Limit (music)
Encyclopedia
In music theory
, limit or harmonic limit is a way of characterizing the harmony
found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch
, who used it to give an upper bound
on the complexity of harmony; hence the name. "Roughly speaking, the larger the limit number, the more harmonically complex and potentially dissonant
will the intervals
of the tuning be perceived."
, and Ralph David Hill are among the many microtonalists
to suggest that music has been slowly evolving to employ higher and higher harmonics
in its constructs (see emancipation of the dissonance
). In medieval music
, only chords made of octave
s and perfect fifth
s (involving relationships among the first 3 harmonic
s) were considered consonant. In the West, triadic harmony arose (Contenance Angloise
) around the time of the Renaissance
, and triads
quickly became the fundamental building blocks of Western music. The major
and minor third
s of these triads invoke relationships among the first 5 harmonics.
Around the turn of the 20th century, tetrads
debuted as fundamental building blocks in African-American music. In conventional music theory pedagogy, these seventh chord
s are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5. For example, the dominant 7th chord
in 12-ET
approximates 4:5:6:7, while the major 7th chord
approximates 8:10:12:15.
, intervals between pitches are drawn from the rational numbers. Since Partch, two distinct formulations of the limit concept have emerged: odd limit (generally preferred for the analysis of simultaneous intervals and chords) and prime limit (generally preferred for the analysis of scales
). Odd limit and prime limit n do not include the same intervals even when n is an odd prime.
In Genesis of a Music
, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, modulo octaves. Since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partch's theoretical prediction of the sensory dissonance of intervals (his "One-Footed Bride") are very similar to those of theorists including Hermann von Helmholtz
, William Sethares
, and Paul Erlich
.
.
In the late 1970s, a new genre of music began to take shape on the West coast of the United States, known as the American gamelan school
. Inspired by Indonesian gamelan
, musicians in California and elsewhere began to build their own gamelan instruments, often tuning them in just intonation. The central figure of this movement was the American composer Lou Harrison
. Unlike Partch, who often took scales directly from the harmonic series, the composers of the American Gamelan movement tended to draw scales from the just intonation lattice, in a manner like that used to construct Fokker periodicity blocks
. Such scales often contain ratios with very large numbers, that are nevertheless related by simple intervals to other notes in the scale.
, the simple ratios of just intonation are mapped to nearby irrational approximations. This operation, if successful, does not change the relative harmonic complexity of the different intervals, but it can complicate the use of the harmonic limit concept. Since some chords (such as the diminished seventh chord
in 12-ET) have several valid tunings in just intonation, their harmonic limit may be ambiguous.
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, limit or harmonic limit is a way of characterizing the harmony
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...
found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by 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...
, who used it to give an upper bound
Upper bound
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...
on the complexity of harmony; hence the name. "Roughly speaking, the larger the limit number, the more harmonically complex and potentially dissonant
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
will the intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
of the tuning be perceived."
The harmonic series and the evolution of music
Harry Partch, Ivor DarregIvor Darreg
Ivor Darreg was a leading proponent of and composer of microtonal or "xenharmonic" music. He also created a series of experimental musical instruments.Darreg, a contemporary of Harry Partch and a close colleague of John H...
, and Ralph David Hill are among the many microtonalists
Microtonal music
Microtonal music is music using microtones—intervals of less than an equally spaced semitone. Microtonal music can also refer to music which uses intervals not found in the Western system of 12 equal intervals to the octave.-Terminology:...
to suggest that music has been slowly evolving to employ higher and higher harmonics
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
in its constructs (see emancipation of the dissonance
Emancipation of the dissonance
The emancipation of the dissonance was a concept or goal put forth by composer Arnold Schoenberg and others, including his pupil Anton Webern. The phrase first appears in Schoenberg's 1926 essay "Opinion or Insight?". It may be described as a metanarrative to justify atonality...
). In medieval music
Medieval music
Medieval music is Western music written during the Middle Ages. This era begins with the fall of the Roman Empire and ends sometime in the early fifteenth century...
