Harmonics can be achieved in anything that can be fitted to a wave, but they are most commonly used in music. Harmonics are sounds that are produced at multiples of the same frequency as the base sound in music. The fundamental, or first harmonic, is the name given to this base sound.
An example might help to clarify this: the note A4 resonates at 440 Hz/second, for example. The second harmonic is found at the second multiple of this fundamental, or 880 Hz/second. The third harmonic is found at the third multiple of this fundamental, or 1320 Hz/second. The fourth harmonic is found at the fourth multiple of this fundamental, or 1760 Hz/second. This goes on for a long time after the sound is audible to the human ear.
Rather than discussing the harmonics’ actual frequencies, we would instead discuss their relationship to the fundamental in musical terms. As a result, we can consider the second harmonic to be an octave higher than the fundamental. After that, the third harmonic is an octave and a fifth higher than the fundamental. Two octaves above the fundamental is the fourth harmonic. Each time we go up, the harmonics alternate between being a major third and a minor third higher.
When most musical instruments are played, they not only produce a fundamental sound, but also a number of harmonics on top of it. This is best demonstrated listening to a guitar being played, as these other notes will appear on top of the fundamental. Most people don’t hear these harmonics as distinct notes; instead, they’re a combination of sounds that contribute to the timbre of a song. Multiple harmonics are generated on top of a very pure fundamental in a singing bowl, such as those used in Tibet.
Sounds that aren’t harmonics can also layer on top of a fundamental. In music, these are known as inharmonics, and they can be disconcerting or strange to listeners. This strangeness is most noticeable when the frequency is very close to that of a true harmonic, sounding slightly off. The sound is called a partial if it is far enough away from one of the actual harmonics of a fundamental, and it can be used for musical purposes — though the effect is still eerie.
Harmonics can also be layered on top of human voices, resulting in some very interesting results. Overtone singing is the most common term for this, and throat singing refers to a few distinct styles. Throat singing uses the mouth as a resonating chamber, altering the sound as it travels out of the throat with the shape of the mouth and the shape of the tongue to shape it into a harmonic of the fundamental. When done correctly, the harmonics create a piercing whistle reminiscent of birdsong, with the underlying drone of the fundamental still audible, as in the Tuvan style of sygyt.
Understanding harmonics isn’t required for someone who wants to learn to play a musical instrument, but it can help them understand what’s going on. Harmonics are especially important for stringed instruments in developing a dynamic, interesting style. Knowing the physics allows a musician to manipulate them to make the harmonics of an instrument sound the way they want them to.