Figure 2a illustrates the first twelve vibrational modes of an ideal circular membrane, the physical system that timpani use to generate sound. For comparison, Figure 2b shows the first six harmonic partials of a vibrating system like a string or column of air, which produces a true harmonic series.10 4
In Figure 2a, the mode (0,1) is taken as the fundamental mode (f₁) and assigned a ratio of 1. The subsequent modes (e.g., 1.59f₁, 2.14f₁, 2.30f₁, etc.) represent higher frequencies of vibration. Notice that these ratios are not whole-number multiples of the fundamental. In contrast, the ratios in Figure 2b , such as f₁, 2f₁, 3f₁ form a simple integer sequence, characteristic of systems that produce a harmonic series.
Fig. 2a
Fig. 2b
Comparing the Two Systems
-
Ideal circular membrane (timpani):
Ratios → 1, 1.59, 2.14, 2.30, etc.
(includes decimal values, not harmonically related) -
String or air column (harmonic systems):
Ratios → 1, 2, 3, 4, etc.
(whole-number multiples, harmonically related)
In harmonic systems, the overtones are related by simple, whole-number ratios. These harmonically related partials are what allow the human ear to perceive a clear and stable pitch. By contrast, the overtones of a vibrating circular membrane are inharmonic, their frequencies do not align as integer multiples of the fundamental, and thus do not naturally produce a strong sense of pitch.
In short, the timpani’s vibrational system does not share the same harmonic structure as the brasses, winds, or strings of the orchestra.
Visualizing and Hearing the Difference
Figures 2c and 2d show what each overtone series looks and sounds like when:
-
Notated on a musical staff (starting from C2, ~65.4 Hz),
-
And synthesized as sine waves.
Symbols “–” and “+” indicate when a partial is significantly lower or higher than its equal-tempered counterpart.
Each figure includes two sound clips:
-
The overtone series played in sequence.
-
The series played as a complex tone (i.e., all partials added together to form a composite waveform).
Figure 2c – Harmonic Partials (String/Air Column)
Partials 1–12
Fig. 2c
Figure 2d – Inharmonic Modes (Circular Membrane)
Modes 1–12
Fig. 2d
Takeaway: Harmonicity vs. Inharmonicity
The blending of harmonic partials (integer multiples of the fundamental) produces a strong sense of pitch, a phenomenon known as harmonicity. Inharmonicity occurs when the overtones deviate from whole-number relationships. This disrupts pitch perception and alters the tone quality.
In comparing the two series above, it becomes clear that timpani, which rely on inharmonic circular membrane modes, do not share the same acoustical “genetic code” as harmonic instruments like strings, winds, and brass. Yet, as you’ll continue to explore, timpani can still be tempered and tuned in such a way that the illusion of pitch is preserved.