Below are the first four modes sound like when they are combined together into a complex tone or composite waveform.
Since the first four partials of a vibrating circular membrane are in-harmonic, the question still remains, how do timpani produce musical pitch?
The science behind how timpani produce sounds which constitute musical pitch has been a fascination of scientists and physicists for well over a century.
The definition of an ideal circular membrane is as follows:
an absolutely round membrane, infinitely thin, perfectly flexible, completely homogeneous, evenly and uniformly tensioned where the outer circular edge of the membrane constitutes a fixed boundary condition in an in vacuo state (in a vacuum). 7
The fact that a timpano head cannot conform exactly to and ideal membrane actually helps the instrument produce musical pitch. Factors such as the thickness and stiffness of the head alter some of the higher partials helping shift the head into vibrating with near harmonic relationships. This is only one of the factors which in a sense helps tune the instrument.
The most important contributing factor however, is air. Humans do not live in a vacuum. A timpano head has a very large physical footprint compared to that of a vibrating string. It vibrates in a sea of air and the friction caused by the interaction with the air mass outside of bowl, significantly lowers the frequency of the lower modes and the resonance. The volume and stiffness of the air enclosed in the bowl also interact with the head. All of this air working against the head is called air loading or simply loading and is believed to be the main contributing factor for why the drum produces a near harmonic series.1 The volume and stiffness of air inside of the bowl, the air mass outside of the bowl and the physical properties of the head all work together to create a single vibrating system to produce a spectrum of about five or six near or quasi harmonic partials when the head is properly cleared. Even though this narrow quasi harmonic spectrum does not match an exact harmonic series, the human ear (and brain) correlates the spectrum into a sound that we hear as having pitch simply because it is close enough to the real thing.
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