![Undertone series on C](https://upload.wikimedia.org/wikipedia/commons/f/f2/Undertone_series_on_C.png)
In
music
Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
, the undertone series or subharmonic series is a sequence of
notes
Note, notes, or NOTE may refer to:
Music and entertainment
* Musical note, a pitched sound (or a symbol for a sound) in music
* Notes (album), ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian
* ''Notes'', a common (yet unofficial) sho ...
that results from
inverting the intervals of the
overtone series
A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a ''fundamental frequency''.
Pitched musical instruments are often based on an acoustic resonator su ...
. While overtones naturally occur with the physical production of music on instruments, undertones must be produced in unusual ways. While the overtone series is based upon arithmetic multiplication of frequencies, resulting in a
harmonic series, the undertone series is based on arithmetic division.
[ Nattiez shows the undertone series on E, as ]Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
(''Handbuch der Harmonielehre'', 10th ed., 1929, p. 4) and D'Indy (''Cours de composition musicale'', vol. I, 1912, p. 100) had done.
Terminology
The
hybrid term ''subharmonic'' is used in
music
Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...
in a few different ways. In its pure sense, the term ''subharmonic'' refers strictly to any member of the subharmonic series (, , , , etc.). When the subharmonic series is used to refer to frequency relationships, it is written with f representing some highest known reference frequency (, , , , etc.). As such, one way to define subharmonics is that they are "... integral submultiples of the fundamental (driving) frequency".
The complex tones of acoustic instruments do not produce partials that resemble the subharmonic series, unless they are played or designed to induce non-linearity. However, such tones can be produced artificially with audio software and electronics. Subharmonics can be contrasted with
harmonics
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
. While harmonics can "... occur in any linear system", there are "... only fairly restricted conditions" that will lead to the "nonlinear phenomenon known as subharmonic generation".
In a second sense, ''subharmonic'' does not relate to the subharmonic series, but instead describes an instrumental technique for lowering the pitch of an acoustic instrument below what would be expected for the resonant frequency of that instrument, such as a violin string that is driven and damped by increased bow pressure to produce a fundamental frequency lower than the normal pitch of the same open string. The human voice can also be forced into a similar driven resonance, also called “undertone singing” (which similarly has nothing to do with the undertone series), to extend the range of the voice below what is normally available. However, the frequency relationships of the component partials of the tone produced by the acoustic instrument or voice played in such a way still resemble the harmonic series, not the subharmonic series. In this sense, ''subharmonic'' is a term created by reflection from the second sense of the term ''harmonic'', which in that sense refers to an instrumental technique for making an instrument's pitch seem higher than normal by eliminating some lower partials by damping the resonator at the antinodes of vibration of those partials (such as placing a finger lightly on a string at certain locations).
In a very loose third sense, ''subharmonic'' is sometimes used or misused to represent any frequency lower than some other known frequency or frequencies, no matter what the frequency relationship is between those frequencies and no matter the method of production.
Methods for producing an undertone series
The overtone series can be produced physically in two ways – either by
overblowing
Overblowing is the manipulation of supplied air through a wind instrument that causes the sounded pitch to jump to a higher one without a fingering change or the operation of a slide. Overblowing may involve a change in the air pressure, in the ...
a
wind instrument
A wind instrument is a musical instrument that contains some type of resonator (usually a tube) in which a column of air is set into vibration by the player blowing into (or over) a mouthpiece set at or near the end of the resonator. The pitc ...
, or by dividing a
monochord
A monochord, also known as sonometer (see below), is an ancient musical and scientific laboratory instrument, involving one (mono-) string ( chord). The term ''monochord'' is sometimes used as the class-name for any musical stringed instrument h ...
string. If a monochord string is lightly damped at the halfway point, then at , then , , etc., then the string will produce the overtone series, which includes the
major triad
In music theory, a major chord is a chord that has a root, a major third, and a perfect fifth. When a chord comprises only these three notes, it is called a major triad. For example, the major triad built on C, called a C major triad, has pitch ...
. If instead, the length of the string is multiplied in the opposite ratios, the undertones series is produced. Similarly, on a wind instrument, if the holes are equally spaced, each successive hole covered will produce the next note in the undertone series.
