Terminology [edit] Partial, harmonic, fundamental, inharmonicity, and overtone[edit] Any complex tone can be described as a combination of many simple periodic waves (i.e., sine waves) or partials, each with its own frequency of vibration, amplitude, and phase.[1] (Fourier analysis) A partial is any of the sine waves by which a complex tone is described. A harmonic (or a harmonic partial) is any of a set of partials that are whole number multiples of a common fundamental frequency.[2] This set includes the fundamental, which is a whole number multiple of itself (1 times itself). Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial.[3] Typical pitched instruments are designed to have partials that are close to being whole-number ratios, 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 mostly inharmonic partials, yet may give the ear a good sense of pitch. Unpitched, or indefinite-pitched instruments, such as cymbals, gongs, or tam-tams make sounds (produce spectra) rich in inharmonic partials. An overtone is any partial except the lowest. Overtone does not imply harmonicity or inharmonicity and has no other special meaning other than to exclude the fundamental. This can lead to numbering confusion when comparing overtones to partials; the first overtone is the second partial. 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 systems[edit] The simplest case to visualise is a vibrating string, as in the illustration; the string has fixed points at each end, and each harmonic mode divides it into 1, 2, 3, 4, etc., equal-sized sections resonating at increasingly higher frequencies.[4] Similar arguments apply to vibrating air columns in wind instruments, although these are complicated by having the possibility of anti-nodes (that is, the air column is closed at one end and open at the other), conical as opposed to cylindrical bores, or end-openings that run the gamut from no flare (bell), cone flare (bell), or exponentially shaped flares (bells). 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 it vibrates against 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 human ears respond to sound nonlinearly, higher harmonics are perceived 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 people hear these distances as the same in the sense of musical interval. In terms of what one hears, each octave in the harmonic series is divided into increasingly smaller and more numerous intervals. The second harmonic, whose frequency is twice 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). An illustration of the harmonic series in musical notation. The numbers above the harmonic indicate the number of cents difference from equal temperament (rounded to the nearest cent). Blue notes are flat and red notes are sharp. Harmonic series as musical notation with intervals between harmonics labeled. Blue notes differ most significantly from equal temperament. You can listen to A2 (110 Hz) and 15 of its partials Staff notation of partials 1, 3, 5, 7, 11, 13, 17, and 19 on C.[5] About this sound Play (help·info) Harmonics and tuning[edit] 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 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 ears just noticeable difference for notes played one after the other (smaller differences are noticeable with notes played simultaneously). Harmonic 12tET Interval Note Variance cents 1 2 4 8 16 prime (octave) C 0 17 minor second C♯, D♭ +5 9 18 major second D +4 19 minor third D♯, E♭ −2 5 10 20 major third E −14 21 fourth F −29 11 22 tritone F♯, G♭ −49 23 +28 3 6 12 24 fifth G +2 25 minor sixth G♯, A♭ −27 13 26 +41 27 major sixth A +6 7 14 28 minor seventh A♯, B♭ −31 29 +30 15 30 major seventh B −12 31 +45 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 (see just intonation). This objective structure is augmented by psychoacoustic phenomena. For example, a perfect fifth, say 200 and 300 Hz (cycles per second), causes a listener to perceive 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 all further nth-order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When one contrasts 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. Note 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 Paul Hindemith in his book The Craft of Musical Composition. Timbre of musical instruments[edit] This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (November 2011) The relative amplitudes (strengths) of the various harmonics primarily determine the timbre of different instruments and sounds, though onset transients, formants, noises, and inharmonicities also play 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 clarinets resonator is cylindrical, the even-numbered harmonics are less present. The saxophones resonator is conical, which allows the even-numbered harmonics to sound more strongly and thus produces a more complex tone. The inharmonic ringing of the instruments metal resonator is even more prominent in the sounds of brass instruments. Human ears tend to group phase-coherent, harmonically-related frequency components into a single sensation. Rather than perceiving the individual partials–harmonic and inharmonic, of a musical tone, humans perceive them together as a tone color or timbre, and the overall pitch is heard as the fundamental of the harmonic series being experienced. If a sound is heard that is made up of even just a few simultaneous sine tones, and if the intervals among those tones form part of a harmonic series, the brain tends to group this input into a sensation of the pitch of the fundamental of that series, even if the fundamental is not present. 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. Interval strength[edit] David Cope (1997) suggests the concept of interval strength,[6] in which an intervals 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 (About this sound play (help·info)) is stronger than an equal-tempered minor third (About this sound play (help·info)), since they approximate a just perfect fifth (About this sound play (help·info)) and just minor third (About this sound 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[edit] Wikimedia Commons has media related to Harmonic series. Fourier series Inharmonicity Klang (music) Otonality and Utonality Piano acoustics Scale of harmonics Stretched tuning Subharmonic Undertone series References[edit] Jump up ^ William Forde Thompson (2008). Music, Thought, and Feeling: Understanding the Psychology of Music. p. 46. ISBN 978-0-19-537707-1. Jump up ^ John R. Pierce (2001). Consonance and Scales. In Perry R. Cook. Music, Cognition, and Computerized Sound. MIT Press. ISBN 978-0-262-53190-0. Jump up ^ Martha Goodway and Jay Scott Odell (1987). The Historical Harpsichord Volume Two: The Metallurgy of 17th- and 18th- Century Music Wire. Pendragon Press. ISBN 978-0-918728-54-8. Jump up ^ Juan G. Roederer (1995). The Physics and Psychophysics of Music. p. 106. ISBN 0-387-94366-8. Jump up ^ Fonville, John. 1991. Ben Johnstons Extended Just Intonation: A Guide for Interpreters, p.121. Perspectives of New Music 29, no. 2 (Summer): 106–37. Jump up ^ Cope, David (1997). Techniques of the Contemporary Composer, p. 40–41. New York, New York: Schirmer Books. ISBN 0-02-864737-8. External links[edit] Interaction of reflected waves on a string is illustrated in a simplified animation A Web-based Multimedia Approach to the Harmonic Series Importance of prime harmonics in music theory Octave Frequency Sweep, Consonance & Dissonance The combined oscillation of a string with several of its lowest harmonics can be seen clearly in an interactive animation at Edward Zobels Zona Land. [show] v t e Acoustics [show] v t e Consonance and dissonance
Posted on: Mon, 19 Jan 2015 05:09:34 +0000
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