Harmonic

In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. For example, if the frequency is f, the harmonics have frequency 2f, 3f, 4f, etc. The harmonics have the property that they are all periodic at the signal frequency, and due to the properties of Fourier series, the sum of the signal and its harmonics is also periodic at that frequency.

Many oscillators, including the human voice, a bowed violin string, or a Cepheid variable star, are more or less periodic, and thus can be decomposed into harmonics.

Phythagoras' discoveries and theories of acoustic principles, harmonics and their mathematical properties are among the most significant in history. His understanding of harmonics, along with the practice of temperament were key components in evolution of tonality in the Western world.

Jean Philippe Rameau (1684-1764) was among the first composers and musical theoreticians who posited the idea that the relationship of natural harmonics (the overtone series) and triads in the syntax of tonality was that which provided the emotional expressiveness in Western music.

Harmonic Realization

Most passive oscillators, such as a plucked guitar string or a struck drum head or struck bell, naturally oscillate at several frequencies known as overtones. When the oscillator is long and thin, such as a guitar string, a trumpet, or a chime, the overtones are still integer multiples of the fundamental frequency. Hence, these devices can mimic the sound of singing and are often incorporated into music. [[Overtones whose frequency is not an integer multiple of the fundamental are called inharmonic and are often perceived as unpleasant.

The untrained human ear typically does not perceive harmonics as separate notes. Instead, they are perceived as the timbre of the tone. In a musical context, overtones that are not exactly integer multiples of the fundamental are known as inharmonics. Inharmonics that are not close to harmonics are known as partials. Bells have more clearly perceptible partials than most instruments. Antique singing bowls are well known for their unique quality of producing multiple harmonic overtones or multiphonics.

The tight relation between overtones and harmonics in music often leads to their being used synonymously in a strictly musical context, but they are counted differently leading to some possible confusion. This chart demonstrates how they are counted:

1f	440 Hz	fundamental frequency	first harmonic
2f	880 Hz	first overtone	second harmonic
3f	1320 Hz	second overtone	third harmonic
4f	1760 Hz	third overtone	fourth harmonic

In many musical instruments, it is possible to play the upper harmonics without the fundamental note being present. In a simple case (e.g. recorder) this has the effect of making the note go up in pitch by an octave; but in more complex cases many other pitch variations are obtained. In some cases it also changes the timbre of the note. This is part of the normal method of obtaining higher notes in wind instruments, where it is called overblowing. The extended technique of playing multiphonics also produces harmonics. On string instruments it is possible to produce very pure sounding notes, called harmonics by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at a unison the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found half way down the highest string of a cello produces the same pitch as lightly fingering the node 1/3 of the way down the second highest string. For the human voice see Overtone singing, which uses harmonics.

Harmonics may be either used or considered as the basis of just intonation systems. Composer Arnold Dreyblatt is able to bring out different harmonics on the single string of his modified double bass by slightly altering his unique bowing technique halfway between hitting and bowing the strings. Composer Lawrence Ball uses harmonics to generate music electronically.

The fundamental frequency is the reciprocal of the period of the periodic phenomenon.

Harmonics on stringed instruments

The following table displays the stop points on a stringed instrument, such as the guitar, at which gentle touching of a string will force it into a harmonic mode when vibrated.

harmonic	stop note	harmonic note	cents	reduced cents
2	octave	P8	1200.0	0.0
3	just perfect fifth	P8 + P5	1902.0	702.0
4	just perfect fourth	2P8	2400.0	0.0
5	just major third	2P8 + just M3	2786.3	386.3
6	just minor third	2P8 + P5	3102.0	702.0
7	septimal minor third	2P8 + septimal m7	3368.8	968.8
8	septimal major second	3P8	3600.0	0.0
9	Pythagorean major second	3P8 + pyth M2	3803.9	203.9
10	just minor whole tone	3P8 + just M3	3986.3	386.3
11	greater unidecimal neutral second	3P8 + just M3 + GUN2	4151.3	551.3
12	lesser unidecimal neutral second	3P8 + P5	4302.0	702.0
13	tridecimal 2/3-tone	3P8 + P5 + T23T	4440.5	840.5
14	2/3-tone	3P8 + P5 + septimal m3	4568.8	968.8
15	septimal (or major) diatonic semitone	3P8 + P5 + just M3	4688.3	1088.3
16	just (or minor) diatonic semitone	4P8	4800.0	0.0

Harmonics, Temperament, Tonality

Frenchcomposer and organist Jean-Phillipe Rameau (1683-1764) published his Traité de l'harmonie in 1722 and this theoretical discourse remains one of the most important documents on the subject of tonality. Unlike theoreticians before him, Rameau looked to science, specifically the overtone series and harmonics, as a way to explain the nature of musical phenomena in relation to the theoretical properties of tonality in Western music. Influenced by the theories of Descartes and Sauveur, Rameau posited that there was a fundamental relationship between the harmonic principles in tonal music and the physics of sound (acoustics.)

He asserted that chords (triads) where the primary elements in music as opposed to melody or themes. His ideas regarding functional harmony, specifically the cadential relationship between the tonic, sub-dominant and dominant chords within a particular key center, became the underlying principles of what would become known as “the common practice” in musical composition in the Western music for three hundred years. The cadential relationship between tonic and dominant triads (as well as secondary dominants) is elemental to the tonal syntax.

Johann Sebastian Bach’s (1685-1750) seminal composition, The Well-Tempered Clavier, which was composed in the same year that Rameau published his Traité de l'harmoni, is the composition in which it could be said that the full establishment of tonal principles were initially manifested. In that composition Bach composed a set of works in all major and minor keys thereby exhibiting the veracity of tonality both theoretically and aesthetically. It should be noted that Equal Temperament did not become a fully accepted method of tuning until after World War I. Bach's tuning/temperament in 1722 was not the tuning that eventually came to be used in Equal Temperament in the early part of the twentieth century.

Notable twentieth century composers, including Paul Hindemith and Olivier Messiaen, predicated their harmonic languages on the physical principles of acoustic phenomenon. Echoing Rameau, Messiaen stated, "The tonic triad, the dominant and the ninth chords are not theories but phenomena that manifest themselves spontaneously around us and that we cannot deny. Resonance (e.i. acoustic resonance) will exist as long as we have ears to listen to what surrounds us."

As tonality emerged as the prevalent syntax of Westerns composers, this "key-centered" music exhibited new and highly evocative expressive dimensions. The understanding of harmonics and the practice of equal-temperament contributed significantly to the emergence on tonality as a highly evocative musical syntax.

References

ISBN links support NWE through referral fees

• Ash, J. Marshall. Studies in harmonic analysis. Washington: Mathematical Association of America, 1976. ISBN 088385113X
• Hewitt, Edwin, Kenneth A. Ross. Abstract harmonic analysis. Berlin: Springer, 1970. ISBN 3540583181
• Swain, Joseph Peter. Harmonic rhythm: analysis and interpretation. Oxford; NY: Oxford University Press, 2002. ISBN 0195150872