String quartet
The term string quartet can refer to either a type of musical composition or a group of four people who play them. Many composers from the mid-18th century onwards wrote string quartets. The associated musical ensemble consists of two violinists ...
s by composers
George Crumb
George Henry Crumb Jr. (24 October 1929 – 6 February 2022) was an American composer of avant-garde contemporary classical music. Early in his life he rejected the widespread modernist usage of serialism, developing a highly personal musical ...
and
Daniel James Wolf
Daniel James Wolf (born September 13, 1961 in Upland, California) is an American composer.
Studies
Wolf studied composition with Gordon Mumma, Alvin Lucier, and La Monte Young, as well as musical tunings with Erv Wilson and Douglas Leedy and e ...
, as well as works by violinist and composer
Mari Kimura
(; born 1962) is a Japanese violinist and composer best known for her use of subharmonics, which, achieved through special bowing techniques, allow pitches below the instrument's normal range. She is credited with "introducing" the use of viol ...
, include undertones, "produced by bowing with great pressure to create pitches below the lowest open string on the instrument." These require string instrument players to bow with sufficient pressure that the strings vibrate in a manner causing the sound waves to modulate and demodulate by the instrument's resonating horn with frequencies corresponding to subharmonics.
The
tritare A tritare is an experimental guitar invented in 2003 by mathematicians Samuel Gaudet and Claude Gauthier of the Université de Moncton of a family of stringed instruments which use Y-shaped strings, instead of the usual linear strings.
Instrument ...
, a guitar with 'Y' shaped strings, cause subharmonics too. This can also be achieved by the
extended technique
In music, extended technique is unconventional, unorthodox, or non-traditional methods of singing or of playing musical instruments employed to obtain unusual sounds or timbres.Burtner, Matthew (2005).Making Noise: Extended Techniques after Exper ...
of crossing two strings as some experimental jazz guitarists have developed. Also
third bridge preparations on guitars cause timbres consisting of sets of high pitched overtones combined with a subharmonic resonant tone of the unplugged part of the string.
Subharmonics can be produced by signal amplification through
loudspeakers
A loudspeaker (commonly referred to as a speaker or speaker driver) is an Acoustical engineering#Electroacoustics, electroacoustic transducer that converts an electrical audio signal into a corresponding sound. A ''speaker system'', also often ...
. They are also a common effect in both digital and analog
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
.
Octave effect
Octave effect boxes are a type of special effects unit which mix the input signal with a synthesised signal whose musical tone is an octave lower or higher than the original. The synthesised octave signal is derived from the original input signa ...
processors, in effect, use the undertone series to create an artificial bass line for an instrument by synthesizing a subharmonic tone at a fixed interval to the input.
Subharmonic synthesizer A subharmonic synthesizer is a device or system that generates subharmonics of an input signal. The ''n''th subharmonic of a signal of fundamental frequency ''F'' is a signal with frequency ''F''/''n''. This differs from ordinary harmonics, where t ...
systems used in audio production and mastering work on the same principle.
By a similar token,
analog synthesizers
An analog (or analogue) synthesizer is a synthesizer that uses analog circuits and analog signals to generate sound electronically.
The earliest analog synthesizers in the 1920s and 1930s, such as the Trautonium, were built with a variety of va ...
such as the
Serge synthesizer
The Serge synthesizer ( Serge Modular or Serge Modular Music System) is an analogue modular synthesizer system originally developed by Serge Tcherepnin, Rich Gold and Randy Cohen at CalArts in late 1972. The first 20 Serge systems (then called "T ...
and many modern
Eurorack
Eurorack is a modular synthesizer format originally specified in 1995 by Doepfer Musikelektronik. It has since grown in popularity, and as of 2022 has become a dominant hardware modular synthesizer format, with over 15,000 modules available from ...
synthesizers can produce undertone series as a side effect of the solid state timing circuits (e.g. the
555 timer IC
The 555 timer IC is an integrated circuit (chip) used in a variety of timer, delay, pulse generation, and oscillator applications. Derivatives provide two ( 556) or four ( 558) timing circuits in one package. The design was first marketed in 197 ...
) in their envelope generators not being able to re-trigger until their cycle is complete.
As an example, sending a clock of period into an envelope generator where the sum of the rise and fall time is greater than and less than would result in an output waveform that tracks at of the frequency of the input clock.
Comparison to the overtone series
![Otonality and utonality 5-limit](https://upload.wikimedia.org/wikipedia/commons/7/7a/Otonality_and_utonality_5-limit.png)
Subharmonic frequencies are frequencies below the fundamental frequency of an oscillator in a ratio of 1/, with a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. For example, if the fundamental frequency of an oscillator is 440 Hz, sub-harmonics include 220 Hz (), ~146.6 Hz () and 110 Hz (). Thus, they are a mirror image of the
harmonic series, the overtone series.
