In Search of the Missing Fundamental: by Richard K. Jones
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Significant Studies

Lord Rayleigh’s theories on the subject of Timpani Harmonicity are generally regarded as being the first and are the basis for most of today’s research. In his studies, he recognised that the perceived pitch or “principal vibration” corresponds to the (1,1) mode, and not mode (0,1) which is the actual fundamental of a circular vibrating membrane. Since mode (1,1) is the sound we hear (and not mode (0,1)), it is referred to as the principal tone and not the fundamental. 4

Arthur H. Benade (1973) measured the sound spectrum of a timpano owned by Cloyd Duff, timpanist of the Cleveland Orchestra and found that in the first ten components (modes) there were near harmonic ratios to a missing fundamental one octave below mode (1,1). 5 Benade’s scientific studies gave credence to a host of earlier writings on the subject i.e P.R Kirby: The Kettle Drums (1930), Charles L. White: Drums Through the Ages (1960), Henry W. Taylor: The Art and Science of Timpani (1964) and James Blades: Percussion Instruments and Their History (1970) which all inferred that a well-tuned timpano must have the principal tone and a strong fifth; if possible, one should hear the octave as and occasionally the third, fifth, seventh and even the double octave. 6

The studies of Thomas D. Rossing and his colleagues (Rossing et al., 1976, 1977, 1982, 1998, 2000) further investigated the vibrational modes of kettledrums and concluded that only the diametric modes are in harmonic relationship to each other. Their works consistently showed that modes (1,1), (2,1), (3,1), (4,1) and (5,1) are respectively in frequency ratios of 1, 1.5, 2, 2.44 (almost 2.5) and 2.9 (almost 3) with mode (1,1). Rossing et al., also contributed significantly to the knowledge of how the bowl influences the sound of the instrument as well as how the sound radiates from the instrument. 7

Rayleigh, Benade and Rossing all studied the effect of air loading on the frequency ratios between the fundamental mode (0,1) and upper modes. Air loading is the effect that the motion of the surrounding mass of air has on 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.

While studying at the Institute of Mechanics, Faculty of Aeronautics and Astronautics, Federal Armed Forces Munich-Neubiberg (1992-1993), A. Breitung pioneered the use of infrared lasers to study the vibrational modes of timpani. He used a large Kolberg timpano with a synthetic head and measured and compared the modes at various frequencies rather than just one frequency as had been done in previous studies. His findings were published in his diploma thesis, Breitung, A., Grundlegende Untersuchungen zur Modalanalyse. Diplomarbeit am Institut für Mechanik, Fakultät für Luft- und Raumfahrttechnik, UniBw München, Neubiberg 1993. His results were also included in Helmut Fleischer and Hugo Fastl’s 2005 publication Vibroakustische Untersuchungen an Paukenfellen.7.1 Breitung’s findings showed that that as the membrane was stretched, the ratios of the lower preferred partials fluctuated only slightly and were consistent enough to retain a sense of harmonicity throughout the range of the drum.

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