Undertone series
Encyclopedia
The undertone series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones must be produced in unusual ways.
, or by dividing a monochord
string. If a monochord string is lightly damped at the halfway point, then at 1/3, then 1/4, 1/5, etc., then the string will produce the overtone series, which includes the major triad
. If instead, the length of the string is doubled in the opposite rations, 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.
In addition, José Sotorrio showed that undertones could be made through the use of a simple oscillator such as a tuning fork
. If that oscillator is gently forced to vibrate against a sheet of paper "it will naturally make contact at various audible modes of vibration." Sotorrio explained that since the tuning fork produces a sine tone, it will normally vibrate at the fundamental frequency (e.g. 440 Hz), but "momentarily", it will make contact only at every other oscillation (220 Hz), or at every third oscillation (147 Hz), and so on. This produces audible "subharmonic spectra", (e.g. below A@440 Hz (1/1), A@220 Hz (1/2), D@147 Hz(1/3), A@110 Hz(1/4), F@88 Hz(1/5), and so on). Sotorrio claims it is possible to sustain these "subspectra" using a sine wave generator through a speaker cone making contact with a flexible (flappable) surface, and also on string instruments "through skillful manipulation of the bow", but that this rarely sustains noticeably beyond the "sub-octave or twelfth".
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
. Sotorrio, however, said that since this minor triad is not built under the fundamental (C) but under the fifth below (F), major and minor tonalities cannot be said to grow out of this 'polarity'. For this idea to be true, the minor triad would be formed under the Fundamental (C), or else the major triad would be built on the fifth above the fundamental (G) in the overtone series.
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 flat, F, D, C, etc.) are struck than those of its overtones. Hemholtaz argued that sympathetic resonance is at least active in under partials as in over partials.
Henry Cowell
in "New Musical Reosurces" (pg 21-23) discusses a "professor Nicolas Garbusov of the Moscow Institue for Musicology" who created an instrument "on which at leats the first nine undertones could be heard without the aid of resonators." The penomenon is described as occurring both in resonators of instruments; "the original sounding body does not produce the undertones but it id 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 relations ships...the subdominant... the minor triad."
, musicologists such as Paul Hindemith
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.
Harry Partch
, on the other hand, argued that the overtone series and the undertone series are equally fundamental, and his concept of Otonality and Utonality
is based on this idea. Similarly, Graham H. Jackson in his book The Spiritual Basis of Musical Harmony (2006) suggests 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. With the help of Rudolf Steiner
’s work, Partch traces the history of these two series, as well as the main other system created by the circle of fifths
, and argues that in hidden form the series are balanced out in Bach
's harmony.
Kathleen Schlesinger, in her 1939 book, The Greek Aulos, pointed out that since the ancient Greek aulos
, 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.
In 1868, Adolf von Thimus showed that an indication by a 1st century Pythagorean, Nicomachus of Gerasa, taken up by Iamblichus 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.
Methods for producing an undertone series
The overtone series can be produced physically in two ways—either by overblowing a wind instrumentWind instrument
A wind instrument is a musical instrument that contains some type of resonator , in which a column of air is set into vibration by the player blowing into a mouthpiece set at the end of the resonator. The pitch of the vibration is determined by the length of the tube and by manual modifications of...
, or by dividing a monochord
Monochord
A monochord is an ancient musical and scientific laboratory instrument. The word "monochord" comes from the Greek and means literally "one string." A misconception of the term lies within its name. Often a monochord has more than one string, most of the time two, one open string and a second string...
string. If a monochord string is lightly damped at the halfway point, then at 1/3, then 1/4, 1/5, etc., then the string will produce the overtone series, which includes the major triad
Major chord
In music theory, a major chord is a chord having a root, a major third, and a perfect fifth. When a chord has these three notes alone, it is called a major triad...
. If instead, the length of the string is doubled in the opposite rations, 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.
In addition, José Sotorrio showed that undertones could be made through the use of a simple oscillator such as a tuning fork
Tuning fork
A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs formed from a U-shaped bar of elastic metal . It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone after waiting a...