, only chords made of octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
s and perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
s (involving relationships among the first 3 harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
s) were considered consonant. In the West, triadic harmony arose (Contenance Angloise
Contenance Angloise
The Contenance Angloise or English manner, is the term used to describe a distinctive style of polyphony developed in fifteenth-century England. It used full, rich harmonies based on the third and sixth. It was highly influential in the fashionable Burgundian court of Philip the Good and as a...
) around the time of the Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
, and triads
Triad (music)
In music and music theory, a triad is a three-note chord that can be stacked in thirds. Its members, when actually stacked in thirds, from lowest pitched tone to highest, are called:* the Root...
quickly became the fundamental building blocks of Western music. The major
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
and minor third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
s of these triads invoke relationships among the first 5 harmonics.
Around the turn of the 20th century, tetrads
Tetrad (music)
A tetrad is a set of four notes in music theory. When these four notes form a tertian chord they are more specifically called a seventh chord, after the diatonic interval from the root of the chord to its fourth note...
debuted as fundamental building blocks in African-American music. In conventional music theory pedagogy, these seventh chord
Seventh chord
A seventh chord is a chord consisting of a triad plus a note forming an interval of a seventh above the chord's root. When not otherwise specified, a "seventh chord" usually means a major triad with an added minor seventh...
s are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5. For example, the dominant 7th chord
Dominant seventh chord
In music theory, a dominant seventh chord, or major minor seventh chord,is a chord composed of a root, major third, perfect fifth, and minor seventh. It can be also viewed as a major triad with an additional minor seventh...
in 12-ET
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
approximates 4:5:6:7, while the major 7th chord
Major seventh chord
In music, a major seventh chord is any nondominant seventh chord where the "third" note is a major third above the root.Most typically, major seventh chord refers to where the "seventh" note is a major seventh above the root . This is more precisely known as the major/major seventh chord, and it...
approximates 8:10:12:15.
Odd-limit and prime-limit
In just intonationJust intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
, intervals between pitches are drawn from the rational numbers. Since Partch, two distinct formulations of the limit concept have emerged: odd limit (generally preferred for the analysis of simultaneous intervals and chords) and prime limit (generally preferred for the analysis of scales
Musical scale
In music, a scale is a sequence of musical notes in ascending and descending order. Most commonly, especially in the context of the common practice period, the notes of a scale will belong to a single key, thus providing material for or being used to conveniently represent part or all of a musical...
). Odd limit and prime limit n do not include the same intervals even when n is an odd prime.
Odd limit
For a positive odd number n, the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than n.In Genesis of a Music
Genesis of a Music
Genesis of a Music is a book first published in 1947 by American microtonal composer Harry Partch .Partch first presents a polemic against both equal temperament and the long history of stagnation in the teaching of music, then goes on to explain his tuning theory based on just intonation, the...
, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, modulo octaves. Since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partch's theoretical prediction of the sensory dissonance of intervals (his "One-Footed Bride") are very similar to those of theorists including Hermann von Helmholtz
Hermann 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...
, William Sethares
William Sethares
William A. Sethares is an American music theorist and professor of Electrical engineering and who is known primarily for his contributions to music theory, including dynamic tonality and a formalization of the source of consonance....
, and Paul Erlich
Paul Erlich
Paul Erlich is a guitarist and music theorist living near Boston, Massachusetts. He is known for his seminal role in developing the theory of regular temperaments, including being the first to discover pajara temperament and its decatonic scales in 22-ET. He holds a Bachelor of Science degree in...
.
Prime limit
For a prime number n, the n-prime-limit contains all rational numbers that can be factored using primes no greater than n. In other words, it is the set of rationals with nominator and denominator both n-smoothSmooth number
In number theory, a smooth number is an integer which factors completely into small prime numbers. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography relying on factorization.-Definition:...
.
In the late 1970s, a new genre of music began to take shape on the West coast of the United States, known as the American gamelan school
American gamelan
American gamelan could refer to both instruments and music; the term has been used to refer to gamelan-style instruments built by Americans, as well as to music written by American composers to be played on gamelan instruments...
