Basic physics of the ideal circular membrane
We begin by analyzing the resonant properties of an ideal circular membrane. although not restricted to a circular shape, many drums feature this property, providing a convenient starting point for discussion.
We solve for the harmonic frequencies of an ideal, thin, homogeneous, stretched, circular membrane using Bessel's functions. we assume the outer circular edge of the membrane constitutes a fixed boundary condition, as with any standard drum. for such a membrane we find the fundamental frequency inversely proportional to the radius, directly proportional to the square root of the tension, and inversely proportional to the square root of the mass per unit area.
Because of the nature of the material and its boundary conditions, the vibrational energy exhibits different observable "modes". each of these modes represents a manner in which the material moves in response to the vibrational energy. we distinguish these modes by noting which areas of the membrane moves, and which areas do not. we find it convenient to designate the stationary points on the surface of the membrane where the material remains in a fixed position, as "nodes". the nodes, in effect, draw boundaries around the material which vibrates. these boundaries, or nodes, consist of three basic types: nodal points, nodal diameters and nodal circles.
For circular membranes, we designate the normal modes of vibration by the notation "(x,y)", where x indicates the number of nodal diameters, and y indicates the number of nodal circles. we leave nodal points out of this discussion for simplicity, due to the rarity of the phenomenon. to see the first 14 modes of an ideal circular membrane, their mode designations, and their relative modal frequency, click here. note that, none of the modal frequencies consist of multiples of the fundamental, and thus do not constitute a harmonic series. note here that two headed drums complicate analysis by introducing coupling between the to resonating membranes. (click here to see some details on that subject). also note the addition of each diametric division of the membrane results in the next harmonic mode (e.g. 3 diametric divisions, third harmonic). whereas the addition of each circular node results in the next odd harmonic (e.g. three circular nodes, fifth harmonic). when considering the circular nodes, always consider the fixed boundary of the membrane itself.
Acoustic properties of the timpani
By modifying or introducing certain design features, we may emphasize particular overtones, and even alter them completely. a carefully tuned classical western timpani is known to have a strong principle note, as well as two or more harmonic overtones, including a prefect fifth, major seventh, and an octave. these overtones come from the (2,1),(3,1), and (1,2) modes respectively. furthermore, recent measurements indicate the modes (1,1), (4,1) and (5,1) have ratios 1, 2.44, and 2.9 respectively. both of these represent frequencies within a semitones, from the ratios 2.5 and 3, respectively. thus the first five nodal diameters (0,1),(1,1),(2,1),(3,1), and (4,1) give the timpani the frequency profile of 1:2:3:4:5:6 -- yielding a strong sense of pitch. the timpani employs several features to alter the overtones of an ideal circular membrane.
The largest factor for the "correction" of the overtones, into a close approximation of a harmonic series, stems from the mass of the air which the membrane vibrates against. the timpani features a large surface area and thus interacts with a large volume of air. this air mass serves to lower the frequencies of the principle modes of vibration. the shape of the timpani's large conical shell exhibits resonance properties of its own. modes with similar shapes interact and reinforce each other, though the medium of the air trapped inside it the timpani. the stiffness of the preferred timpani membrane, raises the frequencies of higher overtones. all of these properties shift the harmonic overtones and result in a close approximation of a harmonic series (from which to designate pitch).
Acoustical properties of the bass drum
With similar design features as the timpani, the bass drum also exhibits these features. a large symphonic bass drum exhibits a near harmonic series in the low frequency range from 32hz to 200hz. however, the ear hears frequencies above 200hz much better than between 32hz - 200hz. inharmonic frequencies above 200hz saturate the bass drum's frequency spectrum, and thus gives the drum an undecernable relative pitch.