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Pitched Musical Instrument s are usually based on a Harmonic Oscillator such as a string or a column of air. Both can and do oscillate at numerous frequencies simultaneously. Because of the self-filtering nature of Resonance , these frequencies are mostly limited to integer multiples of the lowest possible frequency, and such multiples form the harmonic series. DESCRIPTION OF THE HARMONIC SERIES The lowest possible frequency of a Harmonic Oscillator is called its Fundamental Frequency . This frequency determines the musical Pitch or note that is created by vibration over the full length of the string or air column. In nearly every musical instrument, the fundamental note is always accompanied by other, higher-frequency tones that are generally called overtones. In pitched (''i.e.,'' non-percussion) instruments, these shorter, faster Waves are reflected between the two ends of the string or air column. As the reflected waves interact, frequencies whose wavelengths do not divide evenly into the length of the string or air column are suppressed, and the vibrations that persist are called '''harmonics'''. Their wavelengths are 1, 1/2, 1/3, 1/4, 1/5, 1/6, etc. of the length of the string or air column. To better understand this, see Node . Theoretically, these wavelengths produce 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 (2×f, 3×f, 4×f, 5×f, ...). In terms of frequency (measured in cycles per second, or Hertz (Hz)), the difference between consecutive harmonics is therefore constant. But because our ears respond to sound Logarithmically , 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 all ranges. In terms of what we hear, each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals. The second harmonic, twice the frequency of the fundamental, sounds an Octave higher; the third harmonic, three times the frequency of the fundamental, sounds a Perfect Fifth above the second. The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a Perfect Fourth above the third (two octaves above the fundamental). Double the harmonic number means double the frequency (which sounds an octave higher). The combined oscillation of a string with several of its lowest harmonics can be seen clearly in an interactive animation at Edward Zobel's "Zona Land" . For a fundamental of C1, the first 16 harmonics are notated as shown. You can listen to if you have 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 . TERMINOLOGY ''Harmonic vs. partial.'' Harmonics are often called partials. In some contexts, "partial" may refer to an Overtone that is ''not'' an integer multiple of the fundamental frequency, but this can be confusing in wire-stringed instruments where, due to Inharmonicity , ''none'' of the harmonics vibrate at ''exact'' integer multiples of the fundamental. In music, and especially among tuning professionals, the words "harmonic" and "partial" are generally interchangeable. Likewise, many musicians use the term ''overtones'' as a synonym for harmonics. For others, an Overtone may be any frequency that sounds along with the fundamental tone, regardless of its relationship to the Fundamental Frequency . The sound of a cymbal or gong includes overtones that are ''not'' harmonics; that's why the gong's sound doesn't seem to have a very definite pitch compared to the same fundamental note played on a piano. ''Harmonic numbering.'' In most contexts, the fundamental vibration of an oscillating body represents its first harmonic. However, some musicians, tuners, and even developers of piano tuning software do not consider the fundamental to be a harmonic; it is just the fundamental. For them, the harmonic one octave above the fundamental (the second mode of vibration) is the first harmonic or first partial. There are logical arguments for both approaches to numbering, but in this article, the fundamental vibration is referred to as the first harmonic for simplicity. HARMONICS AND TUNING If the first 15 harmonics are Transposed into the span of one Octave , they approximate some of the notes in 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 , and in relation to that scale, many of the harmonics are slightly out of tune, and the 7th, 11th, and 13th harmonics are significantly so. In the late 1930s, composer Paul Hindemith ranked musical intervals according to their relative Dissonance based on these and similar harmonic relationships. Below is a comparison between the first 20 harmonics and their equivalent frequencies in the 12-tone equal-tempered scale. Orange-tinted fields highlight differences greater than 5 Cents , which is the " Just Noticeable Difference " for the human ear. (Because physical characteristics of musical instruments cause significant variations from these theoretical values, they should not be used for tuning without adjusting for those variations.)
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