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Harmonic series (music)

See harmonic series (mathematics) for the (related) mathematical concept.

Pitched musical instruments 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. These oscillations are called 'standing waves' as the wave in the string or air column oscillates to and fro but does not travel along it. Interaction with the surrounding air causes sound waves - travelling waves which allow us to hear the instrument. Because of the self-filtering nature of resonance, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest possible frequency, and such multiples form the harmonic series. This frequency determines the musical pitch or note that is created by vibration over the full length of the string or air column. The simplest case to visualise is a vibrating string, as in the illustration. Similar arguments apply to vibrating air columns in wind instruments. In most pitched musical instruments, the fundamental note (first harmonic) is accompanied by other, higher-frequency tones that are generally called overtones. These 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 (giving the fundamental tone) is twice the length of the string. Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times that of the fundamental. To better understand this, see node. Theoretically, these shorter 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 (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. But because our ears respond to sound logarithmically (frequency ratios, not differences, determine musical intervals), 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 20 harmonics are notated as shown. You can listen to if you have Media help 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 all partial waves ("partials") within a sound that are integer multiples of the fundamental frequency. In music, and especially among tuning professionals, the words "harmonic" and "partial" are generally interchangeable.

In digital signal processing, a "partial" frequency can also refer to a constituent frequency of a sound which might not be harmonically related to the actual harmonics contained in the original sound that is being examined. According to the Fourier theorem, tonally complex signals can be constructed by adding sinewaves of different frequencies, amplitudes and phases. In the case of a discrete time, discrete frequency Fourier transform it requires N/2+1 such partial frequencies to re-construct a given digital signal of N samples.

Likewise, many musicians use the term overtones as a synonym for harmonics, though not all overtones are necessarily harmonic: some are inharmonic or non-harmonic. That is, 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 tam-tam 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. Barbershop quartets use overtone colloquially in reference to the psychoacoustic phenomenon of close harmony. Scientists take the word harmonic very seriously and define it as integer multiples of the fundamental frequency, thereby separating the concept of harmonics from overtones. That is why the first harmonic is the fundamental frequency multiplied by one, and thus are the same frequency.

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 and consistency with the important notion of odd and even harmonics.

Harmonics and tuning

If the 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, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. 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 most of the first 31 harmonics and their closest frequencies in the 12-tone equal-tempered scale. 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.)

Note Variance cent Harmonic
C 0 1 2 4 8 16
C, D +5 17
D +4 9 18
D, E −2 19
E −14 5 10 20
F −29 21
F, G −49 11 22
+28 23
G +2 3 6 12 24
G, A −27 25
+41 13 26
A +6 27
A, B −31 7 14 28
+30 29
B −12 15 30
+45 31

The frequencies of the overtone series, being a range of integral multiples of the fundamental frequency, are naturally related to each other by small whole number ratios and it is these small whole number ratios that are 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 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. All the intervals succumb to similar analysis as been demonstrated by Paul Hindemith in his book, The Craft of Musical Composition.

Timbre of musical instruments

The relative amplitudes of the various harmonics primarily determine the timbre of different instruments and sounds, though formants also have a role. For example, the clarinet and saxophone have similar mouthpieces and reeds, and both produce sound through resonance of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, the even-numbered harmonics are suppressed, which produces a purer tone. The saxophone's resonator is conical, which allows the even-numbered harmonics to sound more strongly and thus produces a more complex tone. Of course, the differences in resonance between the wood of the clarinet and the brass of the saxophone also affect their tones. The inharmonic ringing of the instrument's metal resonator is even more prominent in the sounds of brass instruments.

Our ears tend to resolve 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 pitch as the fundamental of the harmonic series being experienced. If we hear a sound that is made up of even just a few simultaneous tones, and if the intervals among those tones form part of a harmonic series, our brains tend to resolve this input into a sensation of the pitch of the fundamental of that series, even if the fundamental is not sounding. This phenomenon is used to advantage in music recording, especially with low bass tones that will be reproduced on small speakers.

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. The complex splash of strong, high overtones and metallic ringing sounds from a cymbal almost completely hide its fundamental tone.

Interval strength

David Cope (1997) suggests the concept of interval strength, 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.

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