In Search of the Missing Fundamental: by Richard K. Jones
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Historical Influence

From a physics of vibrating circular membranes perspective, when you strike modern timpani in the center, you excite a different set of modal vibrations than when you strike near the edge; primarily concentric vs diametric modes respectively. Using the concentric modes (or composites) to help produce pitch is not unwarranted, as this is how the heavily loaded head of Tabla produces a sense of pitch57, but will that work for timpani? Interestingly enough, Rosssing et al.,25 Fleischer and Fastl,54  Tronchin,55 and others have documented that the sixth mode of vibration (mode 1,2) and the second mode of vibration (mode 1,1) of an air loaded/baffled membrane (i.e. a modern timpano in this case) is very close to a 2:1 ratio, i.e. the octave.  This sixth mode of vibration, mode 1,2 (which is a composite mode) is also very close in frequency to preferred mode 3,1 which is also the octave above the principal tone mode 1,1 as well.

Mode 1,2 is unique in that it is a combination of both, a single diametric, and two concentric modes. Mode 1,2 does not radiate energy very efficiently; it has somewhat of a quadrupole type behavior. Thus, the mode 1,2 takes a relatively long time to decay compared to mode 0,1, the actual fundamental, which would be excited when struck  slightly off-center. It is in fact mode 1,2 that helps create the “Octave Harmonic” effect used by mid 20th century composers, including Elliot Carter. Is the lack of efficient radiation of mode 1,2 strong enough to produce a viable pitched sound, or just a thud as is does on modern timpani?

Mode 1,2Mode-1,2Figure 2e.1
The sixth mode of vibration mode 1,2, which vibrates with two
concentric modes of vibration and one diametric mode of vibration:
the nodal lines will encompass the entire circumference and diameter of the head

Mode 1,2 in motion
Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State

If the head was struck in the center or slightly off-center, the acoustic effect would be a pitch comprised of a weak principal tone, an octave and possibly a tenth, if the head was loaded properly (i.e., mass/thickness). Because of the presence of the bowl, the concentric modes radiate their energy efficiently, which means that they decay very quickly and are audibly weak; a sound which is considered unmusical by today’s standards. However, since mode 1,2 takes a relatively long time to decay, provided that mode 1,2 was excited to some extent as the head was struck slightly off-center, a period drum with a schalltrichter placed close enough to the head might be able to produce a sound with the acoustic power of these modes enhanced to some extent. If the diameter of the bell were large enough compared to the diameter of the bowl, and placed close enough to the head, the schalltrichter would also influence the actual frequencies of the audible modes.

The acoustic energy of these modes would be directed through the horn and out the bottom of the drum, as well as to the bowl and the horn itself adding extra resonances.  If the sound were strong enough, the spectrum of a quasi-harmonic octave would certainly be musical enough, it just wouldn’t be very resonant or have much sustain. As mentioned earlier, this method of producing pitch (striking the drum dead/off center) is in fact how noted musicologists Edmund Bowles 53 and  John Michael Cooper 29 believe timpani produced pitch until the very end of the 18th century.

This method of striking and playing the drum is in fact still used today by traditional Cuban Timbaleros. Below is a short video showing the traditional approach to tuning Cuban Timbales, and playing the Cuban “danzón” on Timbales Cubanos. Notice where the timbalero is striking the timbales for the primary notes; near dead center on *do* and just slightly off center for *sol* yet there is still strong sense of the key of F. When he strikes the timbales in the more traditional location, the instruments produce a weak, thin sound.

Tutorial on how to tune Cuban Timbales, and play Cuban Danzón with Canelo Vasquez, drummer and director of the Super Lamas Group of Actopan.

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