Timpani Sound Spectra

Intonation and blend are crucial aspects of a player’s sound. For the timpanist, they need to be able to adjust and play the instruments in a such manner that the sound spectra produced has enough harmonicity of the partials that the sound can be recognized by the human ear as one having pitch. This perception of pitch is determined by the amount of strong quasi harmonic partials present in the spectrum of each timpano. This spectrum of quasi harmonic partials varies subtly from note to note and drum to drum, as well as when differing dynamics are played. The spectrum of each timpano also varies when differing mallets and methods of striking the membrane are employed. Needless to say, knowing with what, and how and where to strike the membrane are also important in the tempering process.

The investigation of timpani harmonicity is certainly not a new idea. P. R. Kirby in his book The Kettledrums: A Book for Composers, Conductors, and Kettledrummers, 1930 states:

… The presence of the perfect fifth enormously increases the resonance and the beauty of tone of the drum note. … A well-tuned drum should therefore always have the nominal and its fifth in perfect accord; if possible, the octave as well. On rare occasions I have succeeded in obtaining accurately the third, fifth, seventh, and even the double octave. …The third, however, is almost always flat, but, as both it and the higher harmonics are not very prominent, its flatness does not affect the tone of the drum from a practical point of view. 11

Kirby, in his latter statement is no doubt referring to the tenth above the principal tone as the third, and the tritone above that as the seventh, both being somewhat inharmonic.8 While this does not make a true harmonic series, the question is: is it close enough that the human ear (brain) can correlate these few partials into a complex tone with a pitch center?

In chapter one it was presented that in the middle of the nineteenth century, the German physician and physicist Hermann von Helmholtz (1821-1894) recognized that a musical tone conveying a clear sense of pitch must have several strong harmonic overtones in order to create a harmonic spectrum. Writing on the sensation of tone, he pointed out that tones with a moderately loud series of harmonics up to the sixth partial sound rich and musical. 12

If the modes of vibration of timpani heads are inharmonic, how then do the inharmonic partials in the timpani sound spectra create the effect of “a moderately loud series of strong harmonic overtones” as Helmholtz described?

A number of factors come into play.

  1. membrane integrity: material, mass (thickness) , size (diameter), positioning and tensioning
  2. bowl integrity: roundness – flatness, shape and thickness of the bearing edge
  3. with what, where and how the timpano is struck
  4. the density of the air mass above the timpano membrane
  5. the volume and density of air inside of the bowl
  6. the stiffness and resonance of the air in the bowl
  7. the stiffness of the timpano membrane itself
  8. the material(s) from which the bowl, frame and mechanical parts are made
  9. radiation of the preferred modes in the acoustic environment
  10. operator intervention

All of the above factors influence the frequencies of a small group of vibrating modes called the preferred modes. When these preferred modes are strong enough, aligned properly and decay slowly, the resulting partials create a narrow quasi-harmonic series from which we are able to discern a pitch. The volume of air inside of the bowl (and its viscothermal characteristics), the air outside of the bowl (and its viscothermal characteristics), and the vibrations of the head make up a single system; the three parts are of equal importance in determining the frequencies and overall vibrational shapes (preferred modes) which define the pitch of the instrument.

Viscothermal characteristics include the viscosity of the internal air (the property of its resistance to flow), and the thermal conduction of the internal air (the transmission of its heat energy by conduction). Both influence how the membrane vibrates.  N.B. The viscothermal characteristics of a gas (the confined air in the bowl in this case) are just the opposite of those of a fluid. The viscosity of liquids decreases with an increase in temperature, and the viscosity of gases increases with an increase in temperature. Thus, upon heating, liquids flow more easily, whereas gases flow more sluggishly. When a gas increases in temperature, so does its viscosity. A decrease in temperature lowers its viscosity.49 57

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

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. Mode labels (m,n) count nodal diameters (m) and nodal circles (n). In the vibrating modes of a circular membrane, mode (0,1) is the fundamental partial and mode (1,1) is the second partial. 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. . In many practical cases, 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: 1) adjust the tension of the head at each lug point so that the spectrum is dominated by the preferred modes, and 2) adjust these preferred modes so that they 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. Hyper-resonance (a term I use here) refers to the brief, highly inharmonic, noise-like transient immediately after the strike, before the spectrum settles into a more pitch-salient set of preferred modes. 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. In many practical cases, 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:

  • The principal tone or the perceived pitch is derived from iterations of mode (1,1) which is not the actual fundamental of the membrane.
  • 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), and sometimes (6,1).
  • 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.”
  • The bowl (the sound modifier) functions as a baffled radiator, not as a resonating chamber, however, vibratory aspects of the bowl (collateral color) can contribute to the timbre of the attack.
  • When struck in a region approximately twenty-five percent of the distance in from the bearing edge (lip) ) to the center of the drum, in many practical cases 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.
  • The air enclosed in the bowl acts as a restoring force for the concentric modes trying to return the head to a stationary position. The enclosed air modes also have resonances of their own that interact with modes of the vibrating membrane that have similar shapes.
  • 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 harmonic 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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