In music, the undertone series or subharmonic 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. While the overtone series is based upon arithmetic multiplication of frequencies, resulting in a harmonic series, the undertone series is based on arithmetic division. [1]
The hybrid term subharmonic is used in music in a few different ways. In its pure sense, the term subharmonic refers strictly to any member of the subharmonic series (1⁄1, 1⁄2, 1⁄3, 1⁄4, etc.). When the subharmonic series is used to refer to frequency relationships, it is written with f representing some highest known reference frequency (f⁄1, f⁄2, f⁄3, f⁄4, etc.). As such, one way to define subharmonics is that they are "... integral submultiples of the fundamental (driving) frequency". [2] 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. 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". [2]
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.
The overtone series can be produced physically in two ways – either by overblowing a wind instrument, 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 multiplied in the opposite ratios, the undertones series is produced.
String quartets by composers George Crumb and Daniel James Wolf,[ citation needed ] as well as works by violinist and composer Mari Kimura, [3] include undertones, "produced by bowing with great pressure to create pitches below the lowest open string on the instrument." [4] 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. [5]
The tritare, a guitar with Y-shaped strings, cause subharmonics too. This can also be achieved by the extended technique 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. [6] They are also a common effect in both digital and analog signal processing. Octave effect processors synthesize a subharmonic tone at a fixed interval to the input. Subharmonic synthesizer systems used in audio production and mastering work on the same principle.
By a similar token, analog synthesizers such as the Serge synthesizer and many modern Eurorack synthesizers can produce undertone series as a side effect of the solid state timing circuits (e.g. the 555 timer IC) in their envelope generators not being able to re-trigger until their cycle is complete. [7] As an example, sending a clock of period N into an envelope generator where the sum of the rise and fall time is greater than 2 N and less than 3 N would result in an output waveform that tracks at 1⁄3 of the frequency of the input clock.
Subharmonic frequencies are frequencies below the fundamental frequency of an oscillator in a ratio of 1/n, with n a positive integer. For example, if the fundamental frequency of an oscillator is 440 Hz, sub-harmonics include 220 Hz (1⁄2), ~146.6 Hz (1⁄3) and 110 Hz (1⁄4). Thus, they are a mirror image of the harmonic series, the overtone series.
In the overtone series, if we consider C as the fundamental, the first five notes that follow are: C (one octave higher), G (perfect fifth higher than previous note), C (perfect fourth higher than previous note), E (major third higher than previous note), and G (minor third 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 lower), F (perfect fifth lower than previous note), C (perfect fourth lower than previous note), A♭ (major third lower than previous note), and F (minor third lower than previous note).
| Undertone | 12tET interval | Note | Variance (cents) | Audio | ||||
|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 4 | 8 | 16 | prime (octave) | C | 0 | |
| 17 | major seventh | B | −5 | |||||
| 9 | 18 | minor seventh | A♯, B♭ | −4 | ||||
| 19 | major sixth | A | +2 | |||||
| 5 | 10 | 20 | minor sixth | G♯, A♭ | +14 | |||
| 21 | fifth | G | +29 | |||||
| 11 | 22 | tritone | F♯, G♭ | +49 | ||||
| 23 | −28 | |||||||
| 3 | 6 | 12 | 24 | fourth | F | −2 | ||
| 25 | major third | E | +27 | |||||
| 13 | 26 | −41 | ||||||
| 27 | minor third | D♯, E♭ | −6 | |||||
| 7 | 14 | 28 | major second | D | +31 | |||
| 29 | −30 | |||||||
| 15 | 30 | minor second | C♯, D♭ | +12 | ||||
| 31 | −45 | |||||||
If the first five notes of both series are compared, a pattern is seen:
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. [9] "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." [10] 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. [10]
Hermann von Helmholtz 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 is at least as active in under partials as in over partials. [11] Henry Cowell 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." [12] The phenomenon is described as occurring in resonators of instruments;
-First proposed by Zarlino in Instituzione armoniche (1558)[ page needed ], the undertone series has been appealed to by theorists such as Riemann and D'Indy to explain phenomena such as the minor chord, that they thought the overtone series would not explain. [1] However, while the overtone series occurs naturally as a result of wave propagation and sound acoustics, 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. [15]
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. [16]
Kathleen Schlesinger pointed out, in 1939, 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. [14] 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. [17] Most experiments with undertones to date have focused largely upon improvisation and performance not compositional design (for example the recent use of negative harmony [18] in jazz, popularised by Jacob Collier and stemming from the research of Ernst Levy), although in 1985/86 Jonathan Parry used what he called the Inverse Harmonic Series (identical to the Undertone Series) as one stage in his process of Harmonic Translation. [19]
Harry Partch argued that the overtone series and the undertone series are equally fundamental, and his concepts of Otonality and Utonality is based on this idea. [20]
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". [21] 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's work, Jackson 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.
