A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental frequency, therefore the sum of harmonics is also periodic at that frequency. Harmonic frequencies are equally spaced by the width of the fundamental frequency and can be found by repeatedly adding that frequency. For example, if the fundamental frequency is 25 Hz, the frequencies of the harmonics are: 50 Hz, 75 Hz, 100 Hz, etc.
Many oscillators, including the human voice, a bowed violin string, or a Cepheid variable star, are more or less periodic, and so composed of harmonics.
Most passive oscillators, such as a plucked guitar string or a struck drum head or struck bell, naturally oscillate at not one, but several frequencies known as partials. When the oscillator is long and thin, such as a guitar string, or the column of air in a trumpet, many of the partials are integer multiples of the fundamental frequency; these are called harmonics. Sounds made by long, thin oscillators are for the most part arranged harmonically, and these sounds are generally considered to be musically pleasing. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic and are sometimes perceived as unpleasant.
The untrained human ear typically does not perceive harmonics as separate notes. Rather, a musical note composed of many harmonically related frequencies is perceived as one sound, the quality, or timbre of that sound being a result of the relative strengths of the individual harmonic frequencies. Bells have more clearly perceptible inharmonics than most instruments. Antique singing bowls are well known for their unique quality of producing multiple harmonic partials or multiphonics.
Harmonics and overtones
The tight relation between overtones and harmonics in music often leads to their being used synonymously in a strictly musical context, but they are counted differently leading to some possible confusion. This chart demonstrates how they are counted:
Harmonics are not overtones, when it comes to counting. Even numbered harmonics are odd numbered overtones and vice versa.
In many musical instruments, it is possible to play the upper harmonics without the fundamental note being present. In a simple case (e.g., recorder) this has the effect of making the note go up in pitch by an octave; but in more complex cases many other pitch variations are obtained. In some cases it also changes the timbre of the note. This is part of the normal method of obtaining higher notes in wind instruments, where it is called overblowing. The extended technique of playing multiphonics also produces harmonics. On string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at a unison the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found half way down the highest string of a cello produces the same pitch as lightly fingering the node 1/3 of the way down the second highest string. For the human voice see Overtone singing, which uses harmonics.
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