Harmonic series (music)
A harmonic series (also overtone series) is the sequence of
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
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
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. 1⁄2, 1⁄3, 1⁄4 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
Below is a comparison between the first 31 harmonics and the intervals of
Harmonic | Interval as a ratio | Interval in binary | 12TET interval | Note | Variance cents | ||||
---|---|---|---|---|---|---|---|---|---|
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 (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
This section needs additional citations for verification. (November 2011) |
The relative
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.
See also
- Fourier series
- Klang (music)
- Otonality and Utonality
- Piano acoustics
- Scale of harmonics
- Undertone series
Notes
- ISBN 978-0-19-537707-1.
- ^ 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.
- ISBN 978-0-262-53190-0.
- ISBN 978-0-918728-54-8.
- ^ Riemann 1896, p. 143: "let it be understood, the second overtone is not the third tone of the series, but the second"
- ISBN 0-387-94366-8.
- ISBN 0-07-035874-5.
- JSTOR 833435.
- ISBN 9789401576864.
- ISBN 9781581125955. Cites: Dahlhaus, Carl(1972). "Struktur und Expression bei Alexander Skrjabin", Musik des Ostens, Vol. 6, p. 229.
- ^ .
- ISBN 0-02-864737-8.
Sources
- Riemann, Hugo (1896). Dictionary of Music. Translated by John South Shedlock. London: Augener & Co.
Further reading
- Coul, Manuel Op de. "List of intervals (Compiled)". Huygens-Fokker Foundation centre for microtonal music. Retrieved 2016-06-15.
- Datta, A. K.; Sengupta, R.; Dey, N.; Nag, D. (2006). Experimental Analysis of Shrutis from Performances in Hindustani Music. Kolkata, India: SRD ITC SRA. pp. I–X, 1–103. ISBN 81-903818-0-6. Archived from the originalon 2012-01-18.
- Helmholtz, H. (1865). Die Lehre von dem Tonempfindungen. Zweite ausgabe (in German). Braunschweig: Vieweg und Sohn. pp. I–XII, 1–606. Retrieved 2016-10-12. (see Sensations of Tone)
- IEV (1994). "Electropedia: The World's Online Electrotechnical Vocabulary". International Electrotechnical Commission. Retrieved 2016-06-15.
- Lamb, Horace (1911). . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 12 (11th ed.). Cambridge University Press. pp. 956, 958.
- ISBN 0-306-80106-X. Retrieved 2016-06-15.
- Schouten, J. F. (February 24, 1940). The residue, a new component in subjective sound analysis (PDF). Eindhoven, Holland: Natuurkundig Laboratorium der N. V. Philips' Gloeilampenfabrieken (communicated by Prof. G. Holst at the meeting). pp. 356–65. Retrieved 2016-09-26.
- Волконский, Андрей Михайлович (1998). Основы темперации (in Russian). Композитор, Москва. ISBN 5-85285-184-1. Retrieved 2016-06-15.
- Тюлин, Юрий Николаевич (1966). Беспалова, Н. (ed.). Учение о гармонии [The teaching on harmony] (in Russian) (Издание Третье, Исправленное и Дополненное = Third Edition, Revised and Enlarged ed.). Moscow: Музыка.