Harmonic

In physics, acoustics, and telecommunications, a harmonic represents a sinusoidal wave whose frequency is an exact positive integer multiple of the fundamental frequency of a periodic signal. This fundamental frequency is additionally referred to as the 1st harmonic, while subsequent harmonics are designated as higher harmonics. Given that all harmonics exhibit periodicity at the fundamental frequency, their collective summation similarly maintains periodicity at the same frequency. Collectively, these harmonics constitute a harmonic series.

This concept finds application across diverse disciplines, including music, physics, acoustics, electronic power transmission, and radio technology. For instance, considering a fundamental frequency of 50 Hz, which is typical for AC power supplies, the corresponding frequencies for the initial three higher harmonics would be 100 Hz (the 2nd harmonic), 150 Hz (the 3rd harmonic), and 200 Hz (the 4th harmonic). Any superposition of waves at these frequencies will consequently exhibit periodicity at 50 Hz.

An ${\displaystyle \ n}$^th characteristic mode, provided that 36§ ,{\displaystyle \ n>1\ ,}, will exhibit non-vibrating nodes. For instance, the §4849§rd ^{characteristic} mode possesses nodes at §6465§§6667§L{\displaystyle \ {\tfrac {1}{3}}\ L\ } and §9697§§9899§L,{\displaystyle \ {\tfrac {2}{3}}\ L\ ,}, where L{\displaystyle \ L\ } represents the string's length. Indeed, any n${\displaystyle \ n}$th ^{characteristic} mode, where nn{\displaystyle \ n\ } is not a multiple of 3, will not feature nodes at these specific points. Conversely, these other characteristic modes will be vibrating at the locations §195196§§197198§L{\displaystyle \ {\tfrac {1}{3}}\ L\ } and §227228§§229230§L.{\displaystyle \ {\tfrac {2}{3}}\ L~.}. Should a performer gently touch one of these specified locations, these alternative characteristic modes will be suppressed. Consequently, the tonal harmonics originating from these alternative characteristic modes will also be attenuated. As a result, the tonal harmonics associated with the n${\displaystyle \ n}$th characteristic ^{modes, specifically where} nn{\displaystyle \ n\ } is a multiple of 3, will achieve heightened prominence.

In music, harmonics are employed on both string and wind instruments as a method of sound production, particularly for generating higher notes and, on string instruments, for achieving a distinctive sound quality or "tone colour." When produced on strings with a bow, harmonics exhibit a "glassy," pure timbre. On stringed instruments, harmonics are generated by lightly touching the string at a precise point—without fully depressing it—while simultaneously activating the string (e.g., by plucking or bowing); this action allows the harmonic to resonate, producing a pitch consistently higher than the string's fundamental frequency.

Terminology

Harmonics are sometimes referred to as "overtones," "partials," or "upper partials," and within certain musical contexts, the terms "harmonic," "overtone," and "partial" are often employed synonymously. However, a more precise definition distinguishes "harmonic" as encompassing all pitches within a harmonic series, including the fundamental frequency, whereas "overtone" specifically denotes pitches above the fundamental.

Characteristics

A

whizzing, whistling tonal character distinguishes all harmonics, both natural and artificial, from firmly stopped intervals; therefore, their application in conjunction with these intervals requires meticulous consideration.

The majority of acoustic instruments generate complex tones comprising numerous individual partials (component simple tones or sinusoidal waves), though the unaccustomed human ear generally does not discern these partials as distinct phenomena. Instead, a musical note is perceived as a singular sound, with its quality or timbre determined by the relative amplitudes of its constituent partials. Many acoustic oscillators, including the human voice and bowed violin strings, produce complex tones that exhibit a degree of periodicity. These tones are consequently composed of partials that closely approximate integer multiples of the fundamental frequency, thereby resembling ideal harmonics. For practical purposes, these are often termed "harmonic partials" or simply "harmonics," despite the strict inaccuracy of referring to a  partial  (an actual phenomenon) as a  harmonic  (a theoretical concept).

Oscillators generating harmonic partials typically function similarly to one-dimensional resonators, often characterized by their length and slenderness, such as a guitar string or an air column open at both ends (e.g., in a modern orchestral transverse flute). Wind instruments with an air column open at only one end, including trumpets and clarinets, also produce partials that approximate harmonics. Theoretically, however, these instruments generate only partials corresponding to the odd harmonics. In practical application, no actual acoustic instrument perfectly adheres to the predictions of simplified physical models; for instance, instruments constructed from non-linearly elastic wood rather than metal, or strung with gut instead of brass or steel, tend to exhibit partials that are not exact integer multiples.

Partials with frequencies that are not integer multiples of the fundamental are designated as inharmonic partials. Certain acoustic instruments, such as pianos, pizzicato-plucked strings, vibraphones, marimbas, and specific pure-sounding bells or chimes, emit a combination of harmonic and inharmonic partials yet still convey a definite fundamental pitch to the listener. Antique singing bowls are recognized for their capacity to generate multiple harmonic partials, often referred to as multiphonics. Conversely, other oscillators, including cymbals, drum heads, and the majority of percussion instruments, inherently produce a profusion of inharmonic partials. These instruments do not imply a specific pitch and are consequently unsuitable for melodic or harmonic deployment in the manner of other instrument types.

Expanding upon Sethares (2004), the concept of dynamic tonality introduces pseudo-harmonic partials, wherein the frequency of each partial is precisely aligned with the pitch of a corresponding note within a pseudo-just tuning system, thereby maximizing the consonance between the pseudo-harmonic timbre and the notes of that specific pseudo-just tuning.

