Harmonic
Encyclopedia
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. 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: 50 Hz, 75 Hz, 100 Hz etc.
, a bowed
violin
string, or a Cepheid variable star, are more or less periodic
, and so composed of harmonics.
Most passive oscillators, such as a plucked guitar string or a struck drum head or struck bell, naturally oscillate at not one, but several frequencies known as partials
. When the oscillator is long and thin, such as a guitar string, or the column of air in a trumpet, many of the partials are integer multiples of the fundamental frequency; these are called harmonics. Sounds made by long, thin oscillators are for the most part arranged harmonically, and these sounds are generally considered to be musically pleasing. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic
and are sometimes perceived as unpleasant.
The untrained human ear typically does not perceive harmonics as separate notes. Rather, a musical note composed of many harmonically related frequencies is perceived as one sound, the quality, or timbre
of that sound being a result of the relative strengths of the individual harmonic frequencies. Bells
have more clearly perceptible inharmonics than most instruments. Antique singing bowl
s are well known for their unique quality of producing multiple harmonic partials or multiphonics.
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:
Harmonics are not overtones, when it comes to counting. Even numbered harmonics are odd numbered overtones and vice versa.
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 multiphonic
s 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.
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.
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 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.
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 bow
ing technique halfway between hitting and bowing the strings. Composer Lawrence Ball
uses harmonics to generate music electronically.
Wave
In physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...
is a component frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of the signal that is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
multiple of the fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
, 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. 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
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
, the frequencies of the harmonics are: 50 Hz, 75 Hz, 100 Hz etc.
Characteristics
Many oscillators, including the human voiceHuman voice
The human voice consists of sound made by a human being using the vocal folds for talking, singing, laughing, crying, screaming, etc. Its frequency ranges from about 60 to 7000 Hz. The human voice is specifically that part of human sound production in which the vocal folds are the primary...
, a bowed
Bow (music)
In music, a bow is moved across some part of a musical instrument, causing vibration which the instrument emits as sound. The vast majority of bows are used with string instruments, although some bows are used with musical saws and other bowed idiophones....
violin
Violin
The violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello....
string, or a Cepheid variable star, are more or less periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
, and so composed of harmonics.
Most passive oscillators, such as a plucked guitar string or a struck drum head or struck bell, naturally oscillate at not one, but several frequencies known as partials
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
. When the oscillator is long and thin, such as a guitar string, or the column of air in a trumpet, many of the partials are integer multiples of the fundamental frequency; these are called harmonics. Sounds made by long, thin oscillators are for the most part arranged harmonically, and these sounds are generally considered to be musically pleasing. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....
and are sometimes perceived as unpleasant.
The untrained human ear typically does not perceive harmonics as separate notes. Rather, a musical note composed of many harmonically related frequencies is perceived as one sound, the quality, or timbre
Timbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...
of that sound being a result of the relative strengths of the individual harmonic frequencies. Bells
Bell (instrument)
A bell is a simple sound-making device. The bell is a percussion instrument and an idiophone. Its form is usually a hollow, cup-shaped object, which resonates upon being struck...
have more clearly perceptible inharmonics than most instruments. Antique singing bowl
Singing bowl
Singing bowls are a type of bell, specifically classified as a standing bell. Rather than hanging inverted or attached to a handle, singing bowls sit with the bottom surface resting...
s are well known for their unique quality of producing multiple harmonic partials or multiphonics.
Harmonics and overtones
The tight relation between overtones and harmonics in musicMusic
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
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 |
|---|---|---|---|
| 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 |
Harmonics are not overtones, when it comes to counting. Even numbered harmonics are odd numbered overtones and vice versa.
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
Recorder
The recorder is a woodwind musical instrument of the family known as fipple flutes or internal duct flutes—whistle-like instruments which include the tin whistle. The recorder is end-blown and the mouth of the instrument is constricted by a wooden plug, known as a block or fipple...
