A vibrating circular membrane can, in theory, support an infinite number of modes, each corresponding to a specific pattern of motion. In practice, only a handful of these modes contribute significantly to a timpano’s sustained sense of pitch. These are found among the lower diametric modes, specifically (1,1), (2,1), (3,1), (4,1), and (5,1), and are collectively referred to as the preferred modes. All other audible modes contribute primarily to the instrument’s timbre through a variety of inharmonic overtones. Some of these overtones are more audible than others, particularly within the first several hundred milliseconds of the sound’s attack. The preferred modes will be discussed in greater depth in the next chapter.18 6
The following section (membrane mode information courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State) describes how the first ten modes of an ideal circular membrane behave, and explains how these modes do and do not contribute to the timpano’s sound spectrum.
The First Ten Vibrational Modes of an Ideal Circular Membrane
NOTE: Throughout the following descriptions, the mode notation (d,c) is used, where d is the number of nodal diameters and c is the number of nodal circles (also known as diametric and concentric modes). An ideal circular membrane is defined as an absolutely round membrane, infinitely thin, perfectly flexible, completely homogeneous, and evenly tensioned, with its outer edge forming a fixed boundary condition in an in vacuo state (in a vacuum). This type of membrane exists only in theory.
The (0,1) mode is the fundamental mode of a circular membrane because it has no nodal diameters and one nodal circle at the outer edge. A node is a point (or line or circle, in this case) that remains motionless while the rest of the membrane vibrates.
When a timpano head is struck at the center, the (0,1) mode is strongly excited. In this mode, the membrane behaves much like a monopole source, radiating energy efficiently in all directions. Because it radiates so readily, its energy is quickly converted to sound and decays rapidly. As a result, the (0,1) mode produces a short, pitchless thump when the drum is struck at the center and does not contribute to the sustained pitch quality of the timpano.
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The (1,1) mode The Principal Tone
The second mode, (1,1), has one nodal diameter and one nodal circle at the edge. In an ideal membrane (unaffected by air loading), the frequency of the (1,1) mode is about 1.593 times the frequency of the (0,1) mode.
In the (1,1) mode, the membrane acts like a dipole source, one half of the membrane moves in one direction while the opposite half moves in the other, producing a side‑to‑side motion of air. This mode radiates energy less efficiently than the (0,1) mode, causing it to decay more slowly. Because it “rings” longer, it contributes significantly to the sense of pitch of the timpano. This mode is the first of the preferred modes and is heard as the instrument’s principal tone.
The (2,1) mode
The third mode, (2,1), has two nodal diameters (usually at right angles) and one nodal circle at the edge. In an ideal membrane, its frequency is about 2.135 times that of the (0,1) mode.
This mode behaves like a quadrupole source, which is less efficient at radiating sound than the (1,1) mode and much less efficient than the (0,1) mode. As a result, its vibrational energy decays more slowly, allowing it to contribute to the timpano’s sustained tone quality and perceived pitch. The (2,1) mode is the second preferred mode, and it is often prominent in the timpani spectrum.
The (0,2) mode
The fourth mode, (0,2), contains two concentric nodal circles, one at the outer edge and one at approximately 0.436a (where a is the membrane’s radius) inward from the edge. In an ideal membrane, its frequency is about 2.295 times the (0,1) frequency.
Like the (0,1) mode, the (0,2) mode is excited by striking the membrane at the center. Its radiation pattern is more complex, appearing to behave partly like a monopole and partly like a dipole. The (0,2) mode decays more slowly than the (0,1) mode but more quickly than the (1,1) mode. Consequently, it contributes to the initial thump heard when a drum is struck at the center but does not contribute significantly to sustained pitch when the drum is struck off‑center.
The (3,1) Mode
The fifth mode of a circular membrane is the (3,1) mode which has three nodal diameters and one nodal circle (the outside edge). Like the (1,1) and (2,1) modes, this pattern is a poor radiator of energy and therefore takes longer to decay, and contributes to the harmonicity of the pitch of a timpano. It is the third of the preferred modes.
The (1,2) mode
The sixth mode (1,2) has one nodal diameter and two nodal circles. The frequency of the (1,2) mode (ideal membrane only – one not affected by air loading) is 2.917 times the frequency of the (0,1) mode. As you might expect after looking at the first several modes of the circular membrane, the (1,2) mode does not radiate energy very effectively. It has somewhat of a quadrupole type behavior. Thus, the (1,2) mode takes a relatively long time to decay. However, this mode doesn’t play a dominant role in the sustained harmonicity of modern timpani, but it is believed to have played a significant role in the sound production of early timpani, due to the manner in which the early timpani were played, and because of the thickness of the membrane (head).
The (4,1) Mode
The seventh mode of a circular membrane is the (4,1) mode which has four nodal diameters and one nodal circle (the outside edge). Like the (1,1), (2,1), (3,1) modes, this pattern is a relatively poor radiator or energy and therefore takes longer to decay, and contributes to the sustained harmonicity and sense of pitch of a timpano. This is the fourth of the preferred modes.
The (2,2) Mode
The eighth mode of a circular membrane is the (2,2) mode which has two nodal diameters and two nodal circles. This mode does not contribute to the sustained harmonicity or pitch of a timpano.
The (0,3) Mode
The ninth mode (0,3), shown below has three circular nodes, but no diameter nodes. The frequency of the (0,3) mode (ideal membrane only – one not affected by air loading) is 3.598 times the frequency of the (0,1) mode. Like the (0,1) and (0,2) modes, the (0,3) mode is excited when the membrane is struck at the center. The sound radiation characteristics of the (0,3) mode are rather complicated. This mode is excited when the membrane is struck at the center radiates energy efficiently, and the sound dies away fairly quickly. As a result, it contributes to the thump sound when a drum is hit at the center, but does not contribute much to the sustained harmonicity or pitch of a drum when hit off-center.
The (5,1) Mode
The tenth mode of a circular membrane is the (5,1) mode which has five nodal diameters and one nodal circles at the fixed outer rim. Like the (1,1), (2,1), (3,1) (4,1) modes, this pattern is a poor radiator or energy and therefore takes longer to decay, and contributes to the sustained harmonicity and sense of pitch of a timpano. It is the fifth of the preferred modes.
Inharmonicity and the Question of Pitch
It is important to note that none of the modal frequencies listed above — for example, 1.593, 2.135, 2.295, 2.917, 3.598, and beyond, are simple multiples of the fundamental frequency. In other words, they do not form a harmonic series in the classical sense. A traditional harmonic series would consist of ratios such as 1, 2, 3, 4, etc., but the ratios of membrane modes are decimal values and do not align with integer multiples.
In theory, this would suggest that a timpano should not produce a clear musical pitch any more than any other drum. Yet timpani do produce a perceptible pitch. Understanding how this perceptual phenomenon arises, based on the alignment and relative strength of preferred modes, is the subject of the next chapter.
Takeaway
A circular membrane has many modes, but a timpano’s sustained pitch comes mainly from a small subset: the lower diametric preferred modes, especially (1,1) (the principal tone) and (2,1) (often prominent as the fifth). Center strikes strongly excite concentric modes like (0,1) that radiate and decay quickly (the “thump”), while off-center playing favors preferred modes that decay more slowly and support pitch. Even though membrane mode ratios are inherently inharmonic, a clear timpani pitch can still emerge when the preferred modes dominate and align strongly enough for the ear to hear a stable pitch center, setting up the question explored in the next chapter.