the modes can be calculated mathematically, but the result is rather more complicated than for the bars. Table 4.2 gives the frequencies of each mode relative to the first mode (Fletcher and Rossing, 1999) and the equivalent number of semitones (calculated using the equation given in Appendix 3). As with the bars, none of the modes is an exact number of equal tempered semitones apart or in an integer ratio, and therefore they are not harmonically related. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1291

Categorization

• XREF: Directly relevant to why we hear drums as less ‘pitched’ than strings — a core auditory-perception question. — connects to: Higher drum modes self-compensate and leave kettle volume unchanged; Drum-kit drums are deliberately tuned non-pitched so they fit any musical key; A free-edged cymbal plate produces inharmonic, mixed modes (+1 more)
• QUESTION: What makes a tabla’s density variation manage more harmonic modes? The book flags this as an exception worth chasing.
• corroborated: 4 independent passages support this claim — Eric J. Heller, Why You Hear What You Hear- An…, 1357; David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 1291

Broader Connections

• The tabla achieves near-harmonic modes through membrane density variation #complicates (1.00) — Eric J. Heller, Why You Hear What You Hear- An…, 1358