Harmonics may be achieved in anything that can be fit to a wave, but most often the term is used to describe their use in music. In music, harmonics refer to the sounds which are produced at multiples of the same frequency as a base sound. This base sound is often referred to as the fundamental, or first harmonic.
An example may help demonstrate this more clearly: if the note being produced is A4, for example, it resonates at 440 Hz/second. At the second multiple of this fundamental, or 880 Hz/second, we find the second harmonic. At the third multiple of this fundamental, or 1320 Hz/second, we find the third harmonic. At the fourth multiple of this fundamental, or 1760 Hz/second, we find the fourth harmonic. This continues on well past the sound being audible to the human ear.
Musically, rather than talking about the actual frequencies of the harmonics, we would instead talk about their relationship to the fundamental in musical terms. So we can look at the second harmonic as being an octave above the fundamental. The third harmonic is then an octave and a fifth above the fundamental. The fourth harmonic is two octaves above the fundamental. And so on, with the harmonics alternating between being a major third and a minor third higher each time we go up.
When most musical instruments are played, they generate not only a fundamental sound, but can also generate a number of harmonics on top of this. This is most easily heard by listening to a guitar being played, where you will hear these other notes coming on top of the fundamental. For most people, these harmonics don’t actually sound like distinct notes — instead, it is this combination of sounds that helps give a song its distinct timbre. A singing bowl, such as those used in Tibet, also gives a very good example of multiple harmonics being generated on top of a very pure fundamental.
Sounds can also layer on top of a fundamental that are not actually harmonics. Musically these are usually referred to as inharmonics, and they can sound jarring or strange to listeners. This strangeness is most pronounced if it is very close to the frequency of a true harmonic, sounding slightly off. If it is far enough away from one of the actual harmonics of a fundamental, the sound is instead called a partial, and may be used for a musical purpose — although the effect is still rather eerie.
Harmonics may also be layered on to the human voice, producing some very interesting effects. This is usually referred to as overtone singing, and some specific styles are collected under the term throat singing. Throat singing makes use of the mouth as a resonating chamber, using its shape, and the shape of the tongue, to alter the sound as it travels out of the throat to shape it into a harmonic of the fundamental. When done correctly this can produce some truly amazing sounds — in the case of the Tuvan style of sygyt, the harmonics create a piercing whistle reminiscent of birdsong, with the underlying drone of the fundamental still audible.
An understanding of harmonics is not absolutely essential for someone taking up a musical instrument, but it can be very helpful in better understanding what’s happening. Particularly for stringed instruments, harmonics play a crucial role in developing a dynamic, interesting style. Knowing the physics involved allows a musician to manipulate them accordingly, to make the instrument’s harmonics sound the way they want them to.