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 slightly off-center the acoustic effect would be a pitch comprised of a heavily muted fundamental, 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.

Striking the drum in the center or slightly off-center seems foreign to the modern timpanist because when done on modern timpani with modern heads, the sound is very dull, non resonant, and has very little musical appeal. This happens because modern timpani have larger diameters with deeper bowls, much thinner heads and superior mechanical tolerances. For the very early instruments, the thickness and homogeneity of the heads was the primary factor influencing on the sound. Early timpani heads were produced by vellum and parchment makers. The process of treating animal skins and producing various parchment and vellum products is a millenniums old tradition. Even though the manufacturing principles remained unchanged throughout the history of parchment and vellum making, the end product varied greatly. This variance can be a consequence of things like the maker’s expertise, the quality of the skins and different raw materials used in the production processes, and the desired outcome of the end product. It could also be argued that the wealth or economy of the making process was reflected in the degree of thinness and finish. There is no doubt that various degrees of thinness and finish were produced so there is little question that the timpanists in the wealthy courts of the period could have had any thickness and finish they desired. 30

The head of choice was a less flexible, half tanned hide which was much thicker than standard “finished” vellum products. This was not because of lack of resources, but by design. Thicker heads were needed because of the impact that it had to endure from outside playing.29.1  The thickness made it less resonant so striking it slightly off-center projected the sound better and produced more of a pitch than on the edge due to the poor mechanical tolerances of the tensioning system, small diameter and  shallowness of the bowl.

Other factors to be taken into consideration, of course, are the lack of pitch standards in Europe at that time, which would affect the frequency of the actual fundamental of the smaller drums with thicker natural-skin heads. This lack of pitch standards would also influence the size of the bowl needed to produce the pitch, which helps explain the varying sizes of early timpani found throughout 16th and 17th century Europe.

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