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Harmonic series

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Harmonic series (music)

Prerequisite Knowledge

Hertz, Basic 12EDO Intervals

Overview

The harmonic series describes a pattern of frequencies naturally occurring as a real, physical (not theoretical or psychoacoustic) phenomenon. This phenomenon can be observed in most sounds.

[Image]

(For a diagram of the harmonic series up to the 49th partial, see The Helmholtz-Ellis JI Pitch Notation Legend and Series)

The lowest frequency (or partial) 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 partials called overtones. With practice, one can learn to hear and identify specific overtones:

Learn to hear Harmonics!! (Intros to Just Intonation) by Mannfish

In the context of microtonality, the approach of creating music based on harmonic series relationships is called Just Intonation.

The Harmonic Series In Real Life

Musical Instruments

Many musicians are already familiar with the harmonic series without even realizing it. For example: the natural harmonics of a string instrument (bowed or strummed) and the open notes on a french horn are both manifestations of the harmonic series. Using the cello as an example, the low C string would act as the fundamental when played openly. The first available natural harmonic is C one octave up, then G, C, E, G etc. To play these harmonics, one effectively shortens the length of the string, at ratios that match those in the harmonic series.

Timbre

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 play 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:

Vocalist sings on alternating vowels as harmonic partials are gradually reintroduced

In the video below, individual 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.

This audio was created using Spear: a free, downloadable spectral analysis software allowing users to explore and edit individual frequencies within recorded sounds.

Mathematical Formula and Ratio Notation

The mathematical formula for the harmonic series is simple: each positive-integer multiple of the fundamental frequency represents one overtone. For example, if the fundamental frequency is 100Hz, the partials, in order, will be 100Hz, 200Hz, 300Hz, 400Hz, etc...

Because frequency is exponential, a linear increase in frequency (as in the harmonic series) results in 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, again, 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.)

Musical Intervals As Ratios

Beginning at the seventh partial, intervals in the harmonic series begin to deviate significantly from what is seen commonly in 12EDO. Therefore, these intervals are referenced based on their relative positions in the harmonic series. 12EDO interval nomenclature (minor third, perfect fifth, etc.) fails to accurately represent these relationships. For instance, the harmonic series contains an infinite number of different minor thirds. This is why it is standard in Just Intonation to use ratios such as 6:5 to refer to a specific type of minor third. There are two ways one might think of these ratios.

• 6:5 represents the 6th and 5th partials of the harmonic series
  • [Image]
• 6:5 represents the mathematical relationship between the frequencies of the two pitches
  • [Image]

In reality, these frames of reference are one and the same, given that the harmonic series is itself based on these mathematical relationships.

Interestingly, if one were to record themself clapping a 6:5 polyrhythm (6 eighth notes over an eighth note quintuplet), then speed up the audio by a factor of 1000, one would hear this very same 6:5 minor third. For more information about how pitch is perceived, see [a page that doesn't exist yet on general psychoacoustics which includes a section explaining how at 20hz, we begin to perceive pitch instead of rhythm.]

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.

Advanced Concepts

The Motherchord

The harmonic series is a fractal, in that it contains an infinite number of harmonic series within it. For example, by isolating every numbered partial with a given factor, one finds that the harmonic series manifests within this subset of the original harmonic series. For example, see the diagram below which isolates every partial multiple of 5:

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For more information on this concept, [see the motherchord section in Primodality - does not exist yet]

Prime Partials

Prime-numbered partials are sometime referred to simply as "primes". These partials are of interest because each new prime produces a unique interval not present in any of the lower partials. For more information on this, see: Prime interval

Further Reading

Title	Author	Format	Summary	Bibliographical Entry
The Arithmetic Of Listening	Gann, Kyle	Book		Gann, Kyle. The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician, 2019.
Harmonic Experience: Tonal Harmony From Its Natural Origins to Its Modern Expression	Mathieu, W. A.	Book		Mathieu, W. A. Harmonic Experience: Tonal Harmony From Its Natural Origins to Its Modern
An Introduction To the Harmonic Series And Logarithmic Integrals-For High School Students Up To Researchers	Olaikhan, Ali	Book		(PDF) an introduction to the harmonic series and logarithmic integrals-for high school students up to researchers. Accessed October 21, 2024.

See also[edit | edit source]

• Subharmonic series
• Gallery of just intervals
• Isoharmonic chords
• First Five Octaves of the Harmonic Series
• Overtone scales
• List of octave-reduced harmonics
• Prime harmonic series
• Mike Sheiman's Very Easy Scale Building From The Harmonic Series Page
• 8th Octave Overtone Tuning

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.