The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of

musical composition Musical composition can refer to an Originality, original piece or work of music, either Human voice, vocal or Musical instrument, instrumental, the musical form, structure of a musical piece or to the process of creating or writing a new pie ...

. The technique is a means of ensuring that all 12 notes of the

chromatic scale The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...

are sounded equally often in a piece of music while preventing the emphasis of any one notePerle 1977, 2. through the use of

tone row In music, a tone row or note row ( or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometime ...

s, orderings of the 12

pitch class In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...

es. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. The technique was first devised by Austrian composer

Josef Matthias Hauer Josef Matthias Hauer (March 19, 1883 – September 22, 1959) was an Austrian composer and music theorist. He is best known for developing, independent of and a year or two before Arnold Schoenberg Arnold Schoenberg or Schönberg (13 Septembe ...

, who published his "law of the twelve tones" in 1919. In 1923,

Arnold Schoenberg Arnold Schoenberg or Schönberg (13 September 187413 July 1951) was an Austrian and American composer, music theorist, teacher and writer. He was among the first Modernism (music), modernists who transformed the practice of harmony in 20th-centu ...

(1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "

Second Viennese School The Second Viennese School () was the group of composers that comprised Arnold Schoenberg and his pupils, particularly Alban Berg and Anton Webern, and close associates in early 20th-century Vienna. Their music was initially characterized by late ...

" composers, who were the primary users of the technique in the first decades of its existence. Over time, the technique increased greatly in popularity and eventually became widely influential on Mid 20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such as

Aaron Copland Aaron Copland (, ; November 14, 1900December 2, 1990) was an American composer, critic, writer, teacher, pianist, and conductor of his own and other American music. Copland was referred to by his peers and critics as the "Dean of American Compos ...

and

Igor Stravinsky Igor Fyodorovich Stravinsky ( – 6 April 1971) was a Russian composer and conductor with French citizenship (from 1934) and American citizenship (from 1945). He is widely considered one of the most important and influential 20th-century c ...

, eventually adopted it in their music. Schoenberg himself described the system as a "Method of composing with twelve tones which are related only with one another".Schoenberg 1975, 218. It is commonly considered a form of

serialism In music, serialism is a method of composition using series of pitches, rhythms, dynamics, timbres or other musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though some of his contemporaries were also ...

. Schoenberg's fellow countryman and contemporary Hauer also developed a similar system using unordered

hexachord In music, a hexachord (also hexachordon) is a six- note series, as exhibited in a scale ( hexatonic or hexad) or tone row. The term was adopted in this sense during the Middle Ages and adapted in the 20th century in Milton Babbitt's serial t ...

s or '' tropes''—independent of Schoenberg's development of the twelve-tone technique. Other composers have created systematic use of the chromatic scale, but Schoenberg's method is considered to be most historically and aesthetically significant.

History of use

The twelve-tone technique is most often attributed to Austrian composer

. He recalls using it in 1921 and describing it to pupils two years later. Simultaneously,

was formulating a similar theory in his writings. In the second edition of his book ''Vom Wesen Des Musikalischen'' (On the Essence of Music, 1923), Hauer wrote that the law of the atonal melody requires all twelve tones to be played repeatedly. The method was used during the next twenty years almost exclusively by the composers of the

—

Alban Berg Alban Maria Johannes Berg ( ; ; 9 February 1885 – 24 December 1935) was an Austrian composer of the Second Viennese School. His compositional style combined Romantic lyricism with the twelve-tone technique. Although he left a relatively sma ...

Anton Webern Anton Webern (; 3 December 1883 – 15 September 1945) was an Austrian composer, conductor, and musicologist. His music was among the most radical of its milieu in its lyric poetry, lyrical, poetic concision and use of then novel atonality, aton ...

