In Search of the Missing Fundamental: by Richard K. Jones
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Air Loading

The air that comprises the earth’s atmosphere is made of many different gases. Nitrogen accounts for as much as 78% of the volume while Oxygen accounts for almost 21%. The remaining 1% is composed of such gases as Argon, Carbon Dioxide, Helium and Hydrogen. Water vapor (water in its gaseous state) is also present in the atmosphere in varying amounts, by up to 2%. This atmosphere can be thought of as large ocean of air surrounding the earth.

Air and consequently, our atmosphere, do have weight. The weight of the air is referred to as air density. This weight (air density) decreases as you go up within the atmosphere. When gravity acts on the air, the air exerts a force upon the earth called pressure. Air pressure varies according to temperature. Cold air is more dense than warm air, i.e., it weighs more. As a result, it tends to sink. Warm air, on the other hand, is less dense. Therefore, it weighs less and tends to rise. Meteorologists say that warm air is buoyant. 13

Unlike a vibrating string, which requires a carefully designed coupled surface to propagate its sound, a vibrating timpano heads strongly couples to the surrounding atmosphere, which in turn propagates the sound. Since air does have weight, air density plays an integral role in affecting how a timpano head vibrates. Compared to a vibrating string, a timpano head is considerably larger in size and the surrounding air mass (both inside and outside of the drum) interacts with the vibrating modes substantially. This phenomenon is called air loading and is the main factor responsible for establishing the near harmonic relationship among the preferred modes. Air loading is the effect the weight of the surrounding mass of air has on the motion of 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), (2,1) and plays a significant role in the adjustment of the inharmonic partials. Since air density is not a constant, factors that affect air density (barometric pressure, temperature and humidity) also affect how a timpano head vibrates. The most noticeable effects are slight fluctuations in pitch and changes in the color of the sound when air density changes.

Thomas D. Rossing and his colleagues (Rossing et al., 1976, 1977, 1982, 1998, 2000) summarized the effects of air loading and membrane stiffness as follows: (membrane stiffness is a slight secondary influence on establishing “harmonicity” of the preferred modes) 14

  1. the air loading results in a considerable decrease in the frequency of the lower modes, but the loading is small in the modes of higher frequency
  2. the bending stiffness increase the frequency of the higher modes very slightly, but it is virtually ignorable for the membrane used for a kettledrum
  3. the stiffness to shear is considerable at large amplitude but the resulting forces are of second order and there fore may be considered small at ordinary playing amplitudes of a kettledrum

A crucial component to the production of harmonic timpani pitch is how the vibrating head interacts with the air above and below it. Equalizing the density of the volume of air inside the drum (i.e. air pressure, temperature and moisture content) to that of the density of the air above the head is integral. The interaction between the masses of air inside and outside of the drum contributes significantly to the actual pitch and perceived harmonicity of timpani. Ideally, these masses of air should always have the same density as when the heads were last tempered.

Both the vibrating head and the volume of air inside of the bowl have specific modes of vibration; the air masses inside and outside of the bowl influence how the head will vibrate based on the external air density and the volume and density of air contained within the bowl.

Preferred ModesPreferred Modes (1,1)  (2,1) (3,1) (4,1) (5,1) of the vibrating head.40

Air-Modes-Kettle

Vibrational modes of the air enclosed within the bowl 41

The frequencies of the vibrational modes of the volume of internal air enclosed within the bowl are higher than the frequencies of the membrane modes to which they couple.41  Since the motion of these air modes is dependent on the viscothermal (viscosity and thermal conduction) characteristics of the enclosed air, changes in air density can cause these modes to interact differently with upper partials of the vibrating membrane that have similar frequencies. Consequently, certain notes will tend to sing more on some days than others or the overtones will tend to be sound more in-tune depending on changes in environmental conditions.

With the head being the primary vibrating system, the air mass outside of the drum reacts against the volume, mass and stiffness of the air modes enclosed inside of the bowl adjusting the vibrating modes of the head, in particular, a small group of partials called the preferred modes. When aligned just right, these preferred modes of the air-loaded head will vibrate as a harmonic system giving the drum its sense of pitch, harmonicity,  and projection.

The bowl acts as a baffle (not as a resonator) separating these two air masses preventing them from interacting with each other, much like that of an enclosed speaker system.

The vibrating system is influenced by three factors:

1) the volume and viscothermal characteristics (viscosity and thermal conduction) of the air modes inside of the bowl
2) the density and the viscothermal characteristics of the air pressure outside of the bowl
3) the motion (vibrating modes) of the vibrating head

The three parts are of equal importance in determining the frequencies and overall vibrational shapes (preferred modes) that define the pitch, harmonicity, and to some degree, the color of the instrument. When the conditions of the internal and external air masses begin to differ, or differ significantly, the  preferred modes of the vibrating head do not respond as they did when the head was originally tempered and the timbre, harmonicity and projection of the instrument’s voice is affected.

Head-BowlGraphic courtesy of Gordon Reid

Three Part System Defining
the “Harmonicity” of Timpani Pitch

1) internal air modes
2) external air pressure
3) vibrating membrane

Figure 3e shows the pitch-class equivalents of the harmonic preferred modes of an air loaded 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 as calculated from the principal tone. (click fig. 3e to enlarge) The accompanying sound file includes just the air loaded harmonic preferred modes and does not include the damped fundamental referenced in figure 3e.  N.B. The frequency of the inharmonic fundamental is raised somewhat by the air-loading of the head.

Fig. 3e

I refer to these air loaded modes as the harmonic preferred modes because the ratios set up a harmonic series minus the fundamental (a.k.a. missing fundamental) starting at the principal tone mode (1,1) up to what would be the seventh partial. This in itself is an ideal set of preferred modes because timpani cannot reach true harmonicity, only quasi or near harmonicity. However, it is a standard for which one should strive when tempering heads. One may be able to reach harmonicity on a single note when tempering a head, but harmonicity will change when the pitch of the drum is changed due to factors such as the different resonant air modes in the bowl and the elasticity of the head as well as poor mechanical tolerances of the instrument or changes in barometric pressure and air density.

Figure 3f is a comparison chart showing the impact of air loading on an ideal membrane if the preferred modes were to be brought into a pure harmonic relationship. With a fundamental of A2 110 hz, when placed under the influence of air loading, the principal tone mode (1,1) would drop by 518 cents or the interval of a + perfect fourth. Mode (2,1) would be lowered by 336 cents or the interval of a + minor third. Modes (3,1) and (4,1) would be lowered about 200 cents or the interval of a major second and mode (5,1) about a half of a semitone. In the case of timpani, other factors such as the bowl have an influence as well. N.B. The actual frequency of mode (0,1) would be influenced as well. The stiffness of the air contained inside the bowl raises the frequencies somewhat of the lower concentric modes (0,1), (0,2), (0,3) etc., but since these modes do not contribute to the desired sound spectrum, they will not be displayed other than that of the approximate fundamental.

Fig. 3f

It seems quite amazing that the properties of air could shift the modes that much. Undoubtedly physicists agree because the effects of air loading on timpani has been the subject of numerous scientific studies.

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