Harmonic series (music)
Encyclopedia
Pitched musical instrument
Musical instrument
A musical instrument is a device created or adapted for the purpose of making musical sounds. In principle, any object that produces sound can serve as a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates back to the...

s are often based on an approximate harmonic oscillator
Harmonic oscillator
In 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....

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 each other to form standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...

s. Interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the typical spacing of the resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

s, these frequencies are mostly limited to integer multiples, or harmonic
Harmonic
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...

s, of the lowest frequency, and such multiples form the harmonic series (see harmonic series (mathematics)
Harmonic series (mathematics)
In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

).

The musical pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

of a note is usually perceived as the lowest partial present (the fundamental frequency), which may be the one created by vibration
Vibration
Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.Vibration is occasionally "desirable"...

over the full length of the string or air column, or a higher harmonic chosen by the player. The musical 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 a steady tone from such an instrument is determined by the relative strengths of each harmonic.

### Partial, harmonic, fundamental, inharmonicity, and overtone

Any complex tone "can be described as a combination of many simple periodic waves (i.e., sine waves) or partials, each with its own frequency of vibration, amplitude, and phase."

A partial is any of the sine waves by which a complex tone is described.

A harmonic (or a harmonic partial) is any of a set of partials that are whole number multiples of a common 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...

. This set includes the fundamental, which is a whole number multiple of itself (1 times itself).

Inharmonicity
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....

is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

for each partial.

Typical pitched instruments are designed to have partials that are close to being harmonics, with very low inharmonicity; therefore, in music theory, and in instrument tuning, it is convenient to speak of the partials in those instruments' sounds as harmonics, even if they have some inharmonicity. Other pitched instruments, especially certain percussion instruments, such as marimba
Marimba
The marimba is a musical instrument in the percussion family. It consists of a set of wooden keys or bars with resonators. The bars are struck with mallets to produce musical tones. The keys are arranged as those of a piano, with the accidentals raised vertically and overlapping the natural keys ...

, vibraphone
Vibraphone
The vibraphone, sometimes called the vibraharp or simply the vibes, is a musical instrument in the struck idiophone subfamily of the percussion family....

, tubular bell
Tubular bell
Tubular bells are musical instruments in the percussion family. Each bell is a metal tube, 30–38 mm in diameter, tuned by altering its length. Its standard range is from C4-F5, though many professional instruments reach G5 . Tubular bells are often replaced by studio chimes, which are a smaller...

s, and timpani
Timpani
Timpani, or kettledrums, are musical instruments in the percussion family. A type of drum, they consist of a skin called a head stretched over a large bowl traditionally made of copper. They are played by striking the head with a specialized drum stick called a timpani stick or timpani mallet...

, contain non-harmonic partials, yet give the ear a good sense of pitch. Non-pitched, or indefinite-pitched instruments, such as cymbals, gongs, or tam-tams make sounds rich in inharmonic partials.

An overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

is any partial except the lowest. Overtone does not imply harmonicity or inharmonicity and has no other special meaning other than to exclude the fundamental. This can lead to numbering confusion when comparing overtones to partials; the first overtone is the second partial.

Some electronic instruments, such as theremin
Theremin
The theremin , originally known as the aetherphone/etherophone, thereminophone or termenvox/thereminvox is an early electronic musical instrument controlled without discernible physical contact from the player. It is named after its Russian inventor, Professor Léon Theremin, who patented the device...

s and synthesizer
Synthesizer
A synthesizer is an electronic instrument capable of producing sounds by generating electrical signals of different frequencies. These electrical signals are played through a loudspeaker or set of headphones...

s, can play a pure frequency with no overtones, although synthesizers can also combine frequencies into more complex tones, for example to simulate other instruments. Certain flutes and ocarinas are very nearly without overtones.

## Frequencies, wavelengths, and musical intervals in example systems

The simplest case to visualise is a vibrating string, as in the illustration; the string has fixed points at each end, and each harmonic mode
Normal mode
A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies...

divides it into 1, 2, 3, 4, etc., equal-sized sections resonating at increasingly higher frequencies. Similar arguments apply to vibrating air columns in wind instruments, although these are complicated by having the possibility of anti-nodes (that is, the air column is closed at one end and open at the other) or conical
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

as opposed to cylindrical
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

bore
Bore (wind instruments)
The bore of a wind instrument is its interior chamber that defines a flow path through which air travels and is set into vibration to produce sounds. The shape of the bore has a strong influence on the instruments' timbre.-Bore shapes:...

s.

