Gabor domain analysis of a three spring damped oscillator
Robert James Ferguson
Membranophones are a class of instruments that use stretched membranes to produce sound. Within this class of instrument are drums (number 21 on the Hornbostel-Sachs musical instrument taxonomy) that produce sound from two stretched membranes that enclose a column of air within a cylinder; the membranes or heads (made of Mylar) are clamped to each end, and the cylinder is often made of wood. Typically, the cylinder is ported by a dime-sized hole. The two heads and the column of air are well modelled as springs, and air rushes in and out through the port during sound production. The port and the load of the ambient air have the effect of damping the produced sound.
Vibration of such a system is governed by the acoustic wave equation, and so the study of the associated acoustics may be expected to be well facilitated using the familiar techniques of reflection seismology.
Here, the familiar notions of Vibroseis, Fourier spectrum, cross correlation, and the Gabor spectrum are used to determine the pitches and, more importantly, the relative pitches of a particularly pleasing collection of drums that form a trap set used in jazz and popular music. Pitch determination is interesting in that, for two headed drums, how they should be tuned is not well described, and seismic analysis might be expected, through analysis of a well sounding set, to at least determine to what pitches, and relative pitches they are tuned. It is found that, for this particular set of three drums, the tom tom drum (the smallest) is tuned to approximately 120 Hz, the floor tom (intermediate in size) is tuned to approximately 80 Hz, and the bass drum (the largest) is tuned to approximately 60 Hz. These pitches correspond to the batter heads (the struck heads), and the resonant heads are tuned to similar pitches with the exception of the bass drum. It is found that the relative pitches of the resonant heads correspond to three of the most pleasing of the classical musical intervals octave (tom tom / bass), perfect fifth (tom tom / floor tom), and perfect fourth (floor tom / bass drum).