Elements of what we now call the scientific method developed over many centuries, with a notably clear emphasis on experimentation emerging in the medieval Islamic world (e.g., Ibn al-Haytham in the early 11th century) and becoming more widely codified during the Scientific Revolution in the 16th–17th centuries. 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 of 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?
A fixed boundary condition exists between the bowl and the head, which causes the timpano bowl to become strong acoustic baffle, tightly coupling the head and the bowl at the lip (bearing edge). As this baffle, the bowl functions as a baffled radiator (not a resonator) separating the top of the head from the bottom of the head so that the low frequencies from the top and bottom surfaces cannot interact (see Fig. 3s and 3t). Even though most bowls are vented (or partially leaky through clearances), the bowl still functions as a baffle at low frequencies, strongly reducing top/bottom cancellation. The bowl turns the drum into an effectively monopole-like omnidirectional source at low frequencies that radiates low frequencies well in all directions, which is often confused with the bowl resonating the sound; they are two completely different processes. If the head does not have a baffle, the low frequencies from the top and bottom of the head interfere, and no sound is radiated in the directions lying in the membrane plane. Simply put, they cancel each other limiting the sound.
Presence of the bowl creates a Monopole Omnidirectional Source 39
As the membrane vibrates, the low frequency energy produced by the motion of the top of the head moves the air molecules radiating sound in an outward direction away from the head. The low frequency energy produced by the motion of the bottom of the head disturbs the volume of the internal air exciting internal air modes as the air tries to escape the enclosure. These internal air modes interact with the motion of the head. The strong impedance barrier (at low frequencies) created between the head and the lip of the bowl prevents the low frequency energy from the top of the head from interacting with the air in the cavity below the head. The top and the bottom of the head along with their respective loaded air systems work largely independently of each other and do not normally coincide (see Fig. 3s and 3t); this greatly reduces any true acoustic cavity resonance from behaving like a Helmholtz-type resonator coupled directly to the external air above the head, where the audible acoustic energy is being generated.
Fig 3s
Acoustics and Audio Group at the University of Edinburgh, UK
Figure 3s simulates the internal and external modal patterns of four timpani in an enclosed room struck in sequence. Notice how the bowl separates the motion of the internal air modes from the external radiation patterns of the membrane modes; the two systems of air remain largely separated, but they do interact via the membrane and the bowl acting as a baffle.
Fig 3t
Modal radiation pattern mode 2,1 with a baffle 39
This baffle is also particularly important for reducing the audibility of actual fundamental, mode 0,1, which degrades the perceived pitch of the instrument. When the head has a baffle, mode 0,1 can radiate its energy very efficiently and decay quickly, leaving mode 1,1 (the actual principal tone) strong enough to produce a sense of pitch. The correct amount of air in the bowl works to lengthen the decay times of the preferred modes and shorten the duration of the inharmonic modes.
Without the bowl, the head becomes a dipole source. Dipole sources do not radiate low frequency sounds well. The low frequencies from the top of the head are destructively canceled by opposite phase sound from bottom of the head. Perhaps this is one reason the timbales chromamatiques19 created by Adolphe Sax circa 1859 or the more recent Tour Timp of Marcus De Mowbray and DrumTone Timpani have not gained traction as serious musical instruments.
Adolphe Sax Timbale chromamatique circa 1859
As it is, the bowl isolates the low frequency energy from the underside, thereby leaving the low frequency energy from the upper side to move the air and radiate sound. Since the low frequencies from the monopole source radiate well in all directions, it is perceived that the sound is being amplified or resonated by the bowl, hence the popular misconception that the bowl is resonating the sound of the vibrating head. The video below demonstrates the process with loudspeakers.
Sound Radiation from Dipoles and Quadrupoles
Courtesy of Dan Russell
It is true, however, that mechanical resonances from the bowl and frame exist, but they have been shown to not enhance the pitch of the instrument to any significant degree. If they do, it is completely serendipitous. Contrary to popular belief, the energy output from the vibrating bowl (and frame/parts) doesn’t add all that much to the actual sound output of the instrument. In the frequency range of the preferred modes (65 Hz- 520 Hz), at most, it is less than 1% in terms of actual output level. Most of the SPL (sound pressure level) energy output (<1 dB) from the bowl/frame occurs in the 1K -2K frequency range, which adds a small amount of collateral color to the sound, but it doesn’t support the preferred modes, which are the modes of vibration that give the timpano its sense of pitch. The Fleischer & Fastl studies have shown that there can be as much as 16% of the overall energy being transferred to the bowl, frame and external parts producing what they call “parasitic pitch.” This energy can no longer be used by the vibrating head and in fact, all of these unintended resonances have been shown, more often than not, to actually detract from the sustain of the preferred modes rather than enhance or support them.
One of the important things that the bowl, and the material it is made from, can and does influence, is the amount of mechanical energy loss. Energy created by the vibrating head and the displaced internal air mass is lost through the bowl walls to a certain degree. Once this mechanical energy is lost, it is no longer available for use by the vibrating head. How the bowl reacts to this mechanical energy in turn influences how the head will vibrate contributing to its resonance and sustain, which is often misinterpreted and misunderstood as being resonance generated by the vibrations being emitted from the bowl. To date, most professional timpanists prefer hand-hammered copper artisan bowls on their timpani. The next section “Why copper?” investigates this phenomenon.
Be that as it may, one cannot deny the timpanist the pleasure of indulging in the beauty of sound produced by the various and sundry vibrations and resonances of his or her own instruments, no matter what they hear vibrating. It all adds to the mystique of the instrument.