Unlike one-dimensional vibrating air columns and vibrating strings, vibrating circular membranes are two-dimensional. A two-dimensional circular membrane can have many modes of vibration occurring simultaneously; symmetrical (concentric modes), asymmetrical (diametric modes), and composite modes (which are combinations of the two). With respect to how modern timpani produce pitch, it is the asymmetrical (diametric) modes, and not the symmetrical (concentric) modes, or the composite that produce the sustained sense of pitch. For good timpani sound (one that is rich with near harmonic overtones), you want to mute the inharmonic symmetrical and composite modes as much as possible and project the quasi-harmonic asymmetrical modes. Furthermore, of the many asymmetrical modes that can be generated by a vibrating circular membrane, there are only five or six of these modes that actually contribute to a timpano’s sound spectrum with regard to giving the instrument its harmonicity. These modes are called the preferred modes and are found in the lower diametric modes (1,1), (2,1), (3,1), (4,1), (5,1) and sometimes (6,1).
Preferred Modes (1,1) – (6,1)
The other audible modes (symmetrical, asymmetrical, and composite) are significant because they contribute to envelope and timbre giving the instrument its unique tonal characteristics. However, since the other modes don’t really contribute to timpani harmonicity, they will not be discussed here; the focus will be on how these preferred modes are shifted or coaxed into near harmonicity. Keep in mind that the other modes of vibration still contribute to the overall sound of the instrument as well as does collateral color (also known as parasitic pitch), which are the potential hyper-resonant frequencies generated by the vibrations of the bowl, frame and other mechanical parts of the drum.
N.B. The objective when tempering a timpano is to adjust the tension of the head whereby the vibrations of the membrane emphasize the preferred modes, and at the same time, suppress the inharmonic partials (concentric and composite modes) as much as possible so that the preferred modes can dominate the spectrum. Having the preferred modes dominate the sound spectra is incumbent on striking the head in a manner, and at a location that fully excites the preferred modes and de-emphasizes all other (in-harmonic) modes of vibration.
As a baseline for comparison, cents will be used as the system of measurement when measuring the harmonicity of the preferred modes and resulting partials. Each Equal-Tempered half step has 100 cents and there are 1200 cents in an octave however, Just intervals (e.g. the harmonic series) are not always even multiples of 100 cents like Equal-Tempered intervals. For more information on “musical cents,” visit the following links.
Frequencies for the various permutations of the preferred modes will be translated into musical pitches called pitch-class equivalents and notated on the Grand Staff. Some of the actual frequencies may not correspond exactly to the frequency of the actual pitch and will be notated on the nearest approximate pitch.