Harmonic

This article is about the components of periodic signals. For other uses, see Harmonic (disambiguation).

The nodes of a vibrating string are harmonics.

Two different notations of natural harmonics on the cello. First as sounded (more common), then as fingered (easier to sightread).

The term harmonic in its strictest sense is any member of the harmonic series. The term is employed in various disciplines, including music and acoustics, electronic power transmission, radio technology, etc. It is typically applied when considering the frequencies of repeating signals, such as sinusoidal waves, that happen to relate as whole-numbered multiples. In that case, a harmonic is a signal whose frequency is a whole-numbered multiple of the frequency of some other given signal. For example, in alternating current of 60 cycles per second, "fifth-harmonic distortion" would produce an unwanted additional signal at 300 cycles per second, exactly five times the original frequency.

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. As multiples of the fundamental frequency, successive harmonics can be found by repeatedly adding the fundamental frequency. For example, if the fundamental frequency (first harmonic) is 25 Hz, the frequencies of the next harmonics are: 50 Hz (2nd harmonic), 75 Hz (3rd harmonic), 100 Hz (4th harmonic) etc.

Characteristics

Most acoustic instruments emit complex tones containing many individual partials (component simple tones or sinusoidal waves), but the untrained human ear typically does not perceive those partials as separate phenomena. Rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials.

Many acoustic oscillators, such as the human voice or a bowed violin string, produce complex tones that are more or less periodic, and thus are composed of partials that are near matches to integer multiples of the fundamental frequency and therefore resemble the ideal harmonics and are called "harmonic partials" or simply "harmonics" for convenience (although it's not strictly accurate to call a partial a harmonic, the first being real and the second being ideal). Oscillators that produce harmonic partials behave somewhat like 1-dimensional resonators, and are often long and thin, such as a guitar string or a column of air open at both ends (as with the modern orchestral transverse flute). Wind instruments whose air column is open at only one end, such as trumpets and clarinets, also produce partials resembling harmonics. However they only produce partials matching the odd harmonics, at least in theory. The reality of acoustic instruments is such that none of them behaves as perfectly as the somewhat simplified theoretical models would predict.

Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials.

Some acoustic instruments emit a mix of harmonic and inharmonic partials but still produce an effect on the ear of having a definite fundamental pitch, such as pianos, strings plucked pizzicato, vibraphones, marimbas, and certain pure-sounding bells or chimes. Antique singing bowls are known for producing multiple harmonic partials or multiphonics.

Other oscillators, such as cymbals, drum heads, and other percussion instruments, naturally produce an abundance of inharmonic partials and do not imply any particular pitch, and therefore cannot be used melodically or harmonically in the same way other instruments can.

Harmonics and overtones

An overtone is any frequency higher than the fundamental. 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:

Frequency	Order	Name 1	Name 2	Wave Representation	Molecular Representation
1 · f =   440 Hz	n = 1	fundamental tone	1st harmonic
2 · f =   880 Hz	n = 2	1st overtone	2nd harmonic
3 · f = 1320 Hz	n = 3	2nd overtone	3rd harmonic
4 · f = 1760 Hz	n = 4	3rd overtone	4th 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 halfway 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.

While it is true that electronically produced periodic tones (e.g. square waves or other non-sinusoidal waves) have "harmonics" that are whole number multiples of the fundamental frequency, practical instruments do not all have this characteristic. For example higher "harmonics"' of piano notes are not true harmonics but are "overtones" and can be very sharp, i.e. a higher frequency than given by a pure harmonic series. This is especially true of instruments other than stringed or brass/woodwind ones, e.g., xylophone, drums, bells etc., where not all the overtones have a simple whole number ratio with the fundamental frequency.

The fundamental frequency is the reciprocal of the period of the periodic phenomenon.

 This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".

Harmonics on stringed instruments

Playing a harmonic on a string

The following table displays the stop points on a stringed instrument, such as the guitar (guitar harmonics), at which gentle touching of a string will force it into a harmonic mode when vibrated. String harmonics (flageolet tones) are described as having a "flutelike, silvery quality that can be highly effective as a special color" when used and heard in orchestration. It is unusual to encounter natural harmonics higher than the fifth partial on any stringed instrument except the double bass, on account of its much longer strings.

Harmonic	Stop note	Sounded note relative to open string	Cents above open string	Cents reduced to one octave	Audio
2	octave	octave (P8)	1,200.0	0.0	<phonos file="Perfect octave on C.mid">Play</phonos>
3	just perfect fifth	P8 + just perfect fifth (P5)	1,902.0	702.0	<phonos file="Just perfect fifth on C.mid">Play</phonos>
4	second octave	2P8	2,400.0	0.0	<phonos file="Perfect fifteenth on C.mid">Play</phonos>
5	just major third	2P8 + just major third (M3)	2,786.3	386.3	<phonos file="Just major third on C.mid">Play</phonos>
6	just minor third	2P8 + P5	3,102.0	702.0
7	septimal minor third	2P8 + septimal minor seventh (m7)	3,368.8	968.8	<phonos file="Harmonic seventh on C.mid">Play</phonos>
8	septimal major second	3P8	3,600.0	0.0
9	Pythagorean major second	3P8 + Pythagorean major second (M2)	3,803.9	203.9	<phonos file="Major tone on C.mid">Play</phonos>
10	just minor whole tone	3P8 + just M3	3,986.3	386.3
11	greater undecimal neutral second	3P8 + lesser undecimal tritone	4,151.3	551.3	<phonos file="Eleventh harmonic on C.mid">Play</phonos>
12	lesser undecimal neutral second	3P8 + P5	4,302.0	702.0
13	tridecimal 2/3-tone	3P8 + tridecimal neutral sixth (n6)	4,440.5	840.5	<phonos file="Tridecimal neutral sixth on C.mid">Play</phonos>
14	2/3-tone	3P8 + P5 + septimal minor third (m3)	4,568.8	968.8
15	septimal (or major) diatonic semitone	3P8 + just major seventh (M7)	4,688.3	1,088.3	<phonos file="Just major seventh on C.mid">Play</phonos>
16	just (or minor) diatonic semitone	4P8	4,800.0	0.0

Table

Table of harmonics of a stringed instrument with colored dots indicating which positions can be lightly fingered to generate just intervals up to the 7th harmonic

Artificial harmonics

Although harmonics are most often used on open strings, occasionally a score will call for an artificial harmonic, produced by playing an overtone on a stopped string. As a performance technique, it is accomplished by using two fingers on the fingerboard, the first to shorten the string to the desired fundamental, with the second touching the node corresponding to the appropriate harmonic.

Other information

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.

See also

• Aristoxenus
• Harmonics (electrical power)
• Electronic tuner
• Formant
• Fourier series
• Harmonic oscillator
• Harmony
• Pinch harmonic
• Pure tone
• Pythagorean tuning
• Scale of harmonics
• Spherical harmonics
• Stretched octave
• Subharmonic
• Tap harmonic
• Xenharmonic

External links

• Harmonics, partials and overtones from fundamental frequency
• Discussion of Sciarrino's violin etudes and notation issues
•  
• Harmonics
• Ear and see harmonics on a Piano