In Search of the Missing Fundamental: by Richard K. Jones
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How “They” Vibrate

Harmonic Vibrating Modes of a String or Column of Air

In a composite harmonic waveform, all harmonics operate together to produce just one pitch. Figure 1a  represents how a string or column of air vibrates in segments. Each of these segments is considered to be a harmonic partial with the fundamental frequency being the first partial. The second partial is actually the first harmonic overtone. Each of these partials is the simplest waveform, the sine wave. Since no instrument in the orchestra actually vibrates with just one simple sine wave, we say that the waveform is composite, or in other words, a waveform made up of multiple sine waves all vibrating together simultaneously. The subsequent partials above the fundamental have less amplitude (volume) than the fundamental and the human brain blends all of these individual parts together into just one sound.

Fig, 1a

Wavelengths and Frequencies

Figure 1b shows the wave lengths of first six harmonic partials of a complex tone, or one having a composite harmonic waveform. Measuring time from left to right, in the same amount of time, the first partial has the longest wavelength and cycles only once while the sixth partial has the shortest wavelength and cycles six times.  The frequency of each harmonic is in inverse proportion to the size of the fraction. This means that the vibration of equal halves of a string or a column of air produces double the frequency of the whole (and thus sounds an octave higher), the vibration of equal thirds triples the frequency (and therefore sounds an octave and a fifth higher than the fundamental note) and so on. In this chart, each line represents a harmonic partial.

Fig. 1b