Partials, Overtones, and Harmonics

An overtone is defined as any partial that exceeds the lowest partial within a compound tone. The distinctive timbre of an instrument is determined by the relative amplitudes and frequency relationships of its constituent partials. While the terms "overtone" and "partial" are frequently used interchangeably in musical contexts due to their conceptual proximity, their differing enumeration methods can lead to potential ambiguity. In instances where instrumental timbres possess component partials that closely align with a harmonic series (characteristic of most string and wind instruments), as opposed to inharmonic partials (typical of most pitched percussion instruments), it is common, though technically imprecise, to refer to these component partials as "harmonics." This distinction is crucial because harmonics retain their numbering even when absent, whereas partials and overtones are only counted when physically present. The subsequent illustration clarifies the counting conventions for partials, overtones, and harmonics, assuming the presence of all harmonics.

Many musical instruments permit the production of upper harmonics without the simultaneous presence of the fundamental note. In simple cases, such as with a recorder, this technique results in an octave increase in pitch; however, more complex scenarios yield a broader array of pitch variations. Occasionally, this also alters the note's timbre. This constitutes a standard method for achieving higher notes in wind instruments, where it is termed overblowing. The advanced technique of generating multiphonics also produces harmonics. On string instruments, musicians can create exceptionally pure-sounding notes, referred to by string players as harmonics or flageolets, which possess a distinctively ethereal quality in addition to their elevated pitch. Harmonics can be employed to verify the unison tuning of strings that are not inherently tuned to a unison. For example, gently touching the node located precisely halfway along a cello's highest string produces the same pitch as lightly touching the node ⁠ 1 / 3 ⁠ of the second highest string. The human voice can also utilize harmonics, a technique explored in overtone singing.

While electronically generated periodic tones, such as square waves or other non-sinusoidal waveforms, demonstrably possess "harmonics" that are exact integer multiples of the fundamental frequency, this characteristic is not universally observed in practical musical instruments. For instance, the higher "harmonics" produced by piano notes are not true harmonics but rather "overtones" and can exhibit significant sharpness, meaning their frequencies exceed those predicted by a pure harmonic series. This phenomenon is particularly pronounced in instruments other than strings, brass, or woodwinds. Illustrative examples of such instruments include xylophones, drums, bells, and chimes; their overtone frequencies frequently do not form simple integer ratios with the fundamental frequency. (The fundamental frequency is defined as the reciprocal of the longest temporal period within a collection of vibrations associated with a singular periodic phenomenon.)

Specific considerations apply to the acoustic properties of stringed instruments.

Harmonics on stringed instruments can be generated through two primary methods: (1) by adjusting the bow's contact point, or (2) by applying gentle pressure to the string at its nodal points, which correspond to the divisions of its aliquot parts (§1011§ §1213§ {\displaystyle {\tfrac {1}{2}}} , §3435§ §3637§ {\displaystyle {\tfrac {1}{3}}} , §59<span>60§ §61<semantics>62§ </semantics></span> {\displaystyle {\tfrac {1}{4}}} , among others). The first method involves moving the bow from the standard position for the fundamental note towards the bridge, which can sequentially produce the entire harmonic scale on instruments that are aged and highly resonant. This technique results in an effect known as 'sul ponticello.' Conversely, the second method, which involves lightly pressing an open string with a finger, is more commonly employed. Harmonics produced by this gentle pressure at various nodes of open strings are termed 'natural harmonics.' It is widely recognized by violinists that a greater string length relative to its thickness enables the production of a larger number of upper harmonics.

A subsequent table delineates the specific stop points on a stringed instrument where a delicate touch can induce a harmonic mode upon vibration. String harmonics, also referred to as flageolet tones, are characterized by a "flutelike, silvery quality," which proves highly effective as a distinctive tonal color or timbre within orchestral compositions. Typically, natural harmonics beyond the fifth partial are rarely observed on most stringed instruments, with the notable exception of the double bass, owing to its significantly greater string length.

Artificial Harmonics

Musical scores may occasionally specify an artificial harmonic, which is generated by sounding an overtone on a string that has already been stopped. This performance technique involves the use of two fingers on the fingerboard: one finger shortens the string to establish the intended fundamental pitch, while the second finger lightly touches the node corresponding to the desired harmonic.

Additional Information

Harmonics serve as a foundational element in, or are considered integral to, systems of just intonation. Composer Arnold Dreyblatt exemplifies their practical application by eliciting distinct harmonics from a single string on his modified double bass, achieved through subtle adjustments to his unique bowing technique, which oscillates between striking and bowing the strings. Similarly, composer Lawrence Ball employs harmonics as a method for electronic music generation.

• Aristoxenus – 4th century BC Greek Peripatetic philosopher
• Formant – Spectrum of phonetic resonance in speech production
• Guitar harmonic – String instrument technique
• Harmonics (electrical power) – Sinusoidal wave whose frequency is an integer multiple
• Harmonic oscillator – Physical system that responds to a restoring force proportional to displacement
• Pure tone – Sound with a sinusoidal waveform
• Scale of harmonics
• Stretched octave – Musical interval which is not a perfect harmonicPages displaying short descriptions of redirect targets
• Xenharmonic music – Music that uses a tuning system outside of 12-TET

• Harmonics, partials and overtones from fundamental frequency
• Chisholm, Hugh, editor (1911). "Harmonic" . In Encyclopædia Britannica (11th edition). Cambridge University Press.Source: TORIma Academy Archive