Harmonic series (music)
   HOME

TheInfoList



OR:

The harmonic series (also overtone series) is the sequence of
harmonics In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st harm ...
,
musical tone Traditionally in Western classical music, Western music, a musical tone is a steady periodic function, periodic sound. A musical tone is characterized by its duration (music), duration, pitch (music), pitch, amplitude, intensity (or loudness), an ...
s, or
pure tone In psychoacoustics, a pure tone is a sound with a sinusoidal waveform; that is, a sine wave of constant frequency, phase-shift, and amplitude. By extension, in signal processing a single-frequency tone or pure tone is a purely sinusoidal signal ...
s whose
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
is an
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
multiple of a ''
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
''. Pitched
musical instrument A musical instrument is a device created or adapted to make Music, musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. A person ...
s are often based on an acoustic
resonator A resonator is a device or system that exhibits resonance or resonant behavior. That is, it naturally oscillates with greater amplitude at some frequencies, called resonant frequencies, than at other frequencies. The oscillations in a reso ...
such as a
string String or strings may refer to: *String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian anim ...
or a column of air, which
oscillates Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
at numerous modes simultaneously. As waves travel in both directions along the string or air column, they reinforce and cancel one another to form
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect t ...
s. Interaction with the surrounding air produces audible sound waves, which travel away from the instrument. These frequencies are generally integer multiples, or
harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
s, of the fundamental and such multiples form the harmonic series. The fundamental, which is usually perceived as the lowest
partial Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...
present, is generally perceived as the pitch of a musical tone. The musical
timbre In music, timbre (), also known as tone color or tone quality (from psychoacoustics), is the perceived sound of a musical note, sound or tone. Timbre distinguishes sounds according to their source, such as choir voices and musical instrument ...
of a steady tone from such an instrument is strongly affected by the relative strength of each harmonic.


