Dictionary## Harmonic series |

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*See harmonic series (mathematics) for the related mathematical concept.*
Pitched musical instruments are often based on an approximate harmonic oscillator 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 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 The musical
## Terminology
## Partial, harmonic, fundamental, inharmonicity, and overtoneAny complex tone "can be described as a combination of many simple periodic waves (i.e., sine waves) or A A
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, vibraphone, tubular bells, and timpani, 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 Some electronic instruments, such as theremins and synthesizers, 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 systemsThe simplest case to visualise is a vibrating string, as in the illustration; the string has fixed points ("nodes") at each end, and each harmonic mode divides it into 2, 3, 4, etc., equal-sized sections resonating at increasingly higher frequencies. 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 (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 against which it vibrates 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 series (1×f, 2×f, 3×f, 4×f, 5×f, ...). In terms of frequency (measured in cycles per second, or hertz (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 series 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 higher; the third harmonic, three times the frequency of the fundamental, sounds a For a fundamental of C1, the first 20 harmonics are notated as shown below. You can listen to A2 (110 Hz) and 15 of its partials if you have a media player capable of playing Vorbis files. You can also hear a sweep of the first 20 harmonics of A1 (55 Hz) in QuickTime format by clicking here. ## Harmonics and tuningIf the harmonics are 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 (1/20th 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. For example, a perfect fifth, say 200 and 300 Hz (cycles per second), produces 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, 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. It will be noted 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 ## Timbre of musical instrumentsThe relative amplitudes (strengths) of the various harmonics primarily determine the Our ears tend to group harmonically-related frequency components into a single sensation. Rather than perceiving the individual harmonics of a musical tone, we perceive them together as a tone color or timbre, and we hear the overall Variations in the frequency of harmonics can also affect the ## Interval strengthDavid Cope (1997) suggests the concept of interval strength, Thus, an equal tempered perfect fifth ( play (help·info)) is stronger than an equal tempered minor third( play (help·info)), since they approximate a just perfect fifth ( play (help·info)) and just minor third ( play (help·info)), respectively. The just minor third appears between harmonics 5 and 6 while the just fifth appears lower, between harmonics 2 and 3. ## See also- Inharmonicity
- Klang (music)
- Otonality and Utonality
- Piano acoustics
- Scale of harmonics
- Stretched tuning
## References**^**William Forde Thompson (2008).*Music, Thought, and Feeling: Understanding the Psychology of Music*. p. 46. ISBN 9780195377071. http://www.oup.com/us/catalog/general/subject/Psychology/CognitivePsychology/?view=usa&ci=9780195377071.**^**John R. Pierce (2001). "Consonance and Scales". in Perry R. Cook.*Music, Cognition, and Computerized Sound*. MIT Press. ISBN 9780262531900. http://books.google.com/books?id=L04W8ADtpQ4C&pg=PA169&dq=musical+tone+harmonic+partial+fundamental+integer&lr=&as_brr=3&as_pt=ALLTYPES.**^**Martha Goodway and Jay Scott Odell (1987).*The Historical Harpsichord Volume Two: The Metallurgy of 17th- and 18th- Century Music Wire*. Pendragon Press. ISBN 9780918728548. http://books.google.com/books?id=sE1mk8ed1dkC&pg=PA93&dq=inharmonicity+defined+partial+frequencies&lr=&as_brr=3&as_pt=ALLTYPES.**^**Juan G. Roederer (1995).*The Physics and Psychophysics of Music*. p. 106. ISBN 0387943668.**^**Cope, David (1997).*Techniques of the Contemporary Composer*, p.40–41. New York, New York: Schirmer Books. ISBN 0-02-864737-8.
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Harmonic series". Allthough most Wikipedia articles provide accurate information accuracy can not be guaranteed. |

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