Chapter One: An Acoustics Primer

8. What are wave shapes, and what is spectral content?

The shape of a wave is directly related to its spectral content, that is, the particular frequencies, amplitudes, and phases of its components. Spectral content is the primary factor in our perception of timbre, or tone color. We are familiar with the fact that white light, when properly refracted, can be broken down into its component colors, as in the rainbow. Similarly, a complex sound wave is the composite of multiple frequencies.

So far, we have made several references to sine waves, so-called because they follow the plotted shape of the mathematical sine function. A perfect sine wave or its cosine cousin produces a single frequency known as the fundamental. Once any deviation is introduced into the sinusoidal shape (but not its basic period), additional frequencies, known as harmonic partials, are produced.

Partials refer to frequencies that are generated by a simple or complex waveform. In real-world sound, these are not necessarily harmonic partials. Harmonics or harmonic partials are integer (whole-number) multiples of the fundamental frequency (ƒ) (so 1ƒ, 2ƒ, 3ƒ, 4ƒ…). Overtones are the harmonics above the fundamental. By convention, we usually refer to the fundamental as partial 1, since it is 1ƒ. The first few harmonic partials are the fundamental, the octave above, the perfect fifth above that, two octaves above, two octaves plus a major third, and two octaves plus a perfect fifth, as shown below for the pitch "A." After the 8th partial, the pitches begin to grow ever closer and do not necessarily correspond to equal-tempered pitches, as shown in the chart. In fact, even the fifths and thirds are slightly offset from their equal-tempered frequencies. You may note that the first few pitches correspond to the harmonic nodes of a violin (or any vibrating) string.

Partials that deviate in frequency from the ideal arithmetic multiples of the fundamental are referred to as inharmonic partials and a general characterization of the deviation is called inharmonicity. Artificially generating inharmonic partials requires instability in a periodic wave, creating aperiodicity. Inharmonicity is common in the real world—piano strings, particularly toward either end of the keyboard, produce a small amount of inharmonicity, whereas bells produce a much greater amount.