In most pitched musical instruments, the fundamental (first harmonic) is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency wave
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...

s occur with varying prominence and give each instrument its characteristic tone quality. The fact that a string is fixed at each end means that the longest allowed wavelength on the string (giving the fundamental frequency) is twice the length of the string (one round trip, with a half cycle fitting between the nodes at the two ends). Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times that of the fundamental.

Theoretically, these shorter wavelengths correspond to vibrations at frequencies that are 2, 3, 4, 5, 6, etc., times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies. (See inharmonicity
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....

and stretched tuning
Stretched tuning
Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital electric pianos , and some sample-based synthesizers based on these instruments, to accommodate the natural inharmonicity of their vibrating elements...

for alterations specific to wire-stringed instruments and certain electric pianos.) However, those alterations are small, and except for precise, highly specialized tuning, it is reasonable to think of the frequencies of the harmonic series as integer multiples of the fundamental frequency.

The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f, 5×f, ...). In terms of frequency (measured in cycles per second, or hertz
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....

(Hz) where f is the fundamental frequency), the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because our ears respond to sound nonlinearly, we perceive higher harmonics as "closer together" than lower ones. On the other hand, the 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"...

series is a geometric progression
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

(2×f, 4×f, 8×f, 16×f, ...), and we hear these distances as "the same" in the sense of musical interval. In terms of what we hear, each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals.

The second harmonic (or first overtone), twice the frequency of the fundamental, sounds 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"...

higher; the third harmonic, three times the frequency of the fundamental, sounds a 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...

above the second. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...

above the third (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher). The combined oscillation of a string with several of its lowest harmonics can be seen clearly in an interactive animation at Edward Zobel's "Zona Land".

## Harmonics and tuning

If the harmonics are transposed
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...

into the span of one 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"...

, they approximate some of the notes in what the West
Western world
The Western world, also known as the West and the Occident , is a term referring to the countries of Western Europe , the countries of the Americas, as well all countries of Northern and Central Europe, Australia and New Zealand...

has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal semitone
Minor second
In modern Western tonal music theory a minor second is the interval between two notes on adjacent staff positions, or having adjacent note letters, whose alterations cause them to be one semitone or half-step apart, such as B and C or C and D....

s, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. In the late 1930s, composer Paul Hindemith
Paul Hindemith
Paul Hindemith was a German composer, violist, violinist, teacher, music theorist and conductor.- Biography :Born in Hanau, near Frankfurt, Hindemith was taught the violin as a child...

ranked musical intervals according to their relative dissonance
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...

based on these and similar harmonic relationships.

Below is a comparison between the first 31 harmonics and the intervals of 12-tone equal temperament (12tET), transposed into the span of one octave. Tinted fields highlight differences greater than 5 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

(1/20th of a semitone), which is the human ear's "just noticeable difference
Just noticeable difference
In psychophysics, a just noticeable difference, customarily abbreviated with lowercase letters as jnd, is the smallest detectable difference between a starting and secondary level of a particular sensory stimulus...

" for notes played one after the other (smaller differences are noticeable with notes played simultaneously).

Harmonic 12tET Interval Note Variance cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...

1 2 4 8 16 prime (octave) C 0
17 minor second C, D +5
9 18 major second D +4
19 minor third D, E −2
5 10 20 major third E −14
21 fourth F −29
11 22 tritone F, G −49
23 +28
3 6 12 24 fifth G +2
25 minor sixth G, A −27
13 26 +41
27 major sixth A +6
7 14 28 minor seventh A, B −31
29 +30
15 30 major seventh B −12
31 +45

The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals (see just intonation
Just 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...

). This objective structure is augmented by psychoacoustic phenomena. For example, a perfect fifth, say 200 and 300 Hz (cycles per second), causes a listener to perceive a combination tone
Combination tone
A combination tone, also called a sum tone or a difference tone , can be any of at least three similar psychoacoustic phenomena. When two tones are played simultaneously, a listener can sometimes perceive an additional tone whose frequency is a sum or difference of the two frequencies...

of 100 Hz (the difference between 300 Hz and 200 Hz); that is, an octave below the lower (actual sounding) note. This 100 Hz first order combination tone then interacts with both notes of the interval to produce second order combination tones of 200 (300-100) and 100 (200-100) Hz and, of course, all further nth order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When we contrast this with a dissonant interval such as a tritone (not tempered) with a frequency ratio of 7:5 we get, for example, 700-500=200 (1st order combination tone)and 500-200=300 (2nd order). The rest of the combination tones are octaves of 100 Hz so the 7:5 interval actually contains 4 notes: 100 Hz (and its octaves), 300 Hz, 500 Hz and 700 Hz. Note that the lowest combination tone (100 Hz) is a 17th (2 octaves and a major third) below the lower (actual sounding) note of the tritone. All the intervals succumb to similar analysis as has been demonstrated by Paul Hindemith
Paul Hindemith
Paul Hindemith was a German composer, violist, violinist, teacher, music theorist and conductor.- Biography :Born in Hanau, near Frankfurt, Hindemith was taught the violin as a child...

in his book, The Craft of Musical Composition.