Notes in the series
In the overtone series, if we consider C as the fundamental, the first five notes that follow are: C (one
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
higher), G (
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
higher than previous note), C (
perfect fourth
A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
higher than previous note), E (
major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
higher than previous note), and G (
minor third
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...
higher than previous note).
The pattern occurs in the same manner using the undertone series. Again we will start with C as the fundamental. The first five notes that follow will be: C (one
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
lower), F (
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
lower than previous note), C (
perfect fourth
A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...
lower than previous note), A
(
major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
lower than previous note), and F (
minor third
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...
lower than previous note).
Triads
If the first five notes of both series are compared, a pattern is seen:
*Overtone series: C C G C E G
*Undertone series: C C F C A F
The undertone series in C contains the F minor triad. Elizabeth Godley argued that the minor triad is also implied by the undertone series and is also a naturally occurring thing in
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
. "According to this theory the ''upper'' and not the lower tone of a minor chord is the generating tone on which the unity of the chord is conditioned."
Whereas the major chord consists of a generator with upper major third and perfect fifth, the minor chord consists of a generator with lower major third and fifth.
Resonance
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
observed in ''On the Sensations of Tone'' that the tone of a string tuned to C on a piano changes more noticeably when the notes of its undertone series (C, F, C, A, F, D, C, etc.) are struck than those of its overtones. Helmholtz argued that
sympathetic resonance
Sympathetic resonance or sympathetic vibration is a harmonic phenomenon wherein a passive string or vibratory body responds to external vibrations to which it has a harmonic likeness. The classic example is demonstrated with two similarly-tuned ...
is at least as active in under partials as in over partials.
Henry Cowell
Henry Dixon Cowell (; March 11, 1897 – December 10, 1965) was an American composer, writer, pianist, publisher and teacher. Marchioni, Tonimarie (2012)"Henry Cowell: A Life Stranger Than Fiction" ''The Juilliard Journal''. Retrieved 19 June 202 ...
discusses a "Professor Nicolas Garbusov of the Moscow Institute for Musicology" who created an instrument "on which at least the first nine undertones could be heard without the aid of resonators."
The phenomenon is described as occurring in resonators of instruments;
:"the original sounding body does not produce the undertones but it is difficult to avoid them in resonation ... such resonators under certain circumstances respond to only every other vibration producing a half tone ... even if the resonator responds normally to every vibration ... under other circumstances the body resonates at only every third vibration ... the fact that such underpartials are often audible in music makes them of importance in understanding certain musical relationships ... the subdominant ... the minor triad."
Importance in musical composition
![Istarska lestvica](https://upload.wikimedia.org/wikipedia/commons/7/75/Istarska_lestvica.jpg)
First proposed by
Zarlino
Gioseffo Zarlino (31 January or 22 March 1517 – 4 February 1590) was an Italian music theorist and composer of the Renaissance. He made a large contribution to the theory of counterpoint as well as to musical tuning.
Life and career
Zarlino w ...
in ''Instituzione armoniche'' (1558), the undertone series has been appealed to by theorists such as
Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
and
D'Indy to explain phenomena such as the
minor chord
In music theory, a minor chord is a chord that has a root, a minor third, and a perfect fifth. When a chord comprises only these three notes, it is called a minor triad. For example, the minor triad built on C, called a C minor triad, has pit ...
, that they thought the overtone series would not explain.
However, while the overtone series occurs naturally as a result of wave propagation and sound
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, musicologists such as
Paul Hindemith
Paul Hindemith (; 16 November 189528 December 1963) was a German composer, music theorist, teacher, violist and conductor. He founded the Amar Quartet in 1921, touring extensively in Europe. As a composer, he became a major advocate of the ''Ne ...
considered the undertone series to be a purely theoretical 'intervallic reflection' of the overtone series. This assertion rests on the fact that undertones do not sound simultaneously with its
fundamental tone as the overtone series does.
In 1868, Adolf von Thimus showed that an indication by a 1st-century Pythagorean,
Nicomachus of Gerasa
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
, taken up by
Iamblichus
Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
in the 4th century, and then worked out by von Thimus, revealed that Pythagoras already had a diagram that could fill a page with interlocking over- and undertone series.
