Significant Investigations into the Acoustic Properties of Timpani:
- Lord Rayleigh: John William Strutt (1877)
- P. R. Kirby (1930)
- Henry W. Taylor (1964)
- Arthur H. Benade (1973)
- Thomas D. Rossing (1976, 1977, 1982, 1998, 2000)
- Donald L. Sullivan (1996)
- Lamberto Tronchin (2004)
- Helmut Fleischer and Hugo Fastl (2005, 2008)
What Determines a Timpano’s Sound Spectrum?
Key factors include:
- material, integrity, positioning and tensioning of the membrane
- bowl integrity: roundness – flatness, shape and thickness of the bearing edge
- with what, where and how the timpano is struck
- the density of the air mass above the timpano membrane
- the volume and density of air inside of the bowl
- viscothermal characteristics of both the internal and external air systems
- the stiffness and resonance of the air in the bowl
- the stiffness of the timpano membrane itself
- the material(s) from which the bowl, frame and mechanical parts are made
- acoustic radiation of the modes
The above factors influence the frequencies of a small group of vibrating modes called the preferred modes, which decay slowly 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 comparable 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, mode (1,1), (2,1), (3,1), (4,1), (5,1) and sometimes (6,1). The other audible non-harmonic modes contribute the sound envelope (attack/decay) 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 an ideal circular vibrating membrane (i.e., it effectively adds mass to the vibrating system). 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 for pitch-harmonicity. 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.
Timpani Sound Spectra In a Nutshell:
- The principal tone or the “perceived pitch” is derived from mode (1,1) which is not the actual fundamental of the drum.
- 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).
- 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).
- 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.
- 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.
- 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? The answer is not found in the drum alone, but in the listener: the ear and brain can infer a pitch from the spacing, strength, and timing of partials even when the fundamental component is weak or absent. Chapter four introduces the fundamentals of human pitch perception, how the auditory system extracts pitch from complex tones, how it distinguishes pitch from noise, and why certain spectra produce a stronger sense of pitch than others.