In Search of the Missing Fundamental: by Richard K. Jones
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Resonator or Radiator?

RorR Example

The scientific method evolved in the latter part of the 10th century as a way to distinguish truth from beliefs and biases. This method has proven to be a viable solution for answering questions and explaining things from a neutral or unbiased position, but answers and discoveries happen only if it is applied. Distinguishing fact from traditional beliefs is never an easy task.

It wasn’t until the age of enlightenment that mathematicians were able to apply the new techniques of calculus to their elaborate theories of sound wave propagation. In the nineteenth century the major figures of mathematical/musical acoustics were Helmholtz in Germany, who consolidated the field of physiological acoustics, and Lord Rayleigh in England, who combined previous knowledge with his own copious contributions to the field in his monumental work The Theory of Sound. In the twentieth century there have been numerous scholars who have pursued the studies of musical acoustics with many studies focused on percussion alone. Thomas D. Rossing and Helmut Fleischer and Hugo Fastl in particular have devoted numerous studies to the acoustical properties of timpani and both have contributed greatly to our knowledge of how the instrument functions acoustically.

In the percussion community, there remain a number of misconceptions; ideas that are perhaps based on historical beliefs and traditions, not scientific fact. With regard to timpani, one of those is the belief that the timpano bowl functions as a sound chamber or acoustic cavity resonator (Helmholtz resonator) similar to the resonator tubes on keyboard percussion instruments. Reputable performers, authors and instrument manufacturers have promulgated this theory in numerous publications on the subject; perhaps they simply jumped on the bandwagon based on historic hearsay. The misconception of the bowl being a true air cavity resonator perhaps started innocently enough as many the early instrument developers and musicians did not have an understanding of musical acoustics and assumed through deduction that the timpani bowl served the same function as that of the body of many instruments. It wasn’t until 1877 that Lord Rayleigh properly identified and documented the function of the bowl as it related to the acoustic properties of timpani.

The bowl clearly functions as a system modifier, but what is that function? Why do so many percussionists associate the bowl with resonance?

Percussionists/timpanists have a special interest and concern for how their instruments vibrate; vibrations (and not necessarily those associated with pitch) are the very essence of what they listen to in the sound of their instruments. For timpani in particular, vibrations can and do contribute to resonance, but all vibrations do not create the harmonic resonant frequencies necessary for the enhancement of pitch.

So, what is resonance?

Resonance is oscillation induced in a physical system when it is affected by another system that is itself oscillating at the right frequency. 37

But why then is a timpano bowl not a resonator or a sound chamber? The bowl vibrates because of resonant frequencies from the head doesn’t it? What is a resonator anyway?

The simplest definition of an acoustic resonator cavity is:

A hollow chamber or cavity with dimensions chosen to permit internal resonant oscillation of acoustical waves of specific frequencies. 38

It is true that a timpano bowl can be viewed as a hollow air cavity interacting with a membrane vibrating at specific frequencies, therefore it would seem that it fits the description of a hollow acoustic chamber, which would permit resonant oscillation. With respect to the timpano bowl, it is in fact the volume of air contained within the bowl, and the associated air modes that create any actual resonance. However, since the volume of the bowl remains the same while the head is tuned over the range of a sixth (or more), a true acoustic resonance corresponding to the fundamental resonant air mode and that of the principal tone could only exist for perhaps one frequency of the shifting principal tone. In fact, the various resonant air modes enclosed within the bowl are considerably higher than that of the principal tone in any range of the drum; some of these resonant air modes can and do interact with some of the higher partials in the spectrum resulting in notes with more pronounced partials (both harmonic and inharmonic) (Rossing). The result is the perception of certain notes that sing or project better than others because some of the resonant air modes help to fine-tune the upper preferred modes aligning them into a more harmonic structure.

As Benade states:

The varying degree in perfection in preserving the correct air-to-membrane relationship is what explains the observation by musicians that every drum plays best at one particular frequency in its range of usability. 20

So what is the function of the bowl?

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