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as f0, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f1, the first harmonic.
A harmonic series is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency.
In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.
In music, there are two common meanings for tuning:
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a harmonic series.
An overtone is any resonant frequency above the fundamental frequency of a sound. In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental is the lowest pitch. While the fundamental is usually heard most prominently, overtones are actually present in any pitch except a true sine wave. The relative volume or amplitude of various overtone partials is one of the key identifying features of timbre, or the individual characteristic of a sound.
In music, timbre, also known as tone color or tone quality, is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and musical instruments. It also enables listeners to distinguish different instruments in the same category.
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency.
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1, that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency.
Piano acoustics is the set of physical properties of the piano that affect its sound. It is an area of study within musical acoustics.
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. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.
Piano tuning is the act of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the vibration interaction among notes, which is different for every piano, thus in practice requiring slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament.
Musical acoustics or music acoustics is a multidisciplinary field that combines knowledge from physics, psychophysics, organology, physiology, music theory, ethnomusicology, signal processing and instrument building, among other disciplines. As a branch of acoustics, it is concerned with researching and describing the physics of music – how sounds are employed to make music. Examples of areas of study are the function of musical instruments, the human voice, computer analysis of melody, and in the clinical use of music in music therapy.
A multiphonic is an extended technique on a monophonic musical instrument in which several notes are produced at once. This includes wind, reed, and brass instruments, as well as the human voice. Multiphonic-like sounds on string instruments, both bowed and hammered, have also been called multiphonics, for lack of better terminology and scarcity of research.
In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unpleasantness, or unacceptability, although there is broad acknowledgement that this depends also on familiarity and musical expertise. The terms form a structural dichotomy in which they define each other by mutual exclusion: a consonance is what is not dissonant, and a dissonance is what is not consonant. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. In casual discourse, as German composer and music theorist Paul Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied". The term sonance has been proposed to encompass or refer indistinctly to the terms consonance and dissonance.
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: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,....
An Otonality is that set of pitches generated by the numerical factors (...identities)...over a numerical constant in the denominator. Conversely, a Utonality is the inversion of an Otonality, a set of pitches with a numerical constant in the numerator over the numerical factors...in the denominator.
Playing a string harmonic is a string instrument technique that uses the nodes of natural harmonics of a musical string to isolate overtones. Playing string harmonics produces high pitched tones, often compared in timbre to a whistle or flute. Overtones can be isolated "by lightly touching the string with the finger instead of pressing it down" against the fingerboard. For some instruments this is a fundamental technique, such as the Chinese guqin, where it is known as fan yin, and the Vietnamese đàn bầu.
The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale. Most fret positions appearing on Non-Western string instruments (lutes) are equal to positions of this scale. Unexpectedly, these fret positions are actually the corresponding undertones of the overtones from the harmonic series. The distance from the nut to the fret is an integer number lower than the distance from the fret to the bridge.
In music, klang is a term sometimes used to translate the German Klang, a highly polysemic word. Technically, the term denotes any periodic sound, especially as opposed to simple periodic sounds. In the German lay usage, it may mean "sound" or "tone", "musical tone", "note", or "timbre"; a chord of three notes is called a Dreiklang, etc.
Among alternative tunings for the guitar, an overtones tuning selects its open-string notes from the overtone sequence of a fundamental note. An example is the open tuning constituted by the first six overtones of the fundamental note C, namely C2-C3-G3-C4-E4-G4.
It seems to me repugnant to good sense to assume a force capable of producing such an inversion. ... [The undertone series] can never have for music the same significance as the overtone series. ... This "undertone series" has no influence on the color of the tone, and lacks the other natural advantages of the overtone series ...
Under-number tonality, or Utonality ("minor"), is the immutable faculty of ratios, which in turn represent an immutable faculty of the human ear.
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