) this has the effect of making the note go up in pitch by an octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
; but in more complex cases many other pitch variations are obtained. In some cases it also changes the timbre
Timbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...
of the note. This is part of the normal method of obtaining higher notes in wind instruments, where it is called overblowing
Overblowing
Overblowing A technique used while playing a wind instrument which, primarily through manipulation of the supplied air , causes the sounded pitch to jump to a higher one...
. The extended technique
Extended technique
Extended techniques are performance techniques used in music to describe unconventional, unorthodox, or non-traditional techniques of singing, or of playing musical instruments to obtain unusual sounds or instrumental timbres....
of playing multiphonic
Multiphonic
Multiphonics is an extended technique in instrumental music in which a monophonic instrument is made to produce several notes at once....
s 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
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
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
Cello
The cello is a bowed string instrument with four strings tuned in perfect fifths. It is a member of the violin family of musical instruments, which also includes the violin, viola, and double bass. Old forms of the instrument in the Baroque era are baryton and viol .A person who plays a cello is...
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
Overtone singing
Overtone singing, also known as overtone chanting, or harmonic singing, is a type of singing in which the singer manipulates the resonances created as air travels from the lungs, past the vocal folds, and out the lips to produce a melody.The partials of a sound wave made by the human voice can be...
, 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
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
is the reciprocal of the period of the periodic phenomenon.
Harmonics on stringed instruments
Guitar
The guitar is a plucked string instrument, usually played with fingers or a pick. The guitar consists of a body with a rigid neck to which the strings, generally six in number, are attached. Guitars are traditionally constructed of various woods and strung with animal gut or, more recently, with...
(guitar harmonics
Guitar harmonics
A guitar harmonic is a musical note played by preventing or amplifying vibration of certain overtones of a guitar string. Music using harmonics can contain very high pitch notes difficult or impossible to reach by fretting...
), at which gentle touching of a string will force it into a harmonic mode when vibrated. String harmonics are described as having a "flutelike, silvery quality that can be highly effective as a special color
Timbre
In music, timbre is the quality of a musical note or sound or tone that distinguishes different types of sound production, such as voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. The physical characteristics of sound that determine the...
" when used and heard in orchestration
Orchestration
Orchestration is the study or practice of writing music for an orchestra or of adapting for orchestra music composed for another medium...
. 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 |
|---|---|---|---|---|
| 2 | octave | octave Octave In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"... (P8) |
1,200.0 | 0.0 |
| 3 | just perfect fifth | P8 + just perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones... (P5) |
1,902.0 | 702.0 |
| 4 | second octave | 2P8 | 2,400.0 | 0.0 |
| 5 | just major third | 2P8 + just major third (M3) | 2,786.3 | 386.3 |
| 6 | just minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major... |
2P8 + P5 | 3,102.0 | 702.0 |
| 7 | septimal minor third | 2P8 + septimal minor seventh Harmonic seventh The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of... (m7) |
3,368.8 | 968.8 |
| 8 | septimal major second | 3P8 | 3,600.0 | 0.0 |
| 9 | Pythagorean major second | 3P8 + Pythagorean major second Pythagorean interval In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa... (M2) |
3,803.9 | 203.9 |
| 10 | just minor whole tone | 3P8 + just M3 | 3,986.3 | 386.3 |
| 11 | greater unidecimal neutral second Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... |
3P8 + lesser undecimal tritone Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... |
4,151.3 | 551.3 |
| 12 | lesser unidecimal neutral second | 3P8 + P5 | 4,302.0 | 702.0 |
| 13 | tridecimal 2/3-tone | 3P8 + tridecimal neutral sixth Neutral sixth A neutral sixth is a musical interval wider than a minor sixth but narrower than a major sixth . Three distinct intervals may be termed neutral sixths:... (n6) |
4,440.5 | 840.5 |
| 14 | 2/3-tone | 3P8 + P5 + septimal minor third Septimal minor third In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5... (m3) |
4,568.8 | 968.8 |
| 15 | septimal (or major) diatonic semitone | 3P8 + just major seventh Major seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two... (M7) |
4,688.3 | 1,088.3 |
| 16 | just (or minor) diatonic semitone | 4P8 | 4,800.0 | 0.0 |
Table
Other information
Harmonics may be either used or considered as the basis of just intonationJust intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
systems. Composer Arnold Dreyblatt
Arnold Dreyblatt
Arnold Dreyblatt is an American composer and visual artist. He studied music with Pauline Oliveros, La Monte Young, Alvin Lucier and media art with Steina and Woody Vasulka. He has been based in Berlin, Germany since 1984...
