What are harmonics? Definition and examples

Harmonics – one of the building blocks of audio theory, defining the richness, character, and perceived pitch of the sounds we hear – are components of frequency which occur alongside the fundamental frequency of a sound.

The fundamental frequency is the lowest resonant frequency made by a sound-generating object. Together, ‘the fundamental’ and its harmonics determine the character of sounds, in terms of concepts such as pitch or timbre (AKA tone colour). Primarily discussed in relation to musical instruments, the principles of harmonics can nevertheless also be extended to voices, loudspeakers, synthesisers, and so on.

The fundamental is the lowest prominent frequency in a tonal sound (a sound with a musical pitch, as opposed to random, unpitched noise). It is otherwise known as the first harmonic, because the harmonic series is numbered using multiples of the fundamental; a fundamental with a frequency of 100 Hz will have harmonics at 200 Hz, 300 Hz, 400 Hz, 500 Hz, and so on. The second harmonic is twice the fundamental, the third harmonic is three times the fundamental, etc.

‘Overtone’ is sometimes used as an interchangeable term for ‘harmonic’, but there is in fact a technical distinction. An overtone is a frequency above the fundamental, while a harmonic is specifically an integer multiple of the fundamental – so the first harmonic is the fundamental itself, whereas the first overtone is normally the second harmonic.

A combination of several frequencies creates a complex waveform through the principle of superposition. In this case, individual frequencies may not be heard separately, with the ear instead perceiving them as one note with a particular sonic character. For example, a strong fundamental with relatively weak upper harmonics may sound smooth or mellow, whereas prominent high-frequency harmonics can sound brighter, sharper, or more penetrating.

The distribution of frequencies is known as the spectrum of a sound, and is the key to timbre. In relation to musical instruments, it is timbre which allows us to distinguish between two instruments playing the same pitch. Whereas a tuning fork produces an almost pure tone, with virtually no harmonics – meaning that it creates a sound which seems sterile or plain – the same note played on a piano or a guitar will feature very different harmonic balances, due to the differences in their construction, materials, and method of playing. 

Similarly, open cylindrical and conical air columns can support a broad range of harmonics, while a tube that is closed at one end will tend to emphasise odd-numbered harmonics. This is partly why instruments with different bore shapes (interior spaces) have different tonal qualities. All of which means that different instruments create distinct sound spectra, even when playing the same note – which the ear in turn interprets as different tone colours.

Audio theory makes a distinction between odd and even harmonics, in terms of their effect on the mood of a sound:

• Even harmonics align with simple musical intervals (like octaves) and are generally seen as blending smoothly with the fundamental, adding a pleasing warmth or ‘thickness’ 
• Odd harmonics create more complex intervals, and may be perceived as adding brightness or edginess.

Additionally, different waveforms show how much harmonic content can impact timbre:

• Sine waves: contain only the fundamental, and sound pure and whistle-like
• Square waves: contain only odd harmonics, often causing a hollow or nasal quality, such as the lower register of a clarinet
• Sawtooth waves: containing both odd and even harmonics, this kind of waveform produces an intense, ‘bright’, and buzzy sound.

Audio engineers can manipulate harmonics in various ways. Equalisation selectively changes the balance of frequencies that are already present; boosting upper harmonics may add a sense of clarity, while removing an unpleasant harmonic can reduce harshness or ringing. Saturation, distortion, and excitation, on the other hand, can generate new harmonics – though, in digital systems, this may have the unwelcome effect of creating inharmonic or metallic artefacts.

Fundamentally, harmonics connect noise-generating objects’ physical behavior with musical perception. The fundamental provides a primary pitch reference, while the harmonic spectrum gives the sound an identity and expressive character. 

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Featured photo by Logan Voss on Unsplash