Harmonic series

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Harmonic series (music)

Prerequisite Knowledge

Hertz, Basic 12EDO Intervals

Overview

The harmonic series describes a pattern of frequencies that naturally occur as a real (not theoretical) physical phenomenon, observable in most sounds.

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The lowest frequency in a given harmonic series is called the fundamental. While the fundamental is generally the main audible pitch of a given sound, the harmonic series contains an infinitely proliferating sequence of higher frequencies called overtones. With practice, one can learn to hear and identify specific overtones.

Mathematical Formula

The mathematical formula for the harmonic series is simple: each whole-integer multiple of the fundamental frequency represents one overtone. For example, if the fundamental frequency is 100Hz, the harmonics, in order, will be 200Hz, 300Hz, 400Hz, etc... Because frequency is exponential, a linear increase in frequency (as in the harmonic series) results in the pitches becoming increasingly dense/close together. (One way to think of this is that an octave represents a doubling in frequency: If the fundamental is 100Hz, the first octave will be at 200Hz, the second one at 400Hz, the third at 800Hz, etc. The number of pitches will double between each consecutive octave.) In the context of microtonality, these mathematical relationships form the foundation of the musical practice known as Just Intonation, where musical intervals are represented based on their relative positions in the harmonic series.

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Commonly Encountered Manifestations

Many musicians are already familiar with the harmonic series without even realizing it. For example: the series of natural harmonics on a cello and the open notes on a French Horn are both manifestations of the harmonic series. Using the violin as an example, the low G string would act as the fundamental when played openly. The first available natural harmonic is G one octave up, then D, G, B, D etc. To play these harmonics, one effectively shortens the length of the string at ratios which match those in the harmonic series: 2:1, 3:1, 4:1, 5:1, etc.

The harmonic series is also responsible for timbre: It is the reason one can hear the difference between a trumpet, violin, or flute, even if they playing the exact same pitch: while the same set of frequencies is (mostly) present in all these sounds, some partials will be more prominent than others depending on the sound source.

Partials are gradually re-introduced to a recording of a vocalist singing on different vowels. A psychoacoustic illusion is created where at a certain point, the sound of individual frequencies is suddenly interpreted by the brain as one, timbrally unique and recognizable sound: the human voice. Also note how the intensity of each partial varies with different vowels.

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Notation and Terminology

The term "partial" describes a specific frequency within the harmonic series *beginning with the fundamental as the first partial.* It is important to remember: the standard way to refer to frequencies in the harmonic series is by using numbered partials (*not* overtones). In other words, if someone refers to the first partial (or the first harmonic), they are referring to the fundamental. The first overtone is the second partial.

Note that the terms overtone and overtone series are not quite synonymous with harmonic and harmonic series, respectively, although interchangeable usage is also attested. Technically speaking, overtone series excludes the starting fundamental, so the 2nd harmonic is the 1st overtone. Because of that distinction, the math of the "overtone series" is off by one. So, "harmonic series" is arguably the preferred standard.

The subharmonic series (or undertone series) is the inversion of the harmonic series: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7... ad infinitum. The words otonal (referring to overtones) and utonal (referring to undertones) are used to refer to the harmonic and subharmonic series, respectively.

Music based on the harmonic series[edit | edit source]

The chord of nature is the name sometimes given to the harmonic series, or the series up to a certain stopping point, regarded as a chord.

Steps between adjacent members of the harmonic series are called "superparticular," and they appear in the form (n+1)/n (e.g. 4/3, 28/27, 33/32).

One might compose with the harmonic series by, for instance:

Tuning to the first several harmonics over one fundamental;
Tuning to an octave-repeating slice of the harmonic series for use as a scale (for instance harmonics 8 though 16, 12 through 24, 20 through 40... see overtone scales);
Tuning to the overtones of the overtones & the undertones of the undertones. (This can produce complex scales such as Harry Partch's 43-tone Monophonic; this kind of thing is more often called "just intonation" than "overtone music".)

Music[edit | edit source]

Richard Burdick

Planetary Ripples ^{[dead link]}

Folkart Slovakia (site)

Various played with Fujara (slovak overtone flute)

Georg Friedrich Haas

Various^[which?]

Dave Hill

Chord Progression on the Harmonic Overtone Series ^{[dead link]} play ^{[dead link]}

Norbert Oldani

Drone Inside An Harmonic Series ^{[dead link]}

Dave Seidel

Threnody ^{[dead link]} play
Owllight ^{[dead link]} play
Palimsest ^{[dead link]} play

William Sethares

Immanent Sphere – detail | play

SoundWell (site)

Various ("Snake" overtone flute)

Spectral Voices (site)

Various (meditative new age with overtone singing)

Stimmhorn (site)

Various (experimental alphorn and yodeling combined with overtone singing)

Karlheinz Stockhausen

Stimmung (1968)
Sternklang (1971)

Cam Taylor

Harmonic series 4-8, 8-16 and 16-32 on the Lumatone (2022)

Chris Vaisvil

Rock Trio in Harmonic Series (2016) – blog | play

Glenn Branca (site)

Symphony No. 3 "Gloria" (1983)

External links[edit | edit source]

Spectral music article on Wikipedia
www.naturton-musik.de ^{[dead link]} - web site dedicated to overtone music (by Austrian composer Johannes Kotschy) - a lot of theory material and practical guides to write music based on the overtone series
Overtone music network - a portal for overtone music.
Oberton-Netzwerk (Xing) ^{[dead link]} - German-speaking group dedicated to overtone music on the social network platform Xing. Microtonal music in general is welcome, too.

Em

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