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In
music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
, limits or harmonic limits are a way of characterizing the
harmony In music, harmony is the concept of combining different sounds in order to create new, distinct musical ideas. Theories of harmony seek to describe or explain the effects created by distinct pitches or tones coinciding with one another; harm ...
found in a piece or
genre Genre () is any style or form 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 fo ...
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 every element of . Dually, a lower bound or minorant of is defined to be an element of that is less ...
on the complexity of harmony; hence the name.


The harmonic series and the evolution of music

Harry Partch, Ivor Darreg, and Ralph David Hill are among the many microtonalists to suggest that music has been slowly evolving to employ higher and higher
harmonics 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 harm ...
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, who styled it ''The Path to the New Music''. The phrase first appears in Schoenberg's 1926 essay "Opinio ...
). In
medieval music Medieval music encompasses the sacred music, sacred and secular music of Western Europe during the Middle Ages, from approximately the 6th to 15th centuries. It is the Dates of classical music eras, first and longest major era of Western class ...
, only chords made of
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
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 f ...
s (involving relationships among the first three
harmonic 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 har ...
s) were considered consonant. In the West, triadic harmony arose (
contenance angloise The ''Contenance angloise'', or English manner, a distinctive style of musical polyphony, developed in History of England, fifteenth-century England. It uses full, rich harmonies based on the Interval (music) , third and sixth. It became highly in ...
) around the time of the
Renaissance The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
, and triads quickly became the fundamental building blocks of Western music. The
major Major most commonly refers to: * Major (rank), a military rank * Academic major, an academic discipline to which an undergraduate student formally commits * People named Major, including given names, surnames, nicknames * Major and minor in musi ...
and
minor third In music theory, a minor third is a interval (music), musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval (music)#Number, interval numb ...
s of these triads invoke relationships among the first five harmonics. Around the turn of the 20th century, tetrads debuted as fundamental building blocks in
African-American music African-American music is a broad term covering a diverse range of musical genres largely developed by African Americans and their African-American culture, culture. Its origins are in musical forms that developed as a result of the Slavery in ...
. In conventional music theory pedagogy, these
seventh chord A seventh chord is a chord (music), chord consisting of a triad (music), triad plus a note forming an interval (music), interval of a Interval (music), seventh above the chord's root (chord), root. When not otherwise specified, a "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 seventh chord Domination or dominant may refer to: Society * World domination, structure where one dominant power governs the planet * Colonialism in which one group (usually a nation) invades another region for material gain or to eliminate competition * Ch ...
in 12-ET approximates 4:5:6:7 (albeit very poorly), 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 a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...
, 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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
. 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 ...
'', 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; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
,
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 consona ...
, and Paul Erlich.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 divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...
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 most commonly refers to: * Major (rank), a military rank * Academic major, an academic discipline to which an undergraduate student formally commits * People named Major, including given names, surnames, nicknames * Major and minor in musi ...
' or '
minor Minor may refer to: Common meanings * Minor (law), a person not under the age of certain legal activities. * Academic minor, a secondary field of study in undergraduate education Mathematics * Minor (graph theory), a relation of one graph to an ...
', in a
tonality Tonality is the arrangement of pitch (music), pitches and / or chord (music), chords of a musical work in a hierarchy of perceived ''relations'', ''stabilities'', ''attractions'', and ''directionality''. In this hierarchy, the single pitch or ...
; 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

For a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
''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 (; ; , ; ) is the traditional musical ensemble, ensemble music of the Javanese people, Javanese, Sundanese people, Sundanese, and Balinese people, Balinese peoples of Indonesia, made up predominantly of percussion instrument, per ...
, 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 for ...
. 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 wha ...
. 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 is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent differe ...
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 p ...
, 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 (chord), root note, together with a minor third, a diminished fifth, and a diminished seventh above the root: (1, 3, 5, 7). For example, the dim ...
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 not ...
*
7-limit tuning 7-limit or septimal Musical tuning#Tuning systems, tunings and interval (music), intervals are musical instrument tunings that have a limit (music), limit of seven: the largest prime number, prime factor contained in the interval ratios betwee ...
*
Numerary nexus In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is called the otonality and the other is called the utonality.Rasch, Rudolph (2000). "A Word or Two on the Tunings of Harry Partch", '' ...
*
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 musical tuning, tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is called the Otonality and Utonality, otonality and the other is called the utonality.Rasch, Rudolph (2000). "A Word or ...
*
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 harmony, harmonic changes that can occur in a microtonal music, microtonal context from notes moving from one ch ...


References


External links


"Limits: Consonance Theory Explained"
''Glen Peterson's Musical Instruments and Tuning Systems''.
"Harmonic Limit"
''Xenharmonic''. {{Musical tuning * Harmony Harry Partch