Terminology


Partial, harmonic, fundamental, inharmonicity, and overtone

A "complex tone" (the sound of a note with a timbre particular to the instrument playing the note) "can be described as a combination of many simple periodic waves (i.e.,
sine wave A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic function, periodic wave whose waveform (shape) is the trigonometric function, trigonometric sine, sine function. In mechanics, as a linear motion over time, this is ''simple ...
s) or ''partials,'' each with its own frequency of
vibration Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...
,
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, and phase". (See also,
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
.) A partial is any of the
sine wave A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic function, periodic wave whose waveform (shape) is the trigonometric function, trigonometric sine, sine function. In mechanics, as a linear motion over time, this is ''simple ...
s (or "simple tones", as Ellis calls them when translating Helmholtz) of which a complex tone is composed, not necessarily with an integer multiple of the lowest harmonic. A harmonic is any member of the harmonic series, an ideal set of
frequencies Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
that are
positive integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
multiples of a common
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'' (abbreviated as 0 or 1 ), is defined as the lowest frequency of a Periodic signal, periodic waveform. In music, the fundamental is the musical pitch (music), pitch of a n ...
. The fundamental is a
harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
because it is one times itself. A harmonic partial is any real partial component of a complex tone that matches (or nearly matches) an ideal harmonic. An inharmonic partial is any partial that does not match an ideal harmonic. ''
Inharmonicity In music, inharmonicity is the degree to which the frequency, frequencies of overtones (also known as Harmonic series (music)#Partial, partials or partial tones) depart from Integer, whole multiples of the fundamental frequency (harmonic seri ...
'' is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial. Many pitched acoustic instruments are designed to have partials that are close to being whole-number ratios with very low inharmonicity; therefore, in
music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
, and in instrument design, it is convenient, although not strictly accurate, to speak of the partials in those instruments' sounds as "harmonics", even though they may have some degree of inharmonicity. The
piano A piano is a keyboard instrument that produces sound when its keys are depressed, activating an Action (music), action mechanism where hammers strike String (music), strings. Modern pianos have a row of 88 black and white keys, tuned to a c ...
, one of the most important instruments of western tradition, contains a certain degree of inharmonicity among the frequencies generated by each string. Other pitched instruments, especially certain
percussion A percussion instrument is a musical instrument that is sounded by being struck or scraped by a percussion mallet, beater including attached or enclosed beaters or Rattle (percussion beater), rattles struck, scraped or rubbed by hand or ...
instruments, such as
marimba The marimba ( ) is a musical instrument in the percussion family that consists of wooden bars that are struck by mallets. Below each bar is a resonator pipe that amplifies particular harmonics of its sound. Compared to the xylophone, the mari ...
,
vibraphone The vibraphone (also called the vibraharp) is a percussion instrument in the metallophone family. It consists of tuned metal bars and is typically played by using Percussion mallet, mallets to strike the bars. A person who plays the vibraphone ...
, tubular bells,
timpani Timpani (; ) or kettledrums (also informally called timps) are musical instruments in the percussion instrument, percussion family. A type of drum categorised as a hemispherical drum, they consist of a Membranophone, membrane called a drumhead, ...
, and singing bowls contain mostly inharmonic partials, yet may give the ear a good sense of pitch because of a few strong partials that resemble harmonics. Unpitched, or indefinite-pitched instruments, such as
cymbal A cymbal is a common percussion instrument. Often used in pairs, cymbals consist of thin, normally round plates of various alloys. The majority of cymbals are of indefinite pitch, although small disc-shaped cymbals based on ancient designs sou ...
s and tam-tams make sounds (produce spectra) that are rich in inharmonic partials and may give no impression of implying any particular pitch. An
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
is any partial above the lowest partial. The term overtone does not imply harmonicity or inharmonicity and has no other special meaning other than to exclude the fundamental. It is mostly the relative strength of the different overtones that give an instrument its particular
timbre In music, timbre (), also known as tone color or tone quality (from psychoacoustics), is the perceived sound of a musical note, sound or tone. Timbre distinguishes sounds according to their source, such as choir voices and musical instrument ...
, tone color, or character. When writing or speaking of overtones and partials numerically, care must be taken to designate each correctly to avoid any confusion of one for the other, so the second overtone may not be the third partial, because it is the second sound in a series. Some electronic instruments, such as
synthesizer A synthesizer (also synthesiser or synth) is an electronic musical instrument that generates audio signals. Synthesizers typically create sounds by generating waveforms through methods including subtractive synthesis, additive synthesis a ...
s, can play a pure frequency with no
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s (a
sine wave A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic function, periodic wave whose waveform (shape) is the trigonometric function, trigonometric sine, sine function. In mechanics, as a linear motion over time, this is ''simple ...
). Synthesizers can also combine pure frequencies into more complex tones, such as to simulate other instruments. Certain flutes and ocarinas are very nearly without overtones.