, and Schoenberg himself. Although, another important composer in this period was

Elisabeth Lutyens Agnes Elisabeth Lutyens, CBE (9 July 190614 April 1983) was an English composer. Early life and education Elisabeth Lutyens was born in London on 9 July 1906. She was one of the five children of Lady Emily Bulwer-Lytton (1874–1964), a me ...

who wrote more than 50 pieces using the serial method. The twelve tone technique was preceded by "freely"

atonal Atonality in its broadest sense is music that lacks a tonal center, or key. ''Atonality'', in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on ...

pieces of 1908–1923 which, though "free", often have as an "integrative element ... a minute intervallic

cell Cell most often refers to: * Cell (biology), the functional basic unit of life * Cellphone, a phone connected to a cellular network * Clandestine cell, a penetration-resistant form of a secret or outlawed organization * Electrochemical cell, a de ...

" which in addition to expansion may be transformed as with a tone row, and in which individual notes may "function as pivotal elements, to permit overlapping statements of a basic cell or the linking of two or more basic cells". The twelve-tone technique was also preceded by "nondodecaphonic serial composition" used independently in the works of

Alexander Scriabin Alexander Nikolayevich Scriabin, scientific transliteration: ''Aleksandr Nikolaevič Skrjabin''; also transliterated variously as Skriabin, Skryabin, and (in French) Scriabine. The composer himselused the French spelling "Scriabine" which was a ...

Béla Bartók Béla Viktor János Bartók (; ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt are regarded as Hunga ...

Carl Ruggles Carl Ruggles (born Charles Sprague Ruggles; March 11, 1876 – October 24, 1971) was an American composer, painter and teacher. His pieces employed "dissonant counterpoint", a term coined by fellow composer and musicologist Charles Seeger to ...

, and others.Perle 1977, 37. Oliver Neighbour argues that Bartók was "the first composer to use a group of twelve notes consciously for a structural purpose", in 1908 with the third of his fourteen bagatelles. "Essentially, Schoenberg and Hauer systematized and defined for their own dodecaphonic purposes a pervasive technical feature of 'modern' musical practice, the

ostinato In music, an ostinato (; derived from the Italian word for ''stubborn'', compare English ''obstinate'') is a motif or phrase that persistently repeats in the same musical voice, frequently in the same pitch. Well-known ostinato-based pieces inc ...

". Additionally, John Covach argues that the strict distinction between the two, emphasized by authors including Perle, is overemphasized:

The distinction often made between Hauer and the Schoenberg school—that the former's music is based on unordered hexachords while the latter's is based on an ordered series—is false: while he did write pieces that could be thought of as "trope pieces", much of Hauer's twelve-tone music employs an ordered series.

The "strict ordering" of the Second Viennese school, on the other hand, "was inevitably tempered by practical considerations: they worked on the basis of an interaction between ordered and unordered pitch collections."Whittall 2008, 24.

Rudolph Reti Rudolph Reti, also Réti (; November 27, 1885 – February 7, 1957), was a musical analyst, composer and pianist. He was the older brother of the chess master Richard Réti, but unlike his brother, Reti did not write his surname with an acute acc ...

, an early proponent, says: "To replace one structural force (tonality) by another (increased thematic oneness) is indeed the fundamental idea behind the twelve-tone technique", arguing it arose out of Schoenberg's frustrations with free atonality,Reti 1958 providing a "positive premise" for atonality. In Hauer's breakthrough piece ''Nomos'', Op. 19 (1919) he used twelve-tone sections to mark out large formal divisions, such as with the opening five statements of the same twelve-tone series, stated in groups of five notes making twelve five-note phrases.

Felix Khuner Felix Khuner (1906 – June 10, 1991) was the second violinist of the Kolisch Quartet. He joined the quartet, then the Wien Quartet, in 1926 when the quartet needed a new second violinist. Khuner was reluctant, but when he visited Rudolf Kolisch ...

contrasted Hauer's more mathematical concept with Schoenberg's more musical approach. Schoenberg's idea in developing the technique was for it to "replace those structural differentiations provided formerly by tonal

harmonies 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 ...

