Harmonic

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).

A harmonic is any member of the harmonic series, a divergent infinite series. Its name derives from the concept of overtones, or harmonics in musical instruments: the wavelengths of the overtones of a vibrating string or a column of air (as with a tuba) are derived from the string's (or air column's) fundamental wavelength. Every term of the series (i.e., the higher harmonics) after the first is the "harmonic mean" of the neighboring terms. The phrase "harmonic mean" likewise derives from music.

The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves. A harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is also called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz.

In music, harmonics are used on string instruments and wind instruments as a way of producing sound on the instrument, particularly to play higher notes and, with strings, obtain notes that have a unique sound quality or "tone colour". On strings, harmonics that are bowed have a "glassy", pure tone. On stringed instruments, harmonics are played by touching (but not fully pressing down the string) at an exact point on the string while sounding the string (plucking, bowing, etc.); this allows the harmonic to sound, a pitch which is always higher than the fundamental frequency of the string.

Terminology

Harmonics may also be called "overtones", "partials" or "upper partials". The difference between "harmonic" and "overtone" is that the term "harmonic" includes all of the notes in a series, including the fundamental frequency (e.g., the open string of a guitar), the term "overtone" only includes the pitches above the fundamental. In some music contexts, the terms "harmonic", "overtone" and "partial" are used fairly interchangeably.

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 one-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. [1] [2] 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.

Partials, overtones, and harmonics

An overtone is any partial higher than the lowest partial in a compound tone. The relative strengths and frequency relationships of the component partials determine the timbre of an instrument. The similarity between the terms overtone and partial sometimes leads to their being loosely used interchangeably in a musical context, but they are counted differently, leading to some possible confusion. In the special case of instrumental timbres whose component partials closely match a harmonic series (such as with most strings and winds) rather than being inharmonic partials (such as with most pitched percussion instruments), it is also convenient to call the component partials "harmonics" but not strictly correct (because harmonics are numbered the same even when missing, while partials and overtones are only counted when present). This chart demonstrates how the three types of names (partial, overtone, and harmonic) are counted (assuming that the harmonics are present):

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

[3]

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 or tone color (timbre) when used and heard in orchestration.[4] 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.[5] Harmonics are widely used in plucked string instruments, such as acoustic guitar, electric guitar and electric bass. On an electric guitar played loudly through a guitar amplifier with distortion, harmonics are more sustained and can be used in guitar solos. In the heavy metal music lead guitar style known as shred guitar, harmonics, both natural and artificial, are widely used.

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  Play 
3 just perfect fifth P8 + just perfect fifth (P5) 1,902.0 702.0  Play 
4 second octave 2P8 2,400.0 0.0  Play 
5 just major third 2P8 + just major third (M3) 2,786.3 386.3  Play 
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  Play 
8 septimal major second 3P8 3,600.0 0.0
9 Pythagorean major second 3P8 + Pythagorean major second (M2) 3,803.9 203.9  Play 
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  Play 
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  Play 
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  Play 
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 (natural harmonics), occasionally a score will call for an artificial harmonic, produced by playing an overtone on an already 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. On fretted instruments, such as an electric guitar, the performer can look at the frets to determine where to stop the string and where to touch the node. On unfretted instruments, such as the violin and related instruments, playing artificial harmonics is an advanced technique, as it requires the performer to find two precise locations on the same string.

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
Violin harmonics
Violin natural harmonic stop points on the A string
Harmonics 110x16
Demonstration of 16 harmonics using electronic sine tones, starting with 110 Hz fundamental, 0.5s each. Note that each harmonic is presented at the same signal level as the fundamental; the sample tones sound louder as they increase in frequency
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References

  1. ↑ Acoustical Society of America - Large grand and small upright pianos by Alexander Galembo and Lola L. Cuddly
  2. ↑ Hanna Järveläinen et al. 1999. "Audibility of Inharmonicity in String Instrument Sounds, and Implications to Digital Sound Systems"
  3. ↑  This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".
  4. ↑ Kennan, Kent and Grantham, Donald (2002/1952). The Technique of Orchestration, p.69. Sixth Edition. ISBN 0-13-040771-2.
  5. ↑ Kennan & Grantham, ibid, p.71.
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