. If that oscillator is gently forced to vibrate against a sheet of paper "it will naturally make contact at various audible modes of vibration." Sotorrio explained that since the tuning fork produces a sine tone, it will normally vibrate at the fundamental frequency (e.g. 440 Hz), but "momentarily", it will make contact only at every other oscillation (220 Hz), or at every third oscillation (147 Hz), and so on. This produces audible "subharmonic spectra", (e.g. below A@440 Hz (1/1), A@220 Hz (1/2), D@147 Hz(1/3), A@110 Hz(1/4), F@88 Hz(1/5), and so on). Sotorrio claims it is possible to sustain these "subspectra" using a sine wave generator through a speaker cone making contact with a flexible (flappable) surface, and also on string instruments "through skillful manipulation of the bow", but that this rarely sustains noticeably beyond the "sub-octave or twelfth".
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
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
. Sotorrio, however, said that since this minor triad is not built under the fundamental (C) but under the fifth below (F), major and minor tonalities cannot be said to grow out of this 'polarity'. For this idea to be true, the minor triad would be formed under the Fundamental (C), or else the major triad would be built on the fifth above the fundamental (G) in the overtone series.
Resonance
Hermann von HelmholtzHermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions to several widely varied areas of modern science...
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 flat, F, D, C, etc.) are struck than those of its overtones. Hemholtaz argued that sympathetic resonance is at least active in under partials as in over partials.
Henry Cowell
Henry Cowell
Henry Cowell was an American composer, music theorist, pianist, teacher, publisher, and impresario. His contribution to the world of music was summed up by Virgil Thomson, writing in the early 1950s:...
in "New Musical Reosurces" (pg 21-23) discusses a "professor Nicolas Garbusov of the Moscow Institue for Musicology" who created an instrument "on which at leats the first nine undertones could be heard without the aid of resonators." The penomenon is described as occurring both in resonators of instruments; "the original sounding body does not produce the undertones but it id 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 relations ships...the subdominant... the minor triad."
Importance in musical composition
While the overtone series occurs naturally as a result of wave propagation and sound acousticsAcoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
, musicologists such as Paul Hindemith
Paul Hindemith
Paul Hindemith was a German composer, violist, violinist, teacher, music theorist and conductor.- Biography :Born in Hanau, near Frankfurt, Hindemith was taught the violin as a child...
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.
Harry Partch
Harry Partch
Harry Partch was an American composer and instrument creator. He was one of the first twentieth-century composers to work extensively and systematically with microtonal scales, writing much of his music for custom-made instruments that he built himself, tuned in 11-limit just intonation.-Early...
, on the other hand, argued that the overtone series and the undertone series are equally fundamental, and his concept of Otonality and Utonality
Otonality and Utonality
Otonality and Utonality are terms introduced by Harry Partch to describe chords whose notes are the overtones or "undertones" of a given fixed tone. For example: 1/1, 2/1, 3/1,.....
is based on this idea. Similarly, Graham H. Jackson in his book The Spiritual Basis of Musical Harmony (2006) suggests 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. With the help of Rudolf Steiner
Rudolf Steiner
Rudolf Joseph Lorenz Steiner was an Austrian philosopher, social reformer, architect, and esotericist. He gained initial recognition as a literary critic and cultural philosopher...
’s work, Partch traces the history of these two series, as well as the main other system created by the circle of fifths
Circle of fifths
In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...
, and argues that in hidden form the series are balanced out in Bach
Johann Sebastian Bach
Johann Sebastian Bach was a German composer, organist, harpsichordist, violist, and violinist whose sacred and secular works for choir, orchestra, and solo instruments drew together the strands of the Baroque period and brought it to its ultimate maturity...
's harmony.
Kathleen Schlesinger, in her 1939 book, The Greek Aulos, pointed out that since the ancient Greek aulos
Aulos
An aulos or tibia was an ancient Greek wind instrument, depicted often in art and also attested by archaeology.An aulete was the musician who performed on an aulos...
, 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.
In 1868, Adolf von Thimus showed that an indication by a 1st century Pythagorean, Nicomachus of Gerasa, taken up by Iamblichus 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.