Pleading the Fifth
When you strike a well-tempered timpano in the normal playing area, you are exciting a set of vibrating membrane modes called the preferred diametric modes; in turn, you are not exciting the primary set of vibrating membrane modes called the concentric modes to any significant degree. Because of the location of where the membrane is struck, the inharmonic concentric modes (symmetrical patterns) radiate their energy very efficiently, which means that their audible sound dies away almost immediately. The more harmonic diametric modes (asymmetrical patterns) are not efficient at radiating their energy so their sound dies away more slowly.
The actual fundamental of the membrane belongs to the primary inharmonic concentric modes. The sound humans hear as being the pitch of the drum (the principal tone) is being generated by an overtone of the secondary set of modes, and not the fundamental of the vibrating system. This main pitch is called the principal tone; from a musical perspective, it should not be referred to as the fundamental because the actual fundamental of the membrane is quite dissonant with respect to the partials that produce the sense of pitch and harmonicity for timpani. In essence, the spectrum of any timpano’s sound is comprised of a group of secondary quasi-harmonic overtones encapsulated within an initial core of noise.
Another interesting phenomenon occurs because of the attached bowl; it creates a baffle, which damps the lower modes of the vibrating head of a timpano to some degree. This is a good thing because it helps to suppress the naturally dissonant fundamental of the vibrating membrane, concentric mode 0,1. However, due to the effect of this baffle, not only are the lower concentric modes damped (i.e. mode 0,1 the dissonant fundamental), but so also is the principal tone, diametric mode 1,1 to some extent.
Fig. 3l
Figure 3l charts the decay times of the lower mode frequencies of a timpano both with and without the bowl. Without the bowl, all of the modes are not very efficient at radiating their energy; therefore they have longer decay times, which seems desirable. However, this includes the inharmonic lower concentric modes as well as the more harmonic preferred modes. With a bowl, most of the lower modes radiate energy more efficiently and consequently they decay much faster. When a timpano is struck a quarter of the distance between the edge and the center, the inharmonic modes will radiate their energy much more efficiently and decay faster leaving the more harmonic preferred modes to dominate the sound spectrum.
Too much damping of this principal tone (mode 1,1) is not a good thing because it can lead to what timpanists refer to as the overbearing fifth and pitch creep. This occurs when mode 1,1 (principal tone) does not have a strong and sustaining presence in the spectrum. The objective when tempering a head (a.k.a. clearing or balancing) is to adjust the tension of the membrane to where the principal tone (mode 1,1) sounds stronger than any of the upper overtones.
With mode 1,1 being the principal tone, the next quasi-harmonic partial is mode 2,1. When vibrating in the (2,1) mode a circular membrane acts much like a quadrupole source. A quadrupole source is less efficient at radiating energy than the (1,1) dipole source, and much less efficient at radiating energy than the (0,1) monopole source. This means that mode 2,1 transforms its vibrational energy into sound energy better than either than mode 1,1 and mode 0,1. Therefore mode 2,1 takes longer to decay than either mode 0,1 or 1,1, and thus contributes significantly to the musical pitch of a drum.
The mistake many people make when tempering heads is listening too intently to mode 2,1, instead of focusing on mode 1,1. This happens naturally because:
1) the effect of the baffle makes mode 2,1 stronger than mode 1,1
2) mode 2,1 radiates its energy less efficiently so the sound it creates takes longer to decay
3) mode 2,1 is a strong component part of the overall perceived pitch
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