Frequencies, wavelengths, and musical intervals in example systems

One of the simplest cases to visualise is a vibrating string, as in the illustration; the string has fixed points at each end, and each harmonic mode divides it into an integer number (1, 2, 3, 4, etc.) of equal-sized sections resonating at increasingly higher frequencies. Similar arguments apply to vibrating air columns in
wind instrument A wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitch ...
s (for example, "the French horn was originally a valveless instrument that could play only the notes of the harmonic series"), 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), conical as opposed to
cylindrical A cylinder () has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a Prism (geometry), prism with a circle as its base. A cylinder may ...
bores, or end-openings that run the
gamut In color reproduction and colorimetry, a gamut, or color gamut , is a convex set containing the colors that can be accurately represented, i.e. reproduced by an output device (e.g. printer or display) or measured by an input device (e.g. cam ...
from no flare, cone flare, or exponentially shaped flares (such as in various bells). In most pitched musical instruments, the fundamental (first harmonic) is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency
wave In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium ...
s occur with varying prominence and give each instrument its characteristic
tone quality In music, timbre (), also known as tone color or tone quality (from psychoacoustics), is the perceived sound of a musical note, sound or tone. Timbre distinguishes sounds according to their source, such as choir voices and musical instruments ...
. The fact that a string is fixed at each end means that the longest allowed wavelength on the string (which gives 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 reciprocal multiples (e.g. , , times) that of the fundamental. Theoretically, these shorter wavelengths correspond to
vibration Vibration () is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the os ...
s at frequencies that are integer multiples of (e.g. 2, 3, 4 times) the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies. (See
inharmonicity In music, inharmonicity is the degree to which the frequency, frequencies of overtones (also known as Harmonic series (music)#Partial, partials or partial tones) depart from Integer, whole multiples of the fundamental frequency (harmonic seri ...
and stretched tuning for alterations specific to wire-stringed instruments and certain
electric piano An electric piano is a musical instrument that has a piano-style musical keyboard, where sound is produced by means of mechanical hammers striking metal strings or reeds or wire tines, which leads to vibrations which are then converted into ele ...
s.) 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 progression An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
(''f'', 2''f'', 3''f'', 4''f'', 5''f'', ...). In terms of frequency (measured in cycles per second, or
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or Cycle per second, cycle) per second. The hertz is an SI derived unit whose formal expression in ter ...
, where ''f'' is the fundamental frequency), the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because human ears respond to sound nonlinearly, higher harmonics are perceived as "closer together" than lower ones. On the other hand, the
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
series is a
geometric progression A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the ''common ratio''. For example, the s ...
(2''f'', 4''f'', 8''f'', 16''f'', ...), and people perceive these distances as " the same" in the sense of musical interval. In terms of what one hears, each successively higher
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
in the harmonic series is divided into increasingly "smaller" and more numerous intervals. The second harmonic, whose frequency is twice the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a
perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval f ...
above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a
perfect fourth A fourth is a interval (music), musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending int ...
above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher).
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
wrote: "The order of the Consonances is natural, and ... the way we count them, starting from unity up to the number six and beyond is founded in nature." However, to quote
Carl Dahlhaus Carl Dahlhaus (10 June 1928 – 13 March 1989) was a German musicologist who was among the leading postwar musicologists of the mid to late 20th-century. #Selected bibliography, A prolific scholar, he had broad interests though his research foc ...
, "the interval-distance of the natural-tone-row [
overtone An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
s] [...], counting up to 20, includes everything from the octave to the quarter tone, (and) useful and useless musical tones. The natural-tone-row [harmonic series] justifies everything, that means, nothing."Sabbagh, Peter (2003). ''The Development of Harmony in Scriabin's Works'', p. 12. Universal. . Cites: Dahlhaus, Carl (1972). "Struktur und Expression bei Alexander Skrjabin", ''Musik des Ostens'', Vol. 6, p. 229.


Harmonics and tuning

If the harmonics are octave displaced and compressed into the span of one
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
, some of them are approximated by the notes of what the
West West is one of the four cardinal directions or points of the compass. It is the opposite direction from east and is the direction in which the Sun sets on the Earth. Etymology The word "west" is a Germanic word passed into some Romance langu ...
has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal
semitone A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...
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 ( ; ; 16 November 189528 December 1963) was a German and American composer, music theorist, teacher, violist and conductor. He founded the Amar Quartet in 1921, touring extensively in Europe. As a composer, he became a major advo ...
ranked musical intervals according to their relative dissonance based on these and similar harmonic relationships. Hindemith, Paul (1942)
''The Craft of Musical Composition: Book 1 – Theoretical Part''
pp. 15ff. Translated by Arthur Mendel (London: Schott & Co; New York: Associated Music Publishers. ). .
Below is a comparison between the first 31 harmonics and the intervals of
12-tone equal temperament 12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are Equal temperament, equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the Twelfth root of two, 12th root of 2 ...
(12TET), octave displaced and compressed into the span of one octave. Tinted fields highlight differences greater than 5 cents ( of a semitone), which is the human ear's " just noticeable difference" for notes played one after the other (smaller differences are noticeable with notes played simultaneously). 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 In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...
). 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 A combination tone (also called resultant tone or subjective tone)Combination Tone
, ''Britanni ...
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 all further nth-order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When one contrasts this with a dissonant interval such as a
tritone In music theory, the tritone is defined as a interval (music), musical interval spanning three adjacent Major second, whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be ...
(not tempered) with a frequency ratio of 7:5 one gets, 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 four notes: 100 Hz (and its octaves), 300 Hz, 500 Hz and 700 Hz. The lowest combination tone (100 Hz) is a seventeenth (two octaves and a
major third In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...
) below the lower (actual sounding) note of the
tritone In music theory, the tritone is defined as a interval (music), musical interval spanning three adjacent Major second, whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be ...
. All the intervals succumb to similar analysis as has been demonstrated by
Paul Hindemith Paul Hindemith ( ; ; 16 November 189528 December 1963) was a German and American composer, music theorist, teacher, violist and conductor. He founded the Amar Quartet in 1921, touring extensively in Europe. As a composer, he became a major advo ...
in his book ''The Craft of Musical Composition'', although he rejected the use of harmonics from the seventh and beyond. The
Mixolydian mode Mixolydian mode may refer to one of three things: the name applied to one of the ancient Greek ''harmoniai'' or ''tonoi'', based on a particular octave species or scale; one of the medieval church modes; or a modern musical mode or diatonic s ...
is consonant with the first 10 harmonics of the harmonic series (the 11th harmonic, a tritone, is not in the Mixolydian mode). The
Ionian mode The Ionian mode is a Mode (music), musical mode or, in modern usage, a diatonic scale also called the major scale. It is named after the Ionians, Ionian Greeks. It is the name assigned by Heinrich Glarean in 1547 to his new Gregorian mode#Authent ...
is consonant with only the first 6 harmonics of the series (the seventh harmonic, a minor seventh, is not in the Ionian mode). The Rishabhapriya ragam is consonant with the first 14 harmonics of the series.


