Theharmonic series (alsoovertone series) is the sequence ofharmonics,musical tones, orpure tones whosefrequency is aninteger multiple of afundamental frequency.
Pitchedmusical instruments are often based on anacousticresonator such as astring or a column of air, whichoscillates at numerousmodes simultaneously. As waves travel in both directions along the string or air column, they reinforce and cancel one another to formstanding waves. Interaction with the surrounding air produces audiblesound waves, which travel away from the instrument. These frequencies are generally integer multiples, orharmonics, of thefundamental and such multiples form theharmonic series.
The fundamental, which is usually perceived as the lowestpartial present, is generally perceived as thepitch of a musical tone. The musicaltimbre of a steady tone from such an instrument is strongly affected by the relative strength of each harmonic.
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 simpleperiodic waves (i.e.,sine waves) orpartials, each with its own frequency ofvibration,amplitude, andphase".[1] (See also,Fourier analysis.)
Apartial is any of thesine waves (or "simple tones", asEllis calls them[2] when translatingHelmholtz) of which a complex tone is composed, not necessarily with an integer multiple of the lowest harmonic.
Aharmonic is any member of the harmonic series, an ideal set offrequencies that arepositive integer multiples of a commonfundamental frequency. Thefundamental is aharmonic because it is one times itself. Aharmonic partial is any real partial component of a complex tone that matches (or nearly matches) an ideal harmonic.[3]
Aninharmonic 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 incents for each partial.[4]
Manypitchedacoustic instruments are designed to have partials that are close to being whole-number ratios with very low inharmonicity; therefore, inmusic 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. Thepiano, 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 certainpercussion instruments, such asmarimba,vibraphone,tubular bells,timpani, andsinging 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 ascymbals andtam-tams make sounds (produce spectra) that are rich in inharmonic partials and may give no impression of implying any particular pitch.
Anovertone 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 particulartimbre, 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]
Someelectronic instruments, such assynthesizers, can play a pure frequency with noovertones (asine 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.
One of the simplest cases to visualise is avibrating string, as in the illustration; the string has fixed points at each end, and each harmonicmode 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 inwind instruments (for example, "the Frenchhorn 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 tocylindricalbores, or end-openings that run thegamut 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-frequencywaves occur with varying prominence and give each instrument its characteristictone 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.1⁄2,1⁄3,1⁄4 times) that of the fundamental.
Theoretically, these shorter wavelengths correspond tovibrations 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. (Seeinharmonicity andstretched tuning for alterations specific to wire-stringed instruments and certainelectric 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 anarithmetic progression (f, 2f, 3f, 4f, 5f, ...). In terms of frequency (measured incycles per second, orhertz, wheref is the fundamental frequency), the difference between consecutive harmonics is therefore constant and equal to the fundamental. But because human ears respond to soundnonlinearly, higher harmonics are perceived as "closer together" than lower ones. On the other hand, theoctave series is ageometric progression (2f, 4f, 8f, 16f, ...), and people perceive these distances as "the same" in the sense ofmusical interval. In terms of what one hears, each successively higheroctave 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 aperfect fifth above the second harmonic. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds aperfect 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 quoteCarl 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]
If the harmonics are octave displaced and compressed into the span of oneoctave, some of them are approximated by the notes of what theWest has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equalsemitones, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. In the late 1930s, composerPaul Hindemith ranked musical intervals according to their relativedissonance based on these and similar harmonic relationships.[11]
The table below compares the frequencies of the first 31 harmonics to their nearest equivalents in12-tone equal temperament(12TET), normalized to one octave. Tinted fields highlight differences greater than 5cents (1⁄20 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).
| Harmonic | Interval as a ratio | Interval in binary | 12TET interval | Note | Variancecents | ||||
|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 4 | 8 | 16 | 1, 2 | 1 | prime (octave) | C | 0 |
| 17 | 17/16 (1.0625) | 1.0001 | minor second | C♯, D♭ | +5 | ||||
| 9 | 18 | 9/8 (1.125) | 1.001 | major second | D | +4 | |||
| 19 | 19/16 (1.1875) | 1.0011 | minor third | D♯, E♭ | −2 | ||||
| 5 | 10 | 20 | 5/4 (1.25) | 1.01 | major third | E | −14 | ||
| 21 | 21/16 (1.3125) | 1.0101 | fourth | F | −29 | ||||
| 11 | 22 | 11/8 (1.375) | 1.011 | tritone | F♯, G♭ | −49 | |||
| 23 | 23/16 (1.4375) | 1.0111 | +28 | ||||||
| 3 | 6 | 12 | 24 | 3/2 (1.5) | 1.1 | fifth | G | +2 | |
| 25 | 25/16 (1.5625) | 1.1001 | minor sixth | G♯, A♭ | −27 | ||||
| 13 | 26 | 13/8 (1.625) | 1.101 | +41 | |||||
| 27 | 27/16 (1.6875) | 1.1011 | major sixth | A | +6 | ||||
| 7 | 14 | 28 | 7/4 (1.75) | 1.11 | minor seventh | A♯, B♭ | −31 | ||
| 29 | 29/16 (1.8125) | 1.1101 | +30 | ||||||
| 15 | 30 | 15/8 (1.875) | 1.111 | major seventh | B | −12 | |||
| 31 | 31/16 (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 (seejust 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 acombination 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 atritone (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 amajor third) below the lower (actual sounding) note of thetritone. All the intervals succumb to similar analysis as has been demonstrated byPaul Hindemith in his bookThe Craft of Musical Composition, although he rejected the use of harmonics from the seventh and beyond.[11]
TheMixolydian mode is consonant with the first 10 harmonics of the harmonic series (the 11th harmonic, a tritone, is not in the Mixolydian mode). TheIonian mode is consonant with only the first 6 harmonics of the series (the seventh harmonic, a minor seventh, is not in the Ionian mode). TheRishabhapriya ragam is consonant with the first 14 harmonics of the series.
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The relativeamplitudes (strengths) of the various harmonics primarily determine thetimbre of different instruments and sounds, though onsettransients,formants,noises, and inharmonicities also play a role. For example, theclarinet andsaxophone have similarmouthpieces andreeds, and both produce sound throughresonance of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, theeven-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. Theinharmonic 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 overallpitch 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 theperceived fundamental pitch. These variations, most clearly documented in the piano and other stringed instruments but also apparent inbrass 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.
David Cope (1997) suggests the concept ofinterval strength,[12] in which an interval's strength, consonance, or stability (seeconsonance 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 (playⓘ) is stronger than an equal-temperedminor third (playⓘ), since they approximate a just perfect fifth (playⓘ) and just minor third (playⓘ), respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3.
Sources
| Acoustical engineering | ||
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| Psychoacoustics | ||
| Audio frequency andpitch | ||
| Acousticians | ||
| Related topics | ||
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| Perfect Consonances | ||
| Imperfect Consonances | ||
| Dissonances | ||