". As such, twelve-tone music is usually

, and treats each of the 12

semitone A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between ...

s of the

with equal importance, as opposed to earlier classical music which had treated some notes as more important than others (particularly the tonic and the dominant note). The technique became widely used by the fifties, taken up by composers such as

Milton Babbitt Milton Byron Babbitt (May 10, 1916 – January 29, 2011) was an American composer, music theorist, mathematician, and teacher. He was a Pulitzer Prize and MacArthur Fellowship recipient, recognized for his serial and electronic music. Biography ...

Luciano Berio Luciano Berio (24 October 1925 – 27 May 2003) was an Italian composer noted for his experimental music, experimental work (in particular his 1968 composition ''Sinfonia (Berio), Sinfonia'' and his series of virtuosic solo pieces titled ''Seque ...

Pierre Boulez Pierre Louis Joseph Boulez (; 26 March 19255 January 2016) was a French composer, conductor and writer, and the founder of several musical institutions. He was one of the dominant figures of post-war contemporary classical music. Born in Montb ...

Luigi Dallapiccola Luigi Dallapiccola (3 February 1904 – 19 February 1975) was an Italian composer known for his lyrical twelve-tone compositions. Biography Dallapiccola was born in Pisino d'Istria (at the time part of Austria-Hungary, current Pazin, Croati ...

Ernst Krenek Ernst Heinrich Krenek (, 23 August 1900 – 22 December 1991) was an Austrian, later American, composer. He explored atonality and other modern styles and wrote a number of books, including ''Music Here and Now'' (1939), a study of Johannes Ock ...

, Riccardo Malipiero, and, after Schoenberg's death,

. Some of these composers extended the technique to control aspects other than the pitches of notes (such as duration, method of attack and so on), thus producing

serial music In music, serialism is a method of composition using series of pitches, rhythms, dynamics, timbres or other musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though some of his contemporaries were also ...

. Some even subjected all elements of music to the serial process.

Charles Wuorinen Charles Peter Wuorinen (, ; June 9, 1938 – March 11, 2020) was an American composer of contemporary classical music based in New York City. He also performed as a pianist and conductor. Wuorinen composed more than 270 works: orchestral music, c ...

said in a 1962 interview that while "most of the Europeans say that they have 'gone beyond' and 'exhausted' the twelve-tone system", in America, "the twelve-tone system has been carefully studied and generalized into an edifice more impressive than any hitherto known." American composer Scott Bradley, best known for his musical scores for works like '' Tom & Jerry'' and '' Droopy Dog'', utilized the 12-tone technique in his work. Bradley described his use thus: An example of Bradley's use of the technique to convey building tension occurs in the ''Tom & Jerry'' short " Puttin' on the Dog", from 1944. In a scene where the mouse, wearing a dog mask, runs across a yard of dogs "in disguise", a chromatic scale represents both the mouse's movements, and the approach of a suspicious dog, mirrored octaves lower. Apart from his work in cartoon scores, Bradley also composed

tone poem A symphonic poem or tone poem is a piece of orchestral music, usually in a single continuous movement (music), movement, which illustrates or evokes the content of a poem, short story, novel, painting, landscape, or other (non-musical) source. T ...

s that were performed in concert in California. Rock guitarist Ron Jarzombek used a twelve-tone system for composing Blotted Science's

extended play An extended play (EP) is a Sound recording and reproduction, musical recording that contains more tracks than a Single (music), single but fewer than an album. Contemporary EPs generally contain up to eight tracks and have a playing time of 1 ...

'' The Animation of Entomology''. He put the notes into a clock and rearranged them to be used that are side by side or consecutive. He called his method "Twelve-Tone in Fragmented Rows."

