In Search of the Missing Fundamental: by Richard K. Jones
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Review 3

Significant Investigations into the Acoustic Properties of Timpani:

  1. Lord Rayleigh: John William Strutt (1877)
  2. P. R. Kirby (1930)
  3. Henry W. Taylor (1964)
  4. Arthur H. Benade (1973)
  5. Thomas D. Rossing (1976, 1977, 1982, 1998, 2000)
  6. Donald L. Sullivan (1996)
  7. Lamberto Tronchin (2004)
  8. Helmut Fleischer and Hugo Fastl (2005, 2008)

What Determines a Timpano’s Sound Spectrum?

  1. material, integrity, positioning and tensioning of the membrane
  2. bowl integrity: roundness – flatness, shape and thickness of the bearing edge
  3. with what, where and how the timpano is struck
  4. the density of the air mass above the timpano membrane
  5. the volume and density of air inside of the bowl
  6. viscothermal characteristics of both the internal and external air systems
  7. the stiffness and resonance of the air in the bowl
  8. the stiffness of the timpano membrane itself
  9. the material(s) from which the bowl, frame and mechanical parts are made
  10. acoustic radiation of the modes

The above factors influence the frequencies of a small group of vibrating modes called the preferred modes, which decay slow enough that the resulting partials create a narrow quasi-harmonic series from which we are able to discern a pitch. The volume and density of the air (air modes) inside of the bowl, the density of the air outside of the bowl, and the membrane make up a single system; the three parts are of equal importance in determining the frequencies and overall vibrational shapes (preferred modes) which define the pitch of the instrument.

What are the Preferred Modes?

Of the many modes that can be generated by a vibrating circular membrane, there are only five or six of these modes that actually contribute to a timpano’s sound spectrum with regard to giving the instrument its harmonicity or sense of pitch. These modes are called the preferred modes and are found in the lower diametric modes (1,1), (2,1), (3,1), (4,1), (5,1) and sometimes (6,1). The other audible non-harmonic modes contribute to envelope and timbre and help give the instrument its unique characteristic color.

What is Air Loading and Why is it Significant?

Air loading is the effect the weight (density) of the surrounding mass of air (both inside and outside of the drum) has on the motion of the timpano head. It lowers the natural frequencies of vibrations from those of a ideal circular vibrating membrane. This effect is strongest for the lower modes especially mode (1,1), and plays a significant role in the adjustment of the inharmonic partials. The amount of the air loading effect is determined by the air density of the environment.

Does The Shape of the Bowl Contribute to the Pitch of Timpani Sound Spectra?

The significant studies have determined that bowl shape is of little importance. It is the volume of air inside of the bowl and not the shape of the bowl that effectively fine tunes the partials created by the preferred modes helping to bring them into a near-harmonic relationship. Most modern timpani bowls seem to have the right volume to optimize the harmonicity of the principal partials. Other factors such as bowl and frame resonance can degrade the harmonicity of the principal partials yet this resonance is still considered a desirable sound trait by some timpanists. The first few hundred milliseconds of the sound include vibrations from the bowl and frame, which can contribute to the color of the attack. Hand-hammered copper bowls are preferred by most professional timpanists. Copper bowls are efficient thermal conductors, which help equalize internal and external air temperatures and pressures. Bowl material, hardness and thickness can affect the amount of energy loss through bowl walls, which helps determine how much energy will be available for the vibrating head. Lip shape defines the boundary conditions of the head affecting the vibrations of the concentric and diametric modes. Suppression of the inharmonic concentric modes is essential for pitch production.

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