In Search of the Missing Fundamental: by Richard K. Jones
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Carter: “Octave Harmonics”

The Science

Can timpani produce true harmonics like many other musical instruments?  No. True harmonics are a result of a vibrating system where the overtones/partials are musical tones, which contain frequencies that are integral multiples of the frequency of a fundamental tone. Since timpani heads do not vibrate with a natural harmonic series, true harmonics are not possible. One must always keep in mind that the spectrum of any timpano’s sound is a lingering quasi-harmonic series without a fundamental, encapsulated within an initial core of noise. Furthermore, the sound we hear as being the pitch of the instrument (the principal tone) is actually not the fundamental of the vibrating head, but rather the second mode of vibration or the second partial.

The overtone series we hear as being the pitch of a timpano starts on what would be the second partial of a quasi-harmonic series; that is an overtone series, without a fundamental, which is only close to being harmonic. It is the missing fundamental and the lack of a true harmonic overtone series that prevents the creation of true harmonics on timpani.  The principal tone (the one we hear as the fundamental pitch of the timpani) is actually the second partial of the vibrating membrane.  The actual fundamental (inharmonic to the timpano’s spectrum) is dampened as much as possible. By trying to excite a harmonic partial above the principal tone mode 1,1, in essence you are trying to elicit a harmonic overtone from a inharmonic partial; this is something that is physically impossible to do. So, what is actually happening when you produce this octave harmonic effect?

Timpani have three modes of vibration occurring simultaneously; symmetrical (concentric modes), asymmetrical (diametric modes), and composite modes (which are combinations of the two). 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. The modes that give timpani their strong sense of pitch are the lower asymmetrical modes and are often referred to as the preferred modes.

Within the asymmetrical preferred modes of an air loaded vibrating membrane, there is a partial with a frequency that is very close to the octave above the principal tone mode 1,1. This is the preferred mode 3,1, which is the fifth partial of a vibrating membrane.  However, the techniques timpanists use to produce the octave harmonic technically do not isolate and emphasize this octave partial above the principal tone (mode 3,1) instead, they uses both the symmetrical and asymmetrical modes (composite) to produce the sound by exciting primarily the sixth mode of vibration, the composite mode 1,2 (fig. 1).  At the same time they are de-emphasizing the preferred modes so that the sound of mode 1,2 can project.

Mode 1,2 is a composite mode consisting of one diametric and two concentric modes of vibration. Rosssing et al., Fleischer and Fastl, and others have documented that the ratio of mode 1,2 and mode 1,1 of an air loaded/baffled membrane (i.e. a modern timpano in this case) is very close to 2:1, the octave.25  This sixth mode of vibration is also very close in frequency to preferred mode 3,1, which is also a 2:1 ratio with the principal tone mode 1,1.

Mode 1,2 is unique in that it is a combination of both, a single diametric, and two concentric modes. Being a composite mode, mode 1,2 does not radiate energy very effectively; it has somewhat of a quadrupole type behavior. Thus, the mode 1,2 takes a relatively long time to decay.

Mode 1,2Mode-1,2Figure 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
mode12

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