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

Figure 3d shows the pitch-class equivalents of the preferred modes of an ideal membrane notated on the Grand Staff with the dampened fundamental (A2 110 Hz) included as a reference. Also included are the ratios and the number of cents for each of the preferred modes of an ideal membrane as calculated by Berg & Stork. (click fig. 3d to enlarge) The accompanying sound file does not include the dampened fundamental referenced in figure 3d, just the preferred modes.

Fig. 3d

 

Interestingly enough, the preferred modes of an ideal membrane set up a harmonic series as if it began on the third partial of a complete series. In this case it is the second partial as well as the fundamental that are missing.

Figure 3e is a visualization of the six preferred modes of an ideal vibrating membrane. Beginning with no modes of vibration, the preferred modes are displayed as each mode from (1,1) to (6,1) is added incrementally. Notice as how the modes are added, the vibrating motion around the perimeter of the membrane increases.

Fig. 3e

Since humans don’t live in a vacuum, the ratios of the partials of the preferred modes of an ideal membrane change when factors such as the air surrounding the head and the timpano bowl are introduced into the equation. The most significant being the air above the head and the air inside of the bowl. This phenomenon is called air loading and is the main factor responsible for establishing the near harmonic relationship among the preferred modes.

When the effects of air loading are introduced, the inharmonic partials of the preferred modes of an ideal membrane are coaxed into a near harmonic sequence, which is very near a harmonic series without a fundamental. The next section explains this phenomenon.

Harmonic Alignment of the Preferred Modes
of an Air Loaded Membrane

Timpani-Pitch-Cents

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