Harmonics are components of a musical sound with well-defined frequency relationships to one another. For a pitch of frequency f, typically measured in units of cycles per second or Hertz (Hz), the nth harmonic has frequency n × f. In this context the frequency f is referred to as the fundamental frequency. Harmonics are closely related to overtones (or equivalently partials), which are defined to be secondary pitches that audibly resonate when a fundamental pitch sounds. The number and strength of these secondary pitches are responsible for the distinct timbres perceived in different instruments or voices. The Overtone Series in music (also called the Harmonic Series at the risk of confusion with the infinite sum of the same name) refers to the sequence of ascending harmonics with frequencies f, 2f, 3f, 4f…. With only a few exceptions, the pitches of the lower harmonics match well with the frequencies of twelve pitches of the equally tempered scale. Further along the overtone series, the pitch spacing becomes very small --- smaller than the traditional half step --- and these upper harmonics, if heard, would sound distinctly out of tune. With the discovery of the overtone series by Jean-Philippe Rameau in the eighteenth century, the notion of musical consonance as the exclusive natural and rational sonic phenomenon --- pursued by mathematicians from Pythagorus to Euler --- began to fade. There is a close physical relationship between the harmonic frequencies and the length of the vibrating medium. This is exploited in the performance practices of musical instruments.
Vibrating Media and the Overtone Series
For vibrating strings (e.g. violins and guitars) and open vibrating air columns (the western concert flute and some organ pipes) the words harmonic, partial, and overtone are essentially synonymous, with a slight difference in the enumeration: the fundamental pitch (frequency f ) is referred to as the 1st harmonic. The first overtone (frequency 2f ) refers to the 2nd harmonic, etc. In stopped air columns (the clarinet and some organ pipes), the overtone series omits certain harmonic frequencies. For vibrating membranes (percussion instruments), overtones may exist at non-harmonic frequencies.
It is therefore a slight abuse of terminology to refer, as is commonly done, to the sequence of harmonics as the overtone series. Physically, the overtone series is seen by observing the motion of a vibrating string of length L and natural frequency f. If forced to vibrate at frequencies n × f (for n = 2,3,…) , n-1 stationary points (nodes) appear along the string, at intervals of L/n. In effect, the string moves as n strings of length L/n joined end to end. String performers utilize this fact by lightly stopping the string at lengths L/2, L/3, etc to produce flute-like harmonic tones (sometimes called flageolet tones).
From the overtone perspective, only lower harmonics are perceptible to the hearer of a fundamental pitch. The first six harmonics are perceived by the modern hearer as in tune within the twelve pitches of the equally tempered scale, in which the octave (the distance between the first and second harmonic) is divided into 12 equal half step intervals. The frequency difference between successive pitches in this twelve-tone system is given by fn+1= 21/12 fn. The 2nd, 4th, 8th, harmonics, at octaves above the fundamental, sound perfectly in tune.. Upper harmonics can sound significantly out of tune however. The 7th harmonic sounds uncomfortably flat compared to its nearest corresponding equal temperament pitch. The 11th harmonic has a frequency almost equidistant between adjacent notes of the equally tempered scale, causing it to sound very out of tune --- likewise for the 13th and 14th harmonics. These considerations are significant for period instrument brass performers, whose instruments, like the so-called natural trumpet, are nothing more than long tubes without the length-changing system of valves of modern trumpets. Performers play tunes on these instruments by producing overtones, typically between the 3rd and 16th in the series. While skillful performers can compensate for the most problematic overtones, composers in the baroque era typically avoided these notes or used their sonic character for special effect. Modern composers have experimented with specially tuned pianos and electronic instruments to directly explore the sonorities of harmonics.
The first twenty-four harmonics are listed in the table with fundamental pitch taken as the A below middle C. Harmonics with frequencies that differ significantly from the equally tempered scale are indicated in bold.
Other uses of the word Harmonic in Mathematics
In mathematics, the word harmonic appears in a number of contexts, all of which trace their origins to the overtone series and associated physical vibrations. A harmonic progression is defined as the term by term reciprocal of an arithmetic progression. For example, the arithmetic sequence a1=1, a2=2, a3=3, … gives rise to the harmonic sequence h1=1, h2=1/2, h3=1/3, ... . In this example, the arithmetic sequence gives the frequency multiples for the overtone series, and the harmonic sequence corresponds to the wavelengths of the respective overtones. The harmonic mean is the reciprocal of the arithmetic mean of reciprocals. For example the harmonic mean of two numbers x and y is defined as 2(1/x + 1/y)–1. The harmonic series in mathematics is the infinite sum 1+1/2+1/3+…, providing the canonical example of a series whose terms approach zero, but nevertheless the sum diverges. The harmonic oscillator is a differential equation whose solutions are sinusoidal functions that can be used to model musical sounds. Harmonic analysis is the study of functions (or signals) by decomposition into fundamental component functions by means of the Fourier transform or other techniques. In the study of complex variables, harmonic functions are generalizations of the sinusoidal functions that model fundamental vibrations.
FURTHER READINGS:
Cohen, H.F. Quantifying Music: The Science of Music at the First Stage of the Scientific Revolution, 1580—1650. Dordrecht: D. Reidel Publishing Company, 1984
Gouk, Penelope. “The Role of Harmonics in the Scientific Revolution,” in The Cambridge History of Western Music Theory, ed. Thomas Christensen. Cambridge: Cambridge University Press, 2002
Johnston, Ben. Suite for Microtonal Piano. Robert Miller, piano; New World Records, 80203-2.
Sundberg, Johan. The Science of Musical Sounds. San Diego: Academic Press, 1991
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