The

frequencies of the

overtones of an

ideal taut string are easy to calculate: they are simply

harmonics, integral multiples of the

fundamental pitch. Real strings follow this model very closely, with only minor deviations from the true

harmonic series caused by stiffness (

inelasticity) and irregular

density, as any

chordophone player who uses

harmonics can attest. The overtones of an ideal

membrane, however, are much more complicated.

All of a string's modes of vibration have one simple kind of node, or place where the string remains stationary: points that divide the string into equal segments. Circular membranes, on the other hand, have two kinds of nodes: diametric nodes which are diameters of the circle, and circular nodes which are circles concentric with the circumference (they both divide the membrane into sections with equal areas). The modes are designated by pairs of numbers the first of which represents the number of diametric nodes and the second the number of circular nodes. Thus the simplest mode (the fundamental), the one in which the entire membrane vibrates up and down and the only node is a circular node at the rim, is called (01). The mode (07) would look a target with 7 concentric circles including the bullseye and the mode (51) would look like a pie cut into 5 slices. In any given mode one section is always moving in the direction opposite from its adjacent sections.

The respective frequencies of these modes of vibration are not harmonics, as is the case with a string, but non-integer multiples of the fundamental given by bessel functions. Here are the frequencies of some of the first modes given in both multiples of the fundamental and cents:

(01) 1.00 0¢

(11) 1.59 803¢

(21) 2.14 1317¢

(02) 2.30 1442¢

(31) 2.65 1687¢

(12) 2.92 1855¢

(41) 3.16 1992¢

Of course in a real membrane the overtones will not have exactly these values and in fact can be made to line up in an approximation of a harmonic series, giving a strong sense of pitch as in a tympani or kettledrum.

(information taken from a __Scientific American__ article titled "The Physics of Kettledrums.")