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Harmonic series ()

From Wikipedia, the free encyclopedia.

This article is about the series in . See harmonic series (mathematics) for the related mathematical concept.

Pitched musical instruments are usually based on some sort of , for example a string or a column of air, which can oscillate at a number of . The integer multiples of the lowest make up the harmonic series.

Contents

1 Description of the harmonic series 2 of musical instruments 3 Register and special effects of musical instruments 4 See also 5 Source 6 External links

Description of the harmonic series [edit]

The lowest of these frequencies is called the fundamental or first partial. This is the note created from the of the full string length (the first transverse vibrational mode) of a stringed instrument or from air compression vibrating through the whole length of a woodwind instrument (the first longitudinal vibrational mode). All of the other frequencies in the harmonic series are integer multiples of the fundamental. The difference in terms of frequency (measured in (Hz)) is the same between all partials, but the responds in a logarithmic fashion, so the higher partials 'closer' together. Put another way: since the harmonic series is an arithmetic series (1f, 2f, 3f, 4f...), and the , or octave series, is a geometric series (f, 2×f, 4×f, 8×f...), this causes the series to divide the octave into increasingly smaller parts as it ascends.

The partial is twice the frequency of the fundamental, which makes it an octave higher. The third harmonic partial, at three the frequency of the fundamental, is a above the second harmonic. Similarly, the fourth harmonic partial is four times the frequency of the fundamental; it is a above the third partial (two above the fundamental). Double the partial number means double the frequency, which in means the 'pitch' is an octave higher. For example, the 6th partial is an octave higher than the 3rd partial.

After that the come thick and fast, getting closer and closer together. Some harmonics correspond very nearly to named pitches of the equal tempered scale; others, for example the 7th harmonic, are significantly off from the equal tempered tones. If you have a player capable of reading Vorbis files (for example Winamp 3), you can listen to A'' (110 Hz) and 15 partials by clicking here. For a fundamental of C', the first 16 harmonics are:

An illustration of the harmonic series as musical notation. Not all the notes are exactly in tune; see text for more details.

1st C' 2nd C 3rd G 4th c 5th e 6th g 7th b-flat 8th c' 9th d' 10th e' 11th f'-sharp 12th g' 13th a' 14th b'-flat 15th b' natural 16th c'' 17th c-sharp''

The 7th, 11th, 13th, and 14th harmonics, and integer multiples thereof, are so out of tune with the values given above that they are not considered part of the normal twelve-tone scale used in Western music. They sound distressingly 'out of tune' when played in concert with a twelve-tone scale.

Since many instruments, and the fundamental intervals of Western , are often considered based on the harmonic series, many scale tuning systems (e.g. ) attempt to build the musical scale largely or entirely on the frequencies of the harmonic series. However, in modern , many notes in the harmonic series are off pitch as indicated above. In fact, all the notes except for the powers of 2 are slightly off pitch in equal temperament, but it is seldom noticeable. Conversely, many, such as Milton Babbitt, hold what Fred Maus (2004) calls a constructionist view, arguing that the essentialist view above is a "normative allegation...which then would label as 'un-natural' (in the , it would appear, of morally perverse) music which is not 'founded' on [the harmonic series]," and asking, "what music, in what sense, ever had been founded on it?" Timbre of musical instruments [edit]

Most instruments produce a number of frequencies in addition to the fundamental of the tone that is sounding. The and placement of different partials determine the timbre of different instruments. determine some of the character of the instrument, but the harmonic are probably the most important effect. For example, close ended instruments (e.g. the , which is considered closed at the reed end) sound only odd numbered harmonics, giving each individual note a purer timbre than a stringed or . It is the strength of higher harmonics in brass instruments that gives them their "brassy", rich, slightly dissonant timbre.

The placement of partials can also affect the perceived fundamental pitch. Not all musical instruments have partials that exactly match the harmonic partials as described here. The partials of , and other, strings are increasingly sharper than perfect harmonics because the strings are stiff, leading to nonlinear, inharmonic effects. See Piano . Register and special effects of musical instruments [edit]

Many instruments are designed to allow higher harmonics to be picked out while the normal fundamental, thus making the instrument sound higher. For example, on most woodwind instruments (clarinet, saxophone, oboe, bassoon, etc.), there is an octave or register key which opens a small hole in the tube, prompting the instrument to oscillate at a higher harmonic partial and giving a higher octave of the instrument. Generally, flautists can access higher harmonics even without a register key simply by blowing with more and thereby forcing the air column to prefer the second vibrational mode; this is also evident when blowing over the lips of bottles.

On brass instruments, the small number of keys only allows a small chromatic range to be played off of any given harmonic, so it is necessary for the musician to play many harmonics to get the full range of the instrument. The different harmonics are accessed by increasing the vibration of the lips against the mouthpiece, essentially by tightening the and blowing the air faster. A brass instrument with no valves (e.g. military bugle) plays only the notes of the harmonic series, making it ideal for bugle calls and little else.

For cylindrical bore brass (e.g. , ), the second harmonic is the lowest playable note. The fundamental is technically playable on a trumpet or trombone, but not in context, as it is extremely difficult to play. On a conical bore brass instrument (e.g. flugelhorn, horn, ) the fundamental is available, but is a somewhat special note called a "" or "pedal note" and is rarely called for in written music. This is probably because the valving system of a brass instrument usually only allows the lowering of the pitch to a below the open sounding pitch, which means that there are five notes above the fundamental that cannot be played.

On a stringed instrument, it is possible to damp the fundamental and thus sound at a higher frequency by using a special fingering technique. By lightly touching the string directly at its midpoint, the musician the string to vibrate in its second transverse mode, sounding an octave above the normal note. This is not so useful as the same note could be sounded by pushing the string all the way to the at this point. However, the touch fingering can be applied at 1/3, 1/4, etc. of the string length to access higher and higher harmonics (the practical limit for this depends on the total length of the string, thus on the size of the instrument). Simply pressing the string to the fingerboard at these positions would not yield the same note as the harmonic. See also [edit]

Harmony Dissonance FM synthesis Overtone Throat of music Mathematics of musical scales Source [edit]

Maus, Fred Everett (2004). "Sexual and Musical Categories", The Pleasure of Modernist Music, p.158. ISBN 1580461433. External links [edit]

The Overtone/Harmonic Series: A path to intervals, scales, and timbre by Reginald Bain A Web-based Multimedia Approach to the Harmonic Series Bells worth listening to: A composer and a sculptor from Melbourne have developed the world's first harmonic bells

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This page was last modified 03:54, 4 August 2005. All text is available under the terms of the GNU Free Documentation License (see Copyrights for details). About Wikipedia Disclaimers