# Harmonic series (music)

Last updated

A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency .

## Contents

Pitched musical instruments are often based on an acoustic resonator such as a string or a column of air, which oscillates at numerous modes simultaneously. At the frequencies of each vibrating mode, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves. Interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the typical spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency, and such multiples form the harmonic series.

The musical pitch of a note is usually perceived as the lowest partial present (the fundamental frequency), which may be the one created by vibration over the full length of the string or air column, or a higher harmonic chosen by the player. The musical timbre 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 waves) or partials, each with its own frequency of vibration, amplitude, and phase". [1] (See also, Fourier analysis.)

A partial is any of the sine waves (or "simple tones", as Ellis calls them [2] 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 that are positive integer multiples of a common fundamental frequency. The fundamental is a harmonic 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. [3]

An inharmonic partial is any partial that does not match an ideal harmonic. Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial. [4]

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, 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, 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 instruments, such as marimba, vibraphone, tubular bells, timpani, 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 cymbals 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 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, 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. [5]

Some electronic instruments, such as synthesizers, can play a pure frequency with no overtones (a sine wave). 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. [6] [ failed verification ] Similar arguments apply to vibrating air columns in wind instruments (for example, "the French horn was originally a valveless instrument that could play only the notes of the harmonic series" [7] ), 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 bores, or end-openings that run the gamut 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 waves 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 (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. 12, 13, 14 times) that of the fundamental.

Theoretically, these shorter wavelengths correspond to vibrations 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 and stretched tuning 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 progression (f, 2f, 3f, 4f, 5f, ...). In terms of frequency (measured in cycles per second, or hertz, 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 series is a geometric progression (2f, 4f, 8f, 16f, ...), and people perceive these distances as "the same" in the sense of musical interval. In terms of what one hears, each octave 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 above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third harmonic (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher).

Marin Mersenne 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." [9] However, to quote Carl Dahlhaus, "the interval-distance of the natural-tone-row [ overtones] [...], 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." [10]

## Harmonics and tuning

If the harmonics are octave displaced and compressed into the span of one octave, some of them are approximated by the notes of what the West has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal semitones, 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 ranked musical intervals according to their relative dissonance based on these and similar harmonic relationships. [11]

Below is a comparison between the first 31 harmonics and the intervals of 12-tone equal temperament (12TET), octave displaced and compressed into the span of one octave. Tinted fields highlight differences greater than 5 cents (120 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).

HarmonicInterval as a ratioInterval in binary12TET intervalNoteVariance cents
1248161, 21prime (octave)C0
1717/16 (1.0625)1.0001minor secondC, D+5
9189/8 (1.125)1.001major secondD+4
1919/16 (1.1875)1.0011minor thirdD, E2
510205/4 (1.25)1.01major thirdE14
2121/16 (1.3125)1.0101fourthF29
112211/8 (1.375)1.011tritoneF, G49
2323/16 (1.4375)1.0111+28
3612243/2 (1.5)1.1fifthG+2
2525/16 (1.5625)1.1001minor sixthG, A27
132613/8 (1.625)1.101+41
2727/16 (1.6875)1.1011major sixthA+6
714287/4 (1.75)1.11minor seventhA, B31
2929/16 (1.8125)1.1101+30
153015/8 (1.875)1.111major seventhB12
3131/16 (1.9375)1.1111+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). 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 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 (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) below the lower (actual sounding) note of the tritone. All the intervals succumb to similar analysis as has been demonstrated by Paul Hindemith in his book The Craft of Musical Composition, although he rejected the use of harmonics from the seventh and beyond. [11]

The Mixolydian mode 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 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 amplitudes (strengths) of the various harmonics primarily determine the timbre of different instruments and sounds, though onset transients, formants, noises, and inharmonicities also play a role. For example, the clarinet and saxophone have similar mouthpieces and reeds, and both produce sound through resonance 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 instruments, 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, [12] in which an interval's strength, consonance, or stability (see consonance and dissonance) 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.

## Notes

1. William Forde Thompson (2008). Music, Thought, and Feeling: Understanding the Psychology of Music. p. 46. ISBN   978-0-19-537707-1.
2. Hermann von Helmholtz (1885). On the Sensations of Tone as a Physiological Basis for the Theory of Music. Translated by Alexander John Ellis (2nd ed.). Longmans, Green. p. 23.
3. John R. Pierce (2001). "Consonance and Scales". In Perry R. Cook (ed.). Music, Cognition, and Computerized Sound. MIT Press. ISBN   978-0-262-53190-0.
4. Martha Goodway and Jay Scott Odell (1987). The Historical Harpsichord Volume Two: The Metallurgy of 17th- and 18th- Century Music Wire. Pendragon Press. ISBN   978-0-918728-54-8.
5. Riemann 1896 , p. 143: "let it be understood, the second overtone is not the third tone of the series, but the second"
6. Roederer, Juan G. (1995). The Physics and Psychophysics of Music. p. 106. ISBN   0-387-94366-8.
7. Kostka, Stefan; Payne, Dorothy (1995). Tonal Harmony (3rd ed.). McGraw-Hill. p. 102. ISBN   0-07-035874-5.
8. Fonville, John (Summer 1991). "Ben Johnston's Extended Just Intonation: A guide for interpreters". Perspectives of New Music . 29 (2): 106–137 (121). doi:10.2307/833435. JSTOR   833435.
9. Cohen, H. F. (2013). Quantifying Music: The science of music at the first stage of scientific revolution 1580–1650. Springer. p. 103. ISBN   9789401576864.
10. Sabbagh, Peter (2003). The Development of Harmony in Scriabin's Works, p. 12. Universal. ISBN   9781581125955. Cites: Dahlhaus, Carl (1972). "Struktur und Expression bei Alexander Skrjabin", Musik des Ostens, Vol. 6, p. 229.
11. 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. ISBN   0901938300). Archived 2014-07-01 at the Wayback Machine .
12. Cope, David (1997). Techniques of the Contemporary Composer, p. 40–41. New York, New York: Schirmer Books. ISBN   0-02-864737-8.

