In Search of the Missing Fundamental: by Richard K. Jones

# Ratios: 2f vs 1.59f …

Figure 2a illustrates twelve initial vibrational modes of an ideal circular membrane, the system which timpani use to generate sound. Figure 2b illustrates the first six partials or vibrating modes of a system with a harmonic series like that of a string of column of air. In Figure 2a, the mode (0,1) is taken to be the fundamental mode f1 and is assigned a value of 1. The ratios (f1 1.59f1 2.14f etc.) of each mode of an ideal circular membrane are not whole number integers as are the ratios (fl f2 f3 etc.) of the partials in the harmonic series graphic Figure 2b.

Fig. 2a

Fig. 2b

The ideal circular membrane ratios are 1, 1.59, 2.14, 2.30 etc. while the vibrating string or column of air ratios are 1, 2, 3, 4 etc.

The vibrating string or column of air ratios (which is the harmonic series) are simple integers or whole number multiples (which means that they don’t include fractions or decimals) of the fundamental frequency. The ideal circular membrane ratios are not simple integers or whole numbers multiples (which means that they do include fractions or decimals) of the fundamental frequency. They are shown here in decimal number form. Another way to phrase it is that partials of the vibrating circular membrane are not harmonically related while that partials of the vibrating string of column of air are harmonically related. Harmonically related partials are able to elicit a sense of pitch in the human auditory system. Non-harmonic partials are not.

Figures 2c and 2d are what the partials of both of these systems look and sound like when they are notated on the musical staff and reproduced as sine waves. Both series which will begin on C2 (ca. 65.4 Hz.) The – and + symbols indicate that the notated pitch is significantly lower or higher, respectively, than the same pitch in the equal tempered tuning system. The first sound clip for each figure is the series in sequence. The second is the series combined into a complex tone or composite wave form.

Fig. 2c

Partials 1-12

Fig. 2d

modes 1-12

It is the blending together of harmonic partials (whole number multiples of the fundamental) that give definite pitch to a sound and create harmonicity. Inharmonicity occurs when the frequencies of the overtones of a fundamental differ from whole-number multiples of the fundamental’s frequency. In comparing the two overtone series and sound clips above, it quickly becomes apparent that timpani which use the vibrating circular membrane modes to generate sound spectra do not share the same genetic code, as it were as do the brasses, winds and strings of the orchestra.

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