## Timbre of musical instruments

The relative amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...

s (strengths) of the various harmonics primarily determine 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 different instruments and sounds, though onset transients, formant
Formant
Formants 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...

s, noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...

s, and inharmonicities also play a role. For example, the clarinet
Clarinet
The clarinet is a musical instrument of woodwind type. The name derives from adding the suffix -et to the Italian word clarino , as the first clarinets had a strident tone similar to that of a trumpet. The instrument has an approximately cylindrical bore, and uses a single reed...

and saxophone
Saxophone
The saxophone is a conical-bore transposing musical instrument that is a member of the woodwind family. Saxophones are usually made of brass and played with a single-reed mouthpiece similar to that of the clarinet. The saxophone was invented by the Belgian instrument maker Adolphe Sax in 1846...

have similar mouthpiece
Mouthpiece (woodwind)
The mouthpiece of a woodwind instrument is that part of the instrument which is placed partly in the player's mouth. Single-reed instruments, capped double-reed instruments, and fipple flutes have mouthpieces while exposed double-reed instruments and open flutes do not.-Single-reed instruments:On...

s and reeds, and both produce sound through resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, the even-numbered harmonics are suppressed, which produces a purer tone. The saxophone's resonator is conical, which allows the even-numbered harmonics to sound more strongly and thus produces a more complex tone. The inharmonic
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....

ringing of the instrument's metal resonator is even more prominent in the sounds of brass instruments.

Human ears tend to group harmonically-related frequency components into a single sensation. Rather than perceiving the individual harmonics of a musical tone, humans perceive them together as a tone color or timbre, and the overall pitch
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

is heard as the fundamental of the harmonic series being experienced. If a sound is heard that is made up of even just a few simultaneous tones, and if the intervals among those tones form part of a harmonic series, the brain tends to group this input into a sensation of the pitch of the fundamental of that series, even if the fundamental is not present
Missing fundamental
A sound is said to have a missing fundamental, suppressed fundamental, or phantom fundamental when its overtones suggest a fundamental frequency but the sound lacks a component at the fundamental frequency itself....

.

Variations in the frequency of harmonics can also affect the perceived fundamental pitch. These variations, most clearly documented in the piano
Piano
The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...

and other stringed instruments but also apparent in brass instrument
Brass instrument
A brass instrument is a musical instrument whose sound is produced by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips...

s, are caused by a combination of metal stiffness and the interaction of the vibrating air or string with the resonating body of the instrument. The complex splash of strong, high overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

s and metallic ringing sounds from a cymbal almost completely hides its fundamental tone.

## Interval strength

David Cope
David Cope
David Cope is an American author, composer, scientist, and professor emeritus of music at the University of California, Santa Cruz...

(1997) suggests the concept of interval strength, in which an interval's strength, consonance, or stability (see consonance and dissonance
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...

) is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law.

Thus, an equal tempered perfect fifth is stronger than an equal tempered minor third , since they approximate a just perfect fifth and just minor third , respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3.

• Inharmonicity
Inharmonicity
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency....

• Klang (music)
Klang (music)
In music, Klang is the, "chord of nature", so called because it is built from the first five partials of the overtone series...

• Otonality and Utonality
Otonality and Utonality
Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones or "undertones" of a given fixed tone. For example: 1/1, 2/1, 3/1,.....

• Piano acoustics
Piano acoustics
Piano acoustics are those physical properties of the piano which affect its acoustics.-String length and mass:The strings of a piano vary in thickness, and therefore in mass per length, with bass strings thicker than treble. A typical range is from 1/30 inch for the highest treble strings to 1/3...

• Scale of harmonics
Scale of harmonics
The 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 ...

• Stretched tuning
Stretched tuning
Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital electric pianos , and some sample-based synthesizers based on these instruments, to accommodate the natural inharmonicity of their vibrating elements...