In
music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, limit or harmonic limit is a way of characterizing the
harmony
In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. However ...
found in a piece or
genre
Genre () is any form or type of communication in any mode (written, spoken, digital, artistic, etc.) with socially-agreed-upon conventions developed over time. In popular usage, it normally describes a category of literature, music, or other for ...
of music, or the harmonies that can be made using a particular
scale. The term ''limit'' was introduced by
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 ...
, who used it to give an
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an element ...
on the complexity of harmony; hence the name.
The harmonic series and the evolution of music
Harry Partch,
Ivor Darreg
Ivor Darreg (May 5, 1917 – February 12, 1994) was an American composer and leading proponent of microtonal or "xenharmonic" music. He also created a series of experimental musical instruments.
Biography
Darreg, a contemporary of Harry Partch ...
, and Ralph David Hill are among the many
microtonalists to suggest that music has been slowly evolving to employ higher and higher
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 ...
in its constructs (see
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 me ...
). In
medieval music
Medieval music encompasses the sacred and secular music of Western Europe during the Middle Ages, from approximately the 6th to 15th centuries. It is the first and longest major era of Western classical music and followed by the Renaissance ...
, only chords made of
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 ...
s and
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 ...
s (involving relationships among the first three
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 ...
s) were considered consonant. In the West, triadic harmony arose (
contenance angloise
The ''Contenance angloise'', or English manner, is a distinctive style of polyphony developed in fifteenth-century England which uses full, rich harmonies based on the third and sixth. It was highly influential in the fashionable Burgundian court ...
) around the time of the
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
, and
triads quickly became the fundamental building blocks of Western music. The
major
Major (commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicators ...
and
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 ...
s of these triads invoke relationships among the first five harmonics.
Around the turn of the 20th century,
tetrads
Tetrad ('group of 4') or tetrade may refer to:
* Tetrad (area), an area 2 km x 2 km square
* Tetrad (astronomy), four total lunar eclipses within two years
* Tetrad (chromosomal formation)
* Tetrad (general relativity), or frame field
** Tetrad fo ...
debuted as fundamental building blocks in
African-American music
African-American music is an umbrella term covering a diverse range of music and musical genres largely developed by African Americans and their culture. Their origins are in musical forms that first came to be due to the condition of slave ...
. In conventional music theory pedagogy, these
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 dominant seventh chord: a major triad together with a minor ...
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 seventh chord
In music theory, a dominant seventh chord, or major minor seventh chord, is a seventh chord, usually built on the fifth degree of the major scale, and composed of a root, major third, perfect fifth, and minor seventh. Thus it is a major triad tog ...
in
12-ET approximates 4:5:6:7, while the
major seventh chord
In music, a major seventh chord is a seventh chord in which the third is a major third above the root and the seventh is a major seventh above the root. The major seventh chord, sometimes also called a ''Delta chord'', can be written as maj7, M7, , ...
approximates 8:10:12:15.
Odd-limit and prime-limit
In
just intonation
In music, just intonation or pure intonation is the tuning of musical intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to ...
, intervals between pitches are drawn from the
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
. Since Partch, two distinct formulations of the limit concept have emerged: odd limit and prime limit. 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'' is a book first published in 1949 by microtonal composer Harry Partch (1901–1974).
Partch first presents a polemic against both equal temperament and the long history of stagnation in the teaching of music; according t ...
'', 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 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, ...
,
William Sethares
William A. Sethares (born April 19, 1955) is an American music theorist and professor of electrical engineering at the University of Wisconsin. In music, he has contributed to the theory of Dynamic Tonality and provided a formalization of consonan ...
, and
Paul Erlich
Paul Erlich (born 1972) 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 define pajara temperament Accessed 2013 ...
.
[Paul Erlich,]
The Forms of Tonality: A Preview
.
Some Music Theory from Paul Erlich
' (2001), pp. 1–3 (Accessed 29 May 2010).
See , below.
Identity
An identity is each of the
odd number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because
\begin
-2 \cdot 2 &= -4 \\
0 \cdot 2 &= 0 \\
41 ...
s below and including the (odd) limit in a tuning. For example, the identities included in 5-limit tuning are 1, 3, and 5. Each odd number represents a new pitch in the
harmonic series and may thus be considered an identity:
C C G C E G ...
2 4 6 8 10 12 ...
According to Partch: "The number 9, though not a
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, is nevertheless an identity in music, simply because it is an odd number." Partch defines "identity" as "one of the correlatives, '
major
Major (commandant in certain jurisdictions) is a military rank of commissioned officer status, with corresponding ranks existing in many military forces throughout the world. When used unhyphenated and in conjunction with no other indicators ...
' or '
minor
Minor may refer to:
* Minor (law), a person under the age of certain legal activities.
** A person who has not reached the age of majority
* Academic minor, a secondary field of study in undergraduate education
Music theory
*Minor chord
** Barb ...
', in a
tonality
Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is call ...
; one of the odd-number ingredients, one or several or all of which act as a pole of tonality".
''Odentity'' and ''udentity'' are short for ''over-identity'' and ''under-identity'', respectively. According to music software producer Tonalsoft: "An udentity is an identity of an
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 ...
".
Prime limit
![Missing fundamental rectangles](https://upload.wikimedia.org/wikipedia/commons/b/b9/Missing_fundamental_rectangles.png)
For a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''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 numerator and denominator both ''n''-
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
.
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
Gamelan () ( jv, ꦒꦩꦼꦭꦤ꧀, su, ᮌᮙᮨᮜᮔ᮪, ban, ᬕᬫᭂᬮᬦ᭄) is the traditional ensemble music of the Javanese, Sundanese, and Balinese peoples of Indonesia, made up predominantly of percussive instruments. T ...
, 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 Silver Harrison (May 14, 1917 – February 2, 2003) was an American composer, music critic, music theorist, painter, and creator of unique musical instruments. Harrison initially wrote in a dissonant, ultramodernist style similar to his form ...
. 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 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. These are included as the primary subset of what ...
. Such scales often contain ratios with very large numbers, that are nevertheless related by simple intervals to other notes in the scale.
Prime-limit tuning and intervals are often referred to using the term for the
numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
based on the limit. For example, 7-limit tuning and intervals are called septimal, 11-limit is called undecimal, and so on.
Examples
Beyond just intonation
In
musical temperament
In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system. Tempering is the ...
, 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
The diminished seventh chord is a four-note chord (a seventh chord) composed of a root note, together with a minor third, a diminished fifth, and a diminished seventh above the root: (1, 3, 5, 7). For example, the diminished seventh ...
in
12-ET) have several valid tunings in just intonation, their harmonic limit may be ambiguous.
See also
*
3-limit (Pythagorean) tuning
*
Five-limit tuning
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note ...
*
7-limit tuning
7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14:11 ...
*
Numerary nexus
In 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.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthol ...
*
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 ...
*
Tonality diamond
In 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.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", ''Harry Partch: An Anthol ...
*
Tonality flux ''Tonality flux'' is Harry Partch's termPartch, Harry (1949). ''Genesis of a Music'', p.188-190. Da Capo Press . for the kinds of subtle harmonic changes that can occur in a microtonal context from notes moving from one chord to another by tiny in ...
References
External links
"Limits: Consonance Theory Explained" ''Glen Peterson's Musical Instruments and Tuning Systems''.
"Harmonic Limit" ''Xenharmonic''.
{{Musical tuning
*
Harmony
Harry Partch