Timbre of musical instruments

The relative
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
s (strengths) of the various harmonics primarily determine the
timbre In music, timbre (), also known as tone color or tone quality (from psychoacoustics), is the perceived sound of a musical note, sound or tone. Timbre distinguishes sounds according to their source, such as choir voices and musical instrument ...
of different instruments and sounds, though onset transients,
formant In speech science and phonetics, a formant is the broad spectral maximum that results from an acoustic resonance of the human vocal tract. In acoustics, a formant is usually defined as a broad peak, or local maximum, in the spectrum. For harmo ...
s,
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
s, and inharmonicities also play a role. For example, the
clarinet The clarinet is a Single-reed instrument, single-reed musical instrument in the woodwind family, with a nearly cylindrical bore (wind instruments), bore and a flared bell. Clarinets comprise a Family (musical instruments), family of instrume ...
and
saxophone The saxophone (often referred to colloquially as the sax) is a type of single-reed woodwind instrument with a conical body, usually made of brass. As with all single-reed instruments, sound is produced when a reed on a mouthpiece vibrates to p ...
have similar mouthpieces and reeds, and both produce sound through
resonance Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...
of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, the ''even''-numbered harmonics are less present. 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 ringing of the instrument's metal resonator is even more prominent in the sounds of brass instruments. Human ears tend to group phase-coherent, harmonically-related frequency components into a single sensation. Rather than perceiving the individual partials–harmonic and inharmonic, of a musical tone, humans perceive them together as a tone color or timbre, and the overall pitch 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 sine 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. Variations in the frequency of harmonics can also affect the ''perceived'' fundamental pitch. These variations, most clearly documented in the piano and other stringed instruments but also apparent in
brass instrument A brass instrument is a musical instrument that produces sound by Sympathetic resonance, sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips. The term ''labrosone'', from Latin elements meani ...
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.


Interval strength

David Cope (1997) suggests the concept of interval strength, Cope, David (1997). ''Techniques of the Contemporary Composer'', p. 40–41. New York, New York: Schirmer Books. . in which an interval's strength, consonance, or stability (see
consonance and dissonance In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unple ...
) is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also:
Lipps–Meyer law The Lipps–Meyer law, named for Theodor Lipps (1851–1914) and Max Friedrich Meyer (1873–1967), hypothesizes that the closure of melodic intervals is determined by "whether or not the end tone of the interval can be represented by the number ...
. Thus, an equal-tempered perfect fifth () is stronger than an equal-tempered
minor third In music theory, a minor third is a interval (music), musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval (music)#Number, interval numb ...
(), 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.


See also

*
Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
*
Klang (music) In music, ''klang'', or clang, is a term sometimes used to translate the German ''Klang'', a highly polysemic word. Technically, the term denotes any periodic sound, especially as opposed to simple periodic sounds (sine tones). In the German lay ...
* Otonality and Utonality * Piano acoustics * Scale of harmonics * Undertone series


Notes

Sources *


Further reading

* * * (see '' Sensations of Tone'') * * * * * * {{Strings (music)