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

Timpani Sound Spectra In a Nutshell:

  1. The principal tone or the “perceived pitch” is derived from mode (1,1) which is not the actual fundamental of the drum.
  2. Certain modes of vibration contribute to the “harmonicity” of the sound spectrum more than others. These preferred modes are the diametric modes (1,1), (2,1), (3,1), (4,1), (5,1), (6,1).
  3. Air Loading, which is the density of the air mass exerting force in the vicinity of the membrane, substantially lowers the frequencies of the lower preferred modes (1,1), (2,1).
  4. The bowl functions as a baffle, not as a resonating chamber, however, vibratory aspects of the bowl (collateral color) can contribute to the timbre of the attack. When struck in a region approximately twenty-five percent of the distance in from the bearing edge (lip) to the center of the drum, the concentric modes radiate energy efficiently and decay quickly leaving the diametric modes vibrating. The concentric modes do not contribute greatly to the harmonicity of the drum. Resonant frequencies generated by the bowl and frame may contribute to the overall sound, but not to the harmonicity of the instrument.
  5. The air enclosed in the bowl acts as a restoring force for the concentric modes trying to return the head to a stationary position. This enclosed air also has resonances of its own that can interact with modes of the vibrating membrane that have similar shapes.
  6. The adjustment of the preferred modes produce frequencies nearly in the ratios of 1 : 1.5 : 2 : 2.5 : 3 : 3.5 to that of a pitch with a missing fundamental e.g., a series beginning on the second harmonic up to about the seventh harmonic; the missing fundamental effect might be perceived under certain conditions and dynamic levels.

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.

Throughout this discussion, the terms quasi-harmonic or near-harmonic have been used to describe the partials which contribute to the pitch of timpani. For determining and differentiating pitch from noise, the human auditory system uses what can essentially be considered a pattern matching template to assist in the determination. Pitch is determined by the closest match to a pattern of harmonics. Simply put, it tries to match the spectra of the received complex tone to that of a complex tone with a complete harmonic spectrum. If it is able to resolve the partials, i.e., they conform to the harmonic series, then the sound is heard as a pitch (or varying degrees thereof) and not noise. The strength, number and harmonic accuracy are all determining factors.

If the human ear perceives pitch based on the strength, number and accuracy of the harmonic partials present in the sound, how is it that the human ear is able to discern pitch from the presence of only quasi-harmonic or near-harmonic partials? Even if these quasi-harmonic or near-harmonic partials are close to being harmonic, it was also shown that the spectrum of a timpano is built on an approximate series with a missing fundamental frequency. In chapter two, while discussing the timpani spectra of a personal timpani owned by Cleveland Orchestra timpanist Cloyd Duff, Benade wrote:

* The missing fundamental effect might then give you the pitch C2 for the instrument under certain conditions and dynamic levels. 15

What is meant by “the missing fundamental effect”? How can there be pitch if the fundamental is missing? Chapter four will discuss the aspects of human pitch perception they relate to how the human ear perceives timpani harmonicity.


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