Kathleen Schlesinger
Kathleen Schlesinger (1862, in Holywood, Ireland – 1953, in London) was a British music archaeologist and curator of musical instruments at the British Museum.A. R. Meuss, ''Intervals, Scales, Tones and the Concert Pitch C'', 2004 p. 27/ref ...
pointed out, in 1939, that since the ancient Greek
aulos
An ''aulos'' ( grc, αὐλός, plural , ''auloi'') or ''tibia'' (Latin) was an ancient Greek wind instrument, depicted often in art and also attested by archaeology.
Though ''aulos'' is often translated as "flute" or "double flute", it was usu ...
, or reed-blown flute, had holes bored at equal distances, it must have produced a section of the undertone series.
She said that this discovery not only cleared up many riddles about the original Greek modes, but indicated that many ancient systems around the world must have also been based on this principle.
One area of conjecture is that the undertone series might be part of the compositional design phase of the compositional process. The overtone and undertone series can be considered two different arrays, with smaller arrays that contain different major and minor triads. Most experiments with undertones to date have focused largely upon improvisation and performance not compositional design (for example the recent use of negative harmony in jazz, popularised by
Jacob Collier
Jacob Collier (; né Moriarty; born 2 August 1994) is an English singer, songwriter, and multi-instrumentalist. His music incorporates a combination of jazz with elements from many other musical genres, and often features extensive use of reha ...
and stemming from the research of
Ernst Levy
Ernst Levy (18 November 1895 – 19 April 1981) was a Swiss musicologist, composer, pianist and conductor.
Life
Born in Basel, Switzerland, Levy studied with Egon Petri and Raoul Pugno.. Entry: "Ernst Levy".
David Dubal describes him as an "un ...
), although in 1985/8
Jonathan Parryused what he called the Inverse Harmonic Series (identical to the Undertone Series) as one stage in his process of Harmonic Translation.
Harry Partch
Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...
argued that the overtone series and the undertone series are equally fundamental, and his concepts of
Otonality and Utonality
''Otonality'' and ''utonality'' are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone (identity), respectively. For example: , , ,... or , , ,....
Definition ...
is based on this idea.
Similarly, in 2006 G.H. Jackson suggested that the overtone and undertone series must be seen as a real polarity, representing on the one hand the outer "material world" and on the other, our subjective "inner world".
This view is largely based on the fact that the overtone series has been accepted because it can be explained by materialistic science, while the prevailing conviction about the undertone series is that it can only be achieved by taking subjective experience seriously. For instance, the minor triad is usually heard as sad, or at least pensive, because humans habitually hear all chords as based from below. If feelings are instead based on the high "fundamental" of an undertone series, then descending into a minor triad is not felt as melancholy, but rather as overcoming, conquering something. The overtones, by contrast, are then felt as penetrating from outside. Using
Rudolf Steiner
Rudolf Joseph Lorenz Steiner (27 or 25 February 1861 – 30 March 1925) was an Austrian occultist, social reformer, architect, esotericist, and claimed clairvoyant. Steiner gained initial recognition at the end of the nineteenth century as a ...
’s work, Jackson traces the history of these two series, as well as the main other system created by the
circle of fifths
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval ...
, and argues that in hidden form, the series are balanced out in
Bach
Johann Sebastian Bach (28 July 1750) was a German composer and musician of the late Baroque period. He is known for his orchestral music such as the '' Brandenburg Concertos''; instrumental compositions such as the Cello Suites; keyboard w ...
's harmony.
See also
*
*
Harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
*
Missing fundamental
A harmonic sound is said to have a missing fundamental, suppressed fundamental, or phantom fundamental when its overtones suggest a fundamental frequency but the sound lacks a component at the fundamental frequency itself.
The brain perceives the ...
*
Overtone
An overtone is any resonant frequency above the fundamental frequency of a sound. (An overtone may or may not be a harmonic) In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental i ...
*
Riemannian theory
"Riemannian theory" in general refers to the musical theories of German theorist Hugo Riemann (1849–1919). His theoretical writings cover many topics, including musical logic, notation, harmony, melody, phraseology, the history of music theo ...
*
Subharmonic mixer
*
Subharmonic synthesizer A subharmonic synthesizer is a device or system that generates subharmonics of an input signal. The ''n''th subharmonic of a signal of fundamental frequency ''F'' is a signal with frequency ''F''/''n''. This differs from ordinary harmonics, where t ...
References
External links
* , with audio clips
{{Acoustics
Acoustics
Musical tuning