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

When all of the partials (component frequencies) in a complex tone are all integer multiples of the same fundamental frequency, the sound is said to have a harmonic spectrum. Each component of a harmonic spectrum is called a harmonic partial, or simply a harmonic. The sum of all those harmonically related frequencies still results in a periodic wave having the fundamental frequency. The integer multiple frequencies thus fuse “harmoniously” into a composite harmonic wave form and is perceived by the human ear as a single tone.

In the middle of the nineteenth century, the German physician and physicist Hermann von Helmholtz (1821-1894) recognized that a musical tone conveying a clear sense of pitch must have several strong harmonic overtones in order to create a harmonic spectrum. Writing on the sensation of tone, he pointed out that tones with a moderately loud series of harmonics up to the sixth partial sound rich and musical. 1

All instruments in today’s modern orchestra generate complex tones however, not all vibrate with harmonic motion and therefore do not create harmonic partials or create a composite harmonic wave. A good example would be the sounds emitted from cymbals or gongs, which are definitely complex tones but they are not all harmonic. Timpani do not generate harmonic partials either yet they are considered to be a “pitched” percussion instrument.

Of the orchestral instruments that do generate a harmonic spectrum and create a composite harmonic wave (strings, brasses and winds etc.), we hear various pitches according to whether a string or column of air is vibrating as a unit through its whole length or in particular fractions of it. The vibration along the whole length of a string or column of air gives the lowest or fundamental tone.  The vibrations taking place at various fractions of the length produce higher pitches called harmonics or upper partials. The stationary points along a string or column of air (i.e., where the waves cancel each other out) are called nodes or nodal points.

In mathematical terms, 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. The range of notes produces what is called the harmonic series or overtone series.

The examples in the next section show what the harmonic series looks like when it is notated in traditional musical staff notation and what the individual partials sound like. There is also a sound clip of what a harmonic series sounds like when all of the harmonics are added together.  Follow the notes on the grand staff as you listen to each sound clip. Listen closely to the intonation of each partial, especially as they progress higher on the grand staff.  Each note or partial you hear is a simple sine wave.

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