Unlike one-dimensional vibrating air columns and vibrating strings, vibrating circular membranes are graphed as two-dimensional objects. A two-dimensional circular membrane can have many modes of vibration occurring simultaneously; concentric modes (symmetrical), diametric modes (asymmetrical), and composite modes (which are combinations of the two). With respect to how modern timpani produce pitch, it is the diametric modes (asymmetrical), and not the concentric modes (symmetrical), or the composite that produce the sustained sense of pitch. For good timpani sound (one that is rich with near-harmonic overtones), you want to suppress the inharmonic concentric and composite modes as much as possible and project the quasi-harmonic diametric modes. Furthermore, of the many diametric 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 sense of pitch or harmonicity. These modes are called the preferred modes and are found in the lower diametric modes (1,1), (2,1), (3,1), (4,1), (5,1) and sometimes (6,1). These are the same preferred modes introduced on the previous page.
Preferred Modes (1,1) – (6,1)
The other audible modes (concentric, diametric, and composite) are significant because they contribute to the attack transients of the envelope and sustained timbre giving the instrument its unique tonal characteristics. However, since the other modes don’t really contribute to timpani harmonicity, they will not be discussed here; the focus will be on how these preferred modes are shifted or coaxed into near-harmonicity. Keep in mind that the other modes of vibration still contribute to the overall sound of the instrument as well as does collateral color (also known as parasitic pitch), which includes the potential hyper-resonant frequencies generated by unwanted vibrations of the head, and by vibrations of the bowl, frame and other mechanical parts of the drum.
NB The objective when tempering a timpano is to adjust the tension of the head whereby the vibrations of the membrane emphasize the preferred modes, and at the same time, suppress the inharmonic partials (concentric and composite modes) as much as possible so that the preferred modes can dominate the spectrum. Having the preferred modes dominate the sound spectra is incumbent on striking the head in a manner, and at a location that fully excites the preferred modes and de-emphasizes all other (inharmonic) modes of vibration.
As a baseline for comparison, cents will be used as the system of measurement when measuring the harmonicity of the preferred modes and resulting partials. Each Equal-Tempered half step has 100 cents and there are 1200 cents in an octave. However, Just intervals (e.g. the harmonic series) are not always even multiples of 100 cents like Equal-Tempered intervals. For more information on “musical cents,” visit the following links.
Frequencies for the various permutations of the preferred modes will be translated into musical pitches called pitch-class equivalents and notated on the Grand Staff. Some of the actual frequencies may not correspond exactly to the frequency of the actual pitch and will be notated on the nearest approximate pitch.
The Preferred Timpani Modes
Information and charts courtesy of Georgia State University
HyperPhysics
Assuming that these selected modes are excited, the relative frequencies and intervals in cents are given compared to the 1,1-mode. The preferred vibrational modes for timpani are a subset of the modes of a theoretical vibrating circular membrane.
Fig. 3c
The interval values in cents here are calculated from the mode frequencies given by Berg & Stork. They can be compared to equal tempered intervals. NB The actual sounding frequencies of these modes for timpani are affected by air loading.
f0 = mode (1,1) which is the ideal membrane’s principal tone as if it were a timpano. It is also assumed the the membrane is vibrating as well as circular so it will be referred to as an ideal membrane. It is assumed that this is an ideal, uniformly tensioned circular membrane, so it will be referred to as an ideal membrane. Figure 3c shows the mode shapes, the ratios and the number of cents for each of the preferred modes of an ideal membrane as calculated by Berg & Stork. It is important to note that the ratios indicated by Berg & Stork reflect preferred modes, which have not been affected by air loading and without the actual damped fundamental. It is also important to note that these ratios are for the ideal membrane (no air loading), and are presented here as a baseline for comparison with the air loaded membrane discussed in the next section.
Animation of the First Five Preferred Modes
(courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State)
Mode 1,1
Mode 2,1
Mode 3,1
Mode 4,1
Mode 5,1
Figure 3d shows the pitch-class equivalents of the preferred modes of an ideal membrane notated on the Grand Staff with the damped fundamental (A2 110 Hz) included as a reference. Also included are the ratios and the number of cents for each of the preferred modes of an ideal membrane as calculated by Berg & Stork. (click fig. 3d to enlarge) The accompanying sound file does not include the damped fundamental referenced in figure 3d, just the preferred modes.
Fig. 3d
Interestingly enough, the preferred modes of an ideal membrane fall close to the frequency ratios you would get from a harmonic series with a missing fundamental. Here is the unambiguous way to state it: if the ear infers a missing fundamental at frequency f, then the preferred-mode set corresponds approximately to harmonics 2, 3, 4, 5, 6, and 7 of that missing fundamental (i.e., frequencies near 2f, 3f, 4f, 5f, 6f, 7f). In this mapping, mode (1,1) functions as the second partial (≈ 2f), and the “missing” partials are the fundamental (1f) and any very weak or suppressed components that would normally help complete the series.
Figure 3e is a visualization of the six preferred modes of an ideal vibrating membrane. Beginning with no modes of vibration, the preferred modes are displayed as each mode from (1,1) to (6,1) is added incrementally. Notice as how the modes are added, the vibrating motion around the perimeter of the membrane increases.
Fig. 3e
Since humans don’t live in a vacuum, the ratios of the partials of the preferred modes of an ideal membrane change when factors such as the air surrounding the head and the timpano bowl are introduced into the equation. The most significant being the air above the head and the air inside of the bowl. This phenomenon is called air loading and is the main factor responsible for establishing the near-harmonic relationship among the preferred modes.
When the effects of air loading are introduced, the inharmonic partials of the preferred modes of an ideal membrane are coaxed into a near harmonic sequence, which is very near a harmonic series without a fundamental. The next section explains this phenomenon.
Harmonic Alignment of the Preferred Modes
of an Air Loaded Membrane
