In acoustics and telecommunication, a harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the fundamental frequency is f, the harmonics have frequencies f, 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: 25 Hz, 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 thus can be decomposed into harmonics.
Most passive oscillators, such as a plucked guitar string or a struck drum head or struck bell, naturally oscillate at several frequencies known as partials. When the oscillator is long and thin, such as a guitar string, a trumpet, or a chime, the partials are practically integer multiples of the fundamental frequency. Hence, these devices can mimic the sound of singing and are often incorporated into music. Partials whose frequencies are not integer multiples of the fundamental are called inharmonic and are sometimes perceived as unpleasant.
The untrained human ear typically does not perceive harmonics as separate notes. Instead, they are perceived as the timbre of the tone. Bells have more clearly perceptible partials than most instruments. Antique singing bowls are well known for their unique quality of producing multiple harmonic partials or multiphonics.
Harmonics in music are notes which are produced in a special way. They are notes which are produced as part of the "harmonic series".
In physics a harmonic is a wave which is added to the basic fundamental wave. In this article we are talking about sound waves, and we can understand it clearly by looking at the strings of a musical instrument.
When a violinist plays a note on a violin string the string starts to vibrate very fast. This vibration makes the air vibrate and the sound waves travel to our ear so that we can hear it. If the note was absolutely pure the string would move like a sine wave. Sine waves can only be made electronically and they sound very boring to us. The note played on the violin string makes the string vibrate in a very complicated way. There is the basic note (the fundamental), but added to that are lots of other little notes that all add up to a sound in a special way that tells us that it is a violin playing and not a clarinet or a human voice.
The higher the note the faster the string vibrates. An A above middle C (the violinist's A string) vibrates at 440Hz (440 times per second). This is the "fundamental" or "first harmonic". The second harmonic is vibrating twice as fast (ratio 2:1): 880Hz. This gives an A an octave higher. The third harmonic will give a ratio 3:2. This will be an E (an octave and a fifth above the fundamental). The higher the harmonic the quieter it is, but the ratio is always a whole number (not a fraction).
Every note that we hear on an instrument is really a combination of several notes or "harmonics", even although we may not realize that we are hearing more than one note at a time. Play the lowest C on the piano. Now find the next C which is an octave higher. Press this key very slowly so that it does not sound and hold it down. While holding it down play the bottom C again making it loud and very short. The C that is being held silently will now sound. This is because the strings of that C are vibrating a little because it is a harmonic of the low C (they can vibrate because the damper is off the string while the note is being held down). The same can be done holding the next G down, then the next C, then the E. The higher the note the fainter (quieter) the harmonics become. The musical example below shows the notes of a harmonic series in musical notation.