Tone row

The basis of the twelve-tone technique is the ''

'', an ordered arrangement of the twelve notes of the

(the twelve

equal tempered An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system ...

es). There are four

postulates An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...

or preconditions to the technique which apply to the row (also called a ''set'' or ''series''), on which a work or section is based: # The row is a specific ordering of all twelve notes of the chromatic scale (without regard to

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 ...

placement). # No note is repeated within the row. # The row may be subjected to interval-preserving transformations—that is, it may appear in '' inversion'' (denoted I), '' retrograde'' (R), or '' retrograde-inversion'' (RI), in addition to its "original" or ''prime'' form (P). # The row in any of its four transformations may begin on any degree of the chromatic scale; in other words it may be freely transposed. (Transposition being an interval-preserving transformation, this is technically covered already by 3.) Transpositions are indicated by an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

between 0 and 11 denoting the number of semitones: thus, if the original form of the row is denoted P₀, then P₁ denotes its transposition upward by one semitone (similarly I₁ is an upward transposition of the inverted form, R₁ of the retrograde form, and RI₁ of the retrograde-inverted form). (In Hauer's system postulate 3 does not apply.) A particular transformation (prime, inversion, retrograde, retrograde-inversion) together with a choice of transpositional level is referred to as a ''set form'' or ''row form''. Every row thus has up to 48 different row forms. (Some rows have fewer due to

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

; see the sections on ''derived rows'' and ''invariance'' below.)

Example

Suppose the prime form of the row is as follows: : Then the retrograde is the prime form in reverse order: : The inversion is the prime form with the intervals inverted (so that a rising

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 ...

becomes a falling minor third, or equivalently, a rising

major sixth In music theory, a sixth is a musical interval encompassing six note letter names or staff positions (see Interval number for more details), and the major sixth is one of two commonly occurring sixths. It is qualified as ''major'' because it ...

): : And the retrograde inversion is the inverted row in retrograde: : P, R, I and RI can each be started on any of the twelve notes of the

, meaning that 47

permutations In mathematics, a permutation of a Set (mathematics), set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example ...

of the initial tone row can be used, giving a maximum of 48 possible tone rows. However, not all prime series will yield so many variations because transposed transformations may be identical to each other. This is known as ''invariance''. A simple case is the ascending chromatic scale, the retrograde inversion of which is identical to the prime form, and the retrograde of which is identical to the inversion (thus, only 24 forms of this tone row are available). In the above example, as is typical, the retrograde inversion contains three points where the sequence of two pitches are identical to the prime row. Thus the generative power of even the most basic transformations is both unpredictable and inevitable. Motivic development can be driven by such internal consistency.

Application in composition

Note that rules 1–4 above apply to the construction of the row itself, and not to the interpretation of the row in the composition. (Thus, for example, postulate 2 does not mean, contrary to common belief, that no note in a twelve-tone work can be repeated until all twelve have been sounded.) While a row may be expressed literally on the surface as thematic material, it need not be, and may instead govern the pitch structure of the work in more abstract ways. Even when the technique is applied in the most literal manner, with a piece consisting of a sequence of statements of row forms, these statements may appear consecutively, simultaneously, or may overlap, giving rise to

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 ...

. Durations, dynamics and other aspects of music other than the pitch can be freely chosen by the composer, and there are also no general rules about which tone rows should be used at which time (beyond their all being derived from the prime series, as already explained). However, individual composers have constructed more detailed systems in which matters such as these are also governed by systematic rules (see

Topography

Analyst Kathryn Bailey has used the term 'topography' to describe the particular way in which the notes of a row are disposed in her work on the dodecaphonic music of Webern. She identifies two types of topography in Webern's music: block topography and linear topography. The former, which she views as the 'simplest', is defined as follows: 'rows are set one after the other, with all notes sounding in the order prescribed by this succession of rows, regardless of texture'. The latter is more complex: the musical texture 'is the product of several rows progressing simultaneously in as many voices' (note that these 'voices' are not necessarily restricted to individual instruments and therefore cut across the musical texture, operating as more of a background structure).

