Harmonic series

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Harmonic series (music)

Prerequisite Knowledge

Overview

The harmonic series describes a pattern of frequencies naturally occurring as a real, physical (not theoretical or psychoacoustic) phenomenon. Though its manifestation is often imprecise, the harmonic series can be observed in many pitched sounds.

The fundamental is the lowest frequency (or partial) in a given harmonic series. While the fundamental is generally the main audible pitch of a given sound, the harmonic series contains an infinitely proliferating pattern of higher partials, which are called overtones. With practice, one can learn to hear and identify specific overtones:

Learn To Hear Harmonics!! By Mannfish

The approach of creating music based on intervallic relationships derived from the harmonic series relationships is called Just Intonation.

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

Manifestations of the Harmonic Series

Cello Harmonics

Musical Instruments

Many musicians are already familiar with the harmonic series, even if they may not realize it. For example: the natural harmonics of a string instrument (bowed or strummed) and the open notes on a French horn are two manifestations of the harmonic series. Using the cello as an example, the low, open C string acts as the fundamental of its harmonic series. In this case, 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.

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Timbre

The harmonic series is also responsible for timbre: It is how one can easily identify the sound of a trumpet, violin, or electric guitar, even if they play the same pitch. This is because 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. This also applies to speech and singing, as different vowel formants are distinguished by their unique harmonic identity.

In this video, individual harmonic partials are gradually re-introduced to a recording of a singer alternating between different vowels. A psychoacoustic illusion is created where at a certain point, the the individual frequencies are suddenly interpreted by the brain as one, timbrally unique and recognizable sound: the human voice. Also note how the intensity of each partial varies depending on the vowel being sung.

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 ascending order, will be 100Hz, 200Hz, 300Hz, 400Hz, etc...

Because frequency is exponential, the linear relationship between each partial (as demonstrated in the above example) results in partials becoming increasingly dense/close together (like the frets on a guitar). An octave represents a doubling in frequency: If the fundamental is, again, 100Hz, its first octave will be at 200Hz, the second one at 400Hz, the third at 800Hz, etc. With a new partial at every interval of 100Hz, the number of partials will double with each consecutive octave. For more information on the exponential nature of frequency, see Hertz.

Musical Intervals As Ratios

Beginning with the seventh partial, intervals in the harmonic series begin to deviate significantly from those used in 12EDO. 12EDO interval nomenclature (minor third, perfect fifth, etc.) fails to accurately represent these relationships because, for instance, the harmonic series contains an infinite number of different minor thirds. Therefore, 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 main ways in which one might think of these ratios:

6:5 represents the 6th and 5th partials of the harmonic series
6:5 is the mathematical ratio of the two pitches' frequencies (600Hz and 500Hz, for example).

These two frames of reference are fundamentally the same.

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 the same 6:5 minor third described above. For more information about how pitch is perceived, see [psychoacoustics page that does not exist yet].

Terminology

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. Likewise, the standard way to refer to frequencies in the harmonic series is by using numbered partials (beginning with the fundamental as the first partial), not overtones.

Also note that interval ratios are always notated with the greater number first. In order to change the order of the two pitches, one must use the ratio's reciprocal. For example, to represent a major sixth instead of a minor third, one must notate the interval as 10:6, not 5:6.

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 Harmonic Series As A Fractal

The harmonic series contains an infinite number of harmonic series within it. By isolating every numbered partial with a given factor, one finds that an entire harmonic series manifests within this smaller subset of the original harmonic series. [Add example and image]

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

Title	Author
The Arithmetic Of Listening	Gann, Kyle
Harmonic Experience: Tonal Harmony From Its Natural Origins to Its Modern Expression	Mathieu, W. A.
An Introduction To the Harmonic Series And Logarithmic Integrals-For High School Students Up To Researchers	Olaikhan, Ali

User:Em/Harmonic Series

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