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

Rossing et al.

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

The listener needs to learn to listen for, and then harvest mode 1,1 and ignore mode 2,1.  Mode 2,1 should be present in the spectrum, but subdued, while mode 1,1 should have the most presence. This technique can take years of practice to master when tempering by ear alone.

Since mode 2,1 tends to have more energy than mode 1,1, it will add sustain to the sound which is desirable. However, don’t be deceived by this sustain; mode 2,1 can become a wolf in sheep’s clothing  if you follow it too closely.

When the frequency of the principal tone (mode 1,1) is not strong and consistent from lug to lug, the overall strength of perceived pitch is severely diminished and permutations of the more audible mode 2,1 (the fifth) can tend to dominate the spectrum. An overbearing fifth generates a pitch shift once the principal tone (mode 1,1) begins to decay.

The graphic below shows the prominence of mode 2,1 (the fifth) in a normal timpano spectrum. Once the overbearing fifth becomes pervasive, it can quickly mask the other preferred modes because it has the most energy, which makes it more audible with a longer sustain.

Waterfall chart (frequency, time and amplitude) of a timpano sound spectrum

(single struck note) highlighting six preferred modes (1,1), (2,1), (3,1), (4,1), (5,1) and (6,1)
(Fleischer & Fastl)

This overbearing fifth is especially noticeable once the head becomes worn or becomes out of clear. It can also happen if a head has not been tempered properly during the initial mounting and is then continuously played;  this causes it to become unevenly stretched the more it is played. Unlike animal skin heads that have more natural elasticity, Mylar heads tend to develop a stretch memory. Once the head is consistently stretched to where this overbearing fifth is prominent in the spectrum, it is often best to just replace the head and temper the new head properly.

Ironically, what the manufacturers of synthetic timpani heads head don’t publish about their products is that excessive stretching and flexing are the primary cause of deterioration with Mylar. Mylar’s ability to stretch and return to its normal state (tensile strength) makes it a great replacement for natural skin heads. However, the constant fluctuation in tension and beating of the head contributes to an elongation of the material causing it to become brittle and shortening its service life. This is the primary reason why you can’t remount Mylar heads with any great success; the collar crease is too brittle and has lost all malleability. Other culprits contributing to deterioration are heat and humidity, especially direct sunlight and water. Applying a polymeric coating to the head surface occasionally can extend the chemical integrity of the Mylar. Heat guns, hair dryers, irons etc. will all suck the life right out of a head and the dollar$$ out of your wallet.

If a head replacement with correct tempering is not an option, the overbearing fifth can be mitigated to some degree by following Step No. 5 and Step No. 6 found in Chapter Five.

When the head is properly mounted and tempered, the virtual pitch of the preferred modes convinces the human auditory system that it is hearing harmonic pitch even though it is not.