In Search of the Missing Fundamental: by Richard K. Jones
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Timpani Sound Spectra

The normal striking spot for a timpano is approximately twenty-five percent of the distance in from the bearing edge (lip) to the center of the drum. This excites the second partial, mode (1,1) and a series of secondary diametric modes called the preferred modes. Since the drum is not struck dead center, the fundamental mode (0,1) and the secondary concentric modes are not excited to any significant degree. Because of the method in which the drum is struck, the concentric modes radiate their energy more efficiently and consequently decay faster leaving the preferred modes to generate the more audible part of the spectrum. The creation of harmonic pitch on modern timpani is a process of subtractive synthesis via diametric mode promulgation, and concentric mode mitigation.

The harmonic pitch portion of this spectrum is generated by a small number of vibrating partials (diametric modes), which produce a quasi-harmonic series with a missing fundamental one octave below mode 1,1. Figure 3a is the Ideal Harmonic Alignment of the Preferred Modes of an Air Loaded Membrane. The chart shows the mode numbers of this series, the number of cents for each mode, and the pitch-class equivalents as notated on the grand staff for the pitch C3 130.81 Hz. The numbers above the staff show the respective harmonic ratios of the partials to the principal tone mode 1,1. The numbers below the staff are the respective ratios to the missing fundamental. The figures between the staff indicate the number of cents of each of the preferred modes starting from principal tone, mode 1,1 with zero cents. (click graphic to enlarge)

Fig. 3a

Timpani-Pitch-Cents
Ideal Harmonic Alignment of the
Preferred Modes of an Air Loaded Membrane

The objective when tempering a timpano head is to adjust the tension of the head at each lug point where the spectrum is dominated by the preferred modes and have as much of a harmonic alignment as possible. This is considered ideal, but it is not possible to achieve due to the inherent inharmonicity of the membrane modes. I put the term harmonic in italics because the actual ratios of the partials of the preferred modes in the timpani sound spectra are only near harmonic at best, and change  from lug to lug, note to note on each drum. Timpani sound spectra also change dramatically depending on the dynamic level, the method and the mallet used when exciting the partials. Figure 3a will be used as a baseline to show what the actual partials would be if they were to be in true  harmonic alignment.

Figure 3b (courtesy of Fleischer & Fastl) is a waterfall chart (frequency, time and amplitude) of a timpano sound spectrum (single struck note) highlighting six preferred modes (1,1), (2,1), (3,1), (4,1), (5,1) and (6,1). The chart shows that there is a short period of in-harmonicity that is much like noise (hyper-resonance), which is the percussive aspect of the instrument followed by a longer period of harmonicity, which is associated with the pitch aspect of the instrument. Also at play are differing amounts of modal energy radiation. If an instrument radiates energy efficiently, the sound will decay more rapidly. In the case of a timpano, the modal radiative energy is not consistent across the spectrum. Due to the  presence of the bowl and the way it is struck, the heavily damped concentric modes radiate energy more efficiently than do the diametric modes and so their sound decays much faster. This leads to a change in the sound spectrum as the note progresses making it more harmonic the longer the sound sustains. Notice the prominence of mode (2,1) or the perfect fifth in the spectrum. (see Pleading the Fifth)

Fig, 3b

Timpani Sound Spectra In a Nutshell:

  1. The principal tone or the perceived pitch is derived from mode (1,1) which is not the actual fundamental of the drum.
  2. 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).
  3. Air Loading, which is the “sea of air” exerting force in the vicinity of the membrane (both inside and outside of the drum), effectively adds mass to the membrane substantially lowering the frequencies of the lower preferred modes (1,1), (2,1). Factors that affect the density of the internal and external air, i.e., temperature, barometric pressure, humidity play an integral part in the air loading of the membrane, and consequently, how the membrane vibrates within that “sea of air.”
  4. The bowl (the sound modifier) 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.
  5. 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, but they do contribute to the color. Any resonant frequencies generated by the bowl and frame may contribute to the overall sound, but not to the harmonicity of the instrument.
  6. 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.
  7. The proper adjustment of the preferred modes can 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.

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