Harmonic series
From AudioLexic
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. 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.
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[edit] 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 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.
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 20 harmonics are notated as shown. 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.
[edit] Terminology
Harmonic vs. partial. Harmonics are all partial waves ("partials") within a sound that are an integer multiple 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 uses 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.
[edit] 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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Some theorists believe that the overtone series is the basis of the experience of consonance and dissonance, for example, one writer states that
"If two different notes are played simultaneously, the composite sound includes the harmonics of both notes. In musical intervals, one or more of those harmonics are common to both notes. For example, if C3 and G3 are sounding together, their harmonic series intersect at G4 (2nd of G3, 3rd of C3) and G5 (4th of G3, 6th of C3). If these common harmonics are at the same frequency or nearly so in both notes, the composite sound will seem harmonious. If their frequencies differ significantly, we tend to hear the notes as "out of tune" with each other. Tuning involves changing the fundamental pitch of one of the notes to control the relationship between these common harmonics"
However this theory neglects the fact that intervals made up of electronically generated pure sine waves (with no overtone content) are still perceived as consonant or dissonant. 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.
[edit] 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 reed (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. See stretched tuning, inharmonicity, and piano acoustics.
[edit] Register and special effects of musical instruments
In wind instruments, which produce sounds with a resonating air column, the lowest possible note is the fundamental resonance of the entire instrument. For a given length of resonator, only notes in the harmonic series of the resonator can be played clearly: higher notes are played by exciting higher harmonics, which is accomplished by changing the vibration at the reed or mouthpiece. Notes that are not in the harmonic series are played by changing the effective length of the resonator, usually by opening a tone hole in the side of the instrument, as in most woodwinds, or by increasing and decreasing the length of tubing, as in valved or slide brass instruments.
Many wind instruments are designed to allow higher harmonics to be played more easily by damping the normal fundamental resonance. For example, on most woodwind instruments (such as clarinet, saxophone and oboe), an octave key or register key opens a small hole high up on the resonator, prompting the instrument to oscillate at a higher harmonic partial and giving the player easier access to a higher octave of the instrument. Generally, flutists can access higher harmonics even without a register key simply by changing the pressure and angle of the air stream. This is also evident in 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 embouchure and blowing the air faster. A brass instrument of relatively short length with no valves (e.g. military bugle) plays only the lowest notes of the harmonic series, making it ideal for bugle calls. The six- to ten-foot length of the unvalved "natural trumpet", the predominant form of the trumpet from about 1500 to the early 1800s, not only allowed skilled players to play twice as many harmonics (the "Clarino register"), but also produced a fuller, richer tone due to the presence of those harmonics in lower notes.
The fundamental pitches of conical-bore brass instruments (such as flugelhorn or tuba) can be played, but they require significant skill to play with characteristic tone, and they are not often called for in literature.
The physics of cylindrical-bore brass instruments (such as the modern trumpet and trombone) does not actually provide a mode of vibration for the fundamental pitch. However, the higher harmonics help the lips establish a vibration at the fundamental frequency regardless. These "faked fundamentals" are known as pedal tones. Pedal tones have a distinct timbre and require some skill to control. They are called for occasionally in advanced literature, particularly in that of the trombone.
On a stringed instrument, it is possible to damp the fundamental and thus sound at a higher frequency by lightly touching the string at a harmonic node. For example, by touching the string lightly at its midpoint, the musician forces the string to vibrate in its second transverse mode, sounding the second harmonic. Such light-touch fingering can be applied to notes at 1/3, 1/4, etc. of the string length to produce higher harmonics. Simply pressing the string to the fingerboard at those positions would not yield the same note as the harmonic.
[edit] See also
- Harmony
- FM synthesis
- Additive synthesis
- Missing fundamental
- Third bridge
- Pedal tone
- Overtone singing
- Physics of music
- Mathematics of musical scales
- Normalizing the Musical Scale
- List of musical intervals
[edit] Print resources
- Helmholtz, H. von. (1877). Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik. 1877, 6th ed., Braunschweig: Vieweg, 1913; trans. by A.J. Ellis as On the sensations of tone as a physiological basis for the theory of music (1885). Reprinted New York: Dover, 1954. ISBN 0-486-60753-4
- Partch, Harry. Genesis of a Music, 2nd ed. Da capo press, 1974. ISBN 0-306-80106-X
[edit] External links
- The interaction of reflected waves on a string is illustrated in a simplified animation that can be found at Edward Zobel's "Zona Land".
- A Web-based Multimedia Approach to the Harmonic Series
- The Australian Bell website discusses how to tune the normally inharmonic overtones of bells to the harmonic series.
- [1] The importance of prime harmonics in music theory.
- The Discrete Fourier Transform explained (splitting a sound into its partial frequencies)
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