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}

Graphic courtesy of Gordon Reid

Three Part System Defining
the “Harmonicity” of Timpani Pitch
1) internal air modes
2) external air pressure
3) vibrating membrane

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