Sources

## Related Research Articles

The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as f0, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f1, the first harmonic.

In music, there are two common meanings for tuning:

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 harmonic, the other 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. The set of harmonics forms a harmonic series.

An overtone is any resonant frequency above the fundamental frequency of a sound. In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental is the lowest pitch. While the fundamental is usually heard most prominently, overtones are actually present in any pitch except a true sine wave. The relative volume or amplitude of various overtone partials is one of the key identifying features of timbre, or the individual characteristic of a sound.

In music, timbre, also known as tone color or tone quality, is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and musical instruments. It also enables listeners to distinguish different instruments in the same category.

Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies. Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre.

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

The Bohlen–Pierce scale is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the equal-tempered diatonic scale.

Piano acoustics is the set of physical properties of the piano that affect its sound. It is an area of study within musical acoustics.

Piano tuning is the act of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the vibration interaction among notes, which is different for every piano, thus in practice requiring slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament.

Musical acoustics or music acoustics is a multidisciplinary field that combines knowledge from physics, psychophysics, organology, physiology, music theory, ethnomusicology, signal processing and instrument building, among other disciplines. As a branch of acoustics, it is concerned with researching and describing the physics of music – how sounds are employed to make music. Examples of areas of study are the function of musical instruments, the human voice, computer analysis of melody, and in the clinical use of music in music therapy.

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, unpleasantness, or unacceptability, although there is broad acknowledgement that this depends also on familiarity and musical expertise. The terms form a structural dichotomy in which they define each other by mutual exclusion: a consonance is what is not dissonant, and a dissonance is what is not consonant. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. In casual discourse, as German composer and music theorist Paul Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied". The term sonance has been proposed to encompass or refer indistinctly to the terms consonance and dissonance.

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. In stretched tuning, two notes an octave apart, whose fundamental frequencies theoretically have an exact 2:1 ratio, are tuned slightly farther apart. "For a stretched tuning the octave is greater than a factor of 2; for a compressed tuning the octave is smaller than a factor of 2."

In music, the septimal minor third, also called the subminor third or septimal 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. In 24-tone equal temperament five quarter tones approximate the septimal minor third at 250 cents. A septimal minor third is almost exactly two-ninths of an octave, and thus all divisions of the octave into multiples of nine have an almost perfect match to this interval. The septimal major sixth, 12/7, is the inverse of this interval.

In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the process of altering the size of an interval by making it narrower or wider than pure. "Any plan that describes the adjustments to the sizes of some or all of the twelve fifth intervals in the circle of fifths so that they accommodate pure octaves and produce certain sizes of major thirds is called a temperament." Temperament is especially important for keyboard instruments, which typically allow a player to play only the pitches assigned to the various keys, and lack any way to alter pitch of a note in performance. Historically, the use of just intonation, Pythagorean tuning and meantone temperament meant that such instruments could sound "in tune" in one key, or some keys, but would then have more dissonance in other keys.

In music, the undertone series or subharmonic series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones must be produced in unusual ways. While the overtone series is based upon arithmetic multiplication of frequencies, resulting in a harmonic series, the undertone series is based on arithmetic division.

Playing a string harmonic is a string instrument technique that uses the nodes of natural harmonics of a musical string to isolate overtones. Playing string harmonics produces high pitched tones, often compared in timbre to a whistle or flute. Overtones can be isolated "by lightly touching the string with the finger instead of pressing it down" against the fingerboard. For some instruments this is a fundamental technique, such as the Chinese guqin, where it is known as fan yin, and the Vietnamese đàn bầu.

Sympathetic resonance or sympathetic vibration is a harmonic phenomenon wherein a passive string or vibratory body responds to external vibrations to which it has a harmonic likeness. The classic example is demonstrated with two similarly-tuned tuning forks. When one fork is struck and held near the other, vibrations are induced in the unstruck fork, even though there is no physical contact between them. In similar fashion, strings will respond to the vibrations of a tuning fork when sufficient harmonic relations exist between them. The effect is most noticeable when the two bodies are tuned in unison or an octave apart, as there is the greatest similarity in vibrational frequency. Sympathetic resonance is an example of injection locking occurring between coupled oscillators, in this case coupled through vibrating air. In musical instruments, sympathetic resonance can produce both desirable and undesirable effects.

Dynamic tonality is a paradigm for tuning and timbre which generalizes the special relationship between just intonation and the harmonic series to apply to a wider set of pseudo-just tunings and related pseudo-harmonic timbres.

In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3:2, 1.5, and may be approximated by an equal tempered perfect fifth which is 27/12. If the A above middle C is 440 Hz, the perfect fifth above it would be E, at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.