Repetitive back-and-forth movement through a central, or equilibrium, position in which the maximum displacement on one side is equal to the maximum displacement on the other. Each complete vibration takes the same time, the period; the reciprocal of the period is the frequency of vibration. The force that causes the motion is always directed toward the equilibrium position and is directly proportional to the distance from it. A pendulum displays simple harmonic motion; other examples include the electrons in a wire carrying alternating current and the vibrating particles of a medium carrying sound waves.
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In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the frequency is f, the harmonics have frequency 2f, 3f, 4f, etc, as well as f itself. The harmonics have the property that they are all periodic at the signal frequency. Also, due to the properties of Fourier series, the sum of the signal and its harmonics is also periodic at that frequency.
Most passive oscillators, such as a plucked guitar string or a struck drum head or struck bell, naturally oscillate at several frequencies known as overtones. When the oscillator is long and thin, such as a guitar string, a trumpet, or a chime, the overtones are still integer multiples of the fundamental frequency. Hence, these devices can mimic the sound of singing and are often incorporated into music. Overtones whose frequency is not an integer multiple 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. In a musical context, overtones that are not exactly integer multiples of the fundamental are known as inharmonics. Inharmonics that are not close to harmonics are known as partials. Bells have more clearly perceptible partials than most instruments. Antique singing bowls are well known for their unique quality of producing multiple harmonic overtones or multiphonics.
|1f||440 Hz||fundamental frequency||first harmonic|
|2f||880 Hz||first overtone||second harmonic|
|3f||1320 Hz||second overtone||third harmonic|
|4f||1760 Hz||third overtone||fourth harmonic|
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.
Harmonics may be either used or considered as the basis of just intonation systems. Composer Arnold Dreyblatt is able to bring out different harmonics on the single string of his modified double bass by slightly altering his unique bowing technique halfway between hitting and bowing the strings. Composer Lawrence Ball uses harmonics to generate music electronically.
The following table displays the stop points on a stringed instrument, such as the guitar, at which gentle touching of a string will force it into a harmonic mode when vibrated.
|harmonic||stop note||harmonic noteing||cents|| reduced|
|3||just perfect fifth||P8 + P5||1902.0||702.0|
|4||just perfect fourth||2P8||2400.0||0.0|
|5||just major third||2P8 + just M3||2786.3||386.3|
|6||just minor third||2P8 + P5||3102.0||702.0|
|7||septimal minor third||2P8 + septimal m7||3368.8||968.8|
|8||septimal major second||3P8||3600.0||0.0|
|9||Pythagorean major second||3P8 + pyth M2||3803.9||203.9|
|10||just minor whole tone||3P8 + just M3||3986.3||386.3|
|11||greater unidecimal neutral second||3P8 + just M3 + GUN2||4151.3||551.3|
|12||lesser unidecimal neutral second||3P8 + P5||4302.0||702.0|
|13||tridecimal 2/3-tone||3P8 + P5 + T23T||4440.5||840.5|
|14||2/3-tone||3P8 + P5 + septimal m3||4568.8||968.8|
|15||septimal (or major) diatonic semitone||3P8 + P5 + just M3||4688.3||1088.3|
|16||just (or minor) diatonic semitone||4P8||4800.0||0.0|