. Inspired by Indonesian gamelan
Gamelan
A gamelan is a musical ensemble from Indonesia, typically from the islands of Bali or Java, featuring a variety of instruments such as metallophones, xylophones, drums and gongs; bamboo flutes, bowed and plucked strings. Vocalists may also be included....
, musicians in California and elsewhere began to build their own gamelan instruments, often tuning them in just intonation. The central figure of this movement was the American composer Lou Harrison
Lou Harrison
Lou Silver Harrison was an American composer. He was a student of Henry Cowell, Arnold Schoenberg, and K. P. H. Notoprojo Lou Silver Harrison (May 14, 1917 – February 2, 2003) was an American composer. He was a student of Henry Cowell, Arnold Schoenberg, and K. P. H. Notoprojo Lou Silver Harrison...
. Unlike Partch, who often took scales directly from the harmonic series, the composers of the American Gamelan movement tended to draw scales from the just intonation lattice, in a manner like that used to construct Fokker periodicity blocks
Fokker periodicity blocks
Fokker periodicity blocks are a concept in tuning theory used to mathematically relate musical intervals in just intonation to those in equal tuning. They are named after Adriaan Daniël Fokker...
. Such scales often contain ratios with very large numbers, that are nevertheless related by simple intervals to other notes in the scale.
Examples
| ratio | interval | odd-limit | prime-limit | audio |
|---|---|---|---|---|
| 3/2 | perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones... |
3 | 3 | |
| 4/3 | perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there... |
3 | 3 | |
| 5/4 | major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
5 | 5 | |
| 5/2 | major tenth | 5 | 5 | |
| 5/3 | major sixth Major sixth In classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two... |
5 | 5 | |
| 7/5 | lesser septimal tritone Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7... |
7 | 7 | |
| 10/7 | greater septimal tritone | 7 | 7 | |
| 9/8 | major second Major second In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions... |
9 | 3 | |
| 27/16 | Pythagorean major sixth | 27 | 3 | |
| 81/64 | ditone Ditone In music, a ditone is the interval 81:64, which is 407.82 cents. It is the major third in Pythagorean tuning. The ditone is evenly divisible by two major tones and is wider than a just major third by a syntonic comma... |
81 | 3 | |
| 243/128 | Pythagorean major seventh Pythagorean interval In musical tuning theory, a Pythagorean interval is a musical interval with frequency ratio equal to a power of two divided by a power of three, or vice versa... |
243 | 3 |
Beyond just intonation
In musical temperamentMusical temperament
In musical tuning, a temperament is a system of tuning which slightly compromises the pure intervals of just intonation in order to meet other requirements of the system. Most instruments in modern Western music are tuned in the equal temperament system...
, the simple ratios of just intonation are mapped to nearby irrational approximations. This operation, if successful, does not change the relative harmonic complexity of the different intervals, but it can complicate the use of the harmonic limit concept. Since some chords (such as the diminished seventh chord
Diminished seventh chord
A diminished seventh chord is a four note chord that comprises a diminished triad plus the interval of a diminished seventh above the root. Thus it is , or enharmonically , of any major scale; for example, C diminished-seventh would be , or enharmonically...
in 12-ET) have several valid tunings in just intonation, their harmonic limit may be ambiguous.
See also
- AtonalityAtonalityAtonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...
- Microtonal musicMicrotonal musicMicrotonal music is music using microtones—intervals of less than an equally spaced semitone. Microtonal music can also refer to music which uses intervals not found in the Western system of 12 equal intervals to the octave.-Terminology:...
- Musical tuningMusical tuningIn music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
- Otonality and UtonalityOtonality and UtonalityOtonality 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,.....
- Numerary nexusNumerary nexusIn musical tuning, a numerary nexus is an identity shared by two or more interval ratios in their numerator or denominator, with different identities in the other...
- Tonality diamondTonality diamondIn music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality...
External links
- "Limits: Consonance Theory Explained", Glen Peterson's Musical Instruments and Tuning Systems.
- "Harmonic Limit", Xenharmonic.