is able to bring out different harmonics on the single string of his modified double bass
Double bass
The double bass, also called the string bass, upright bass, standup bass or contrabass, is the largest and lowest-pitched bowed string instrument in the modern symphony orchestra, with strings usually tuned to E1, A1, D2 and G2...
by slightly altering his unique bow
Bow (music)
In music, a bow is moved across some part of a musical instrument, causing vibration which the instrument emits as sound. The vast majority of bows are used with string instruments, although some bows are used with musical saws and other bowed idiophones....
ing technique halfway between hitting and bowing the strings. Composer Lawrence Ball
Lawrence Ball
Lawrence Ball is an English musician and composer who currently lives in North London. He produces multi-media compositions, performs in concert, and also works as a private tutor in mathematics, music theory and physics....
uses harmonics to generate music electronically.
See also
- AristoxenusAristoxenusAristoxenus of Tarentum was a Greek Peripatetic philosopher, and a pupil of Aristotle. Most of his writings, which dealt with philosophy, ethics and music, have been lost, but one musical treatise, Elements of Harmony, survives incomplete, as well as some fragments concerning rhythm and...
- Artificial harmonicArtificial harmonicTo produce an artificial harmonic, a stringed instrument player holds down a note on the neck with the non-dominant hand, thereby shortening the vibrational length of the string, uses a finger to lightly touch a point on the string that is an integer divisor of its vibrational length, and plucks or...
- Harmonics (electrical power)Harmonics (electrical power)Harmonics are electric voltages and currents that appear on the electric power system as a result of certain kinds of electric loads. Harmonic frequencies in the power grid are a frequent cause of power quality problems.-Causes:...
- Electronic tunerElectronic tunerThe term electronic tuner can refer to a number of different things, depending which discipline you wish to study.In the Discipline of radio frequency electronics an electronic tuner is a device which tunes across a part of the radio frequency spectrum by the application of a voltage or appropriate...
- FormantFormantFormants are defined by Gunnar Fant as 'the spectral peaks of the sound spectrum |P|' of the voice. In speech science and phonetics, formant is also used to mean an acoustic resonance of the human vocal tract...
- Fourier seriesFourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
- Fundamental frequencyFundamental frequencyThe fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...
- Harmonic oscillatorHarmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
- Harmonic series (music)Harmonic series (music)Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
- HarmonyHarmonyIn music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
- Inharmonic
- Just intonationJust intonationIn music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
- Overtones
- Pinch harmonicPinch harmonicA pinch harmonic or pick harmonic is a guitar technique in which the player's thumb or index finger on the picking hand slightly catches the string after it is picked, canceling the fundamental of the string, and letting one of the overtones dominate. This results in a high pitched sound...
- Pure tonePure toneA pure tone is a tone with a sinusoidal waveshape.A sine wave is characterized by its frequency, the number of cycles per second—or its wavelength, the distance the waveform travels through its medium within a period—and the amplitude, the size of each cycle...
- Pythagorean tuningPythagorean tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
- Scale of harmonicsScale of harmonicsThe scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale . Most fret positions appearing on Non-Western string instruments ...
- Singing bowlSinging bowlSinging bowls are a type of bell, specifically classified as a standing bell. Rather than hanging inverted or attached to a handle, singing bowls sit with the bottom surface resting...
- Stretched octave
- Tap harmonicTap harmonicTap harmonic is a technique used with fretted string instruments, . It is executed by tapping on the actual fret wire most commonly at the 12th fret, but also can be executed by tapping any of the fret wires with proper technique. It can also be done by gently touching the string over the fret wire...
- Xenharmonic