Definitions

# harmonic

[hahr-mon-ik]
harmonic. 1 Physical term describing the vibration in segments of a sound-producing body (see sound). A string vibrates simultaneously in its whole length and in segments of halves, thirds, fourths, etc. These segments form what is known in algebra as a harmonic series or progression, since the rate of vibration of each segment is an integral multiple of the frequency of the whole string, i.e., each segment vibrates respectively twice, three times, four times, etc., as fast as the whole string. The vibration of the whole string produces the fundamental tone, and the segments produce weaker subsidiary tones. A similar phenomenon occurs in an air column in a pipe. At most the first 16 tones in such a series can be heard by the human ear; the character or timbre of a fundamental tone is determined by the number of its subsidiary tones heard and their relative intensity. The subsidiary tones have been loosely called harmonics (as a noun), but they are properly called partials, the fundamental tone being the first partial. They are also called overtones (a synonym for "upper partials"), although this term includes a number of sounds that do not fit in with the harmonic series, and are therefore not considered musical. 2 Term describing the silvery sound produced separately when the fundamental and possibly more partial tones are damped by touching a string at a nodal point. Similarly harmonics are produced separately in an air column by overblowing or in brass wind instruments by the use of valves.

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.

## Characteristics

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 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.

## 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:

 1f 2f 3f 440 Hz fundamental frequency first harmonic 880 Hz first overtone second harmonic 1320 Hz second overtone third harmonic 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 fundamental frequency is the reciprocal of the period of the periodic phenomenon.

## Harmonics on stringed instruments

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
cents
2 octave P8 1200.0 0.0
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