Elisions, Chains, and Cycles

Serial rows can be connected through elision, a term that describes 'the overlapping of two rows that occur in succession, so that one or more notes at the juncture are shared (are played only once to serve both rows)'. When this elision incorporates two or more notes it creates a row chain; when multiple rows are connected by the same elision (typically identified as the same in set-class terms) this creates a row chain cycle, which therefore provides a technique for organising groups of rows.

Properties of transformations

The tone row chosen as the basis of the piece is called the ''prime series'' (P). Untransposed, it is notated as P₀. Given the twelve

es of the chromatic scale, there are 12

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

(479,001,600) tone rows, although this is far higher than the number of ''unique'' tone rows (after taking transformations into account). There are 9,985,920 classes of twelve-tone rows up to equivalence (where two rows are equivalent if one is a transformation of the other). Appearances of P can be transformed from the original in three basic ways: * transposition up or down, giving P_χ. * reversing the order of the pitches, giving the '' retrograde'' (R) * turning each interval direction to its opposite, giving the '' inversion'' (I). The various transformations can be combined. These give rise to a set-complex of forty-eight forms of the set, 12 transpositions of the ''four'' basic forms: P, R, I, RI. The combination of the retrograde and inversion transformations is known as the ''

retrograde inversion In music theory, retrograde inversion is a musical term that literally means "backwards and upside down": "The inverse of the series is sounded in reverse order." Retrograde reverses the order of the motif's pitches: what was the first pitch ...

'' (''RI''). : thus, each cell in the following table lists the result of the transformations, a four-group, in its row and column headers: : However, there are only a few numbers by which one may ''multiply'' a row and still end up with twelve tones. (Multiplication is in any case not interval-preserving.)

Derivation

''Derivation'' is transforming segments of the full chromatic, fewer than 12 pitch classes, to yield a complete set, most commonly using trichords, tetrachords, and hexachords. A derived set can be generated by choosing appropriate transformations of any

trichord In music theory, a trichord () is a group of three different pitch classes found within a larger group. A trichord is a contiguous three-note set from a musical scale or a twelve-tone row. In musical set theory there are twelve trichords given ...

except 0,3,6, the

diminished triad In music theory, a diminished triad is a triad (music), triad consisting of two minor thirds above the root (chord), root. It is a Minor chord, minor triad with a lowered (flat (music), flattened) Fifth (chord), fifth. When using Chord names and ...

. A derived set can also be generated from any

tetrachord In music theory, a tetrachord (; ) is a series of four notes separated by three interval (music), intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion (approx. 498 cent (m ...

that excludes the interval class 4, a

major third In music theory, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four Semitone, half steps or two ...

, between any two elements. The opposite, ''partitioning'', uses methods to create segments from sets, most often through registral difference.

Combinatoriality

Combinatoriality is a side-effect of derived rows where combining different segments or sets such that the pitch class content of the result fulfills certain criteria, usually the combination of hexachords which complete the full chromatic.

Invariance

''Invariant'' formations are also the side effect of derived rows where a segment of a set remains similar or the same under transformation. These may be used as "pivots" between set forms, sometimes used by

and

. ''Invariance'' is defined as the "properties of a set that are preserved under ny givenoperation, as well as those relationships between a set and the so-operationally transformed set that inhere in the operation", a definition very close to that of mathematical invariance.

George Perle George Perle (6 May 1915 – 23 January 2009) was an American composer and music theory, music theorist. As a composer, his music was largely atonality, atonal, using methods similar to the twelve-tone technique of the Second Viennese School. Th ...

describes their use as "pivots" or non-tonal ways of emphasizing certain pitches. Invariant rows are also

combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

and derived.

Cross partition

A ''cross partition'' is an often monophonic or homophonic technique which, "arranges the pitch classes of an aggregate (or a row) into a rectangular design", in which the vertical columns (harmonies) of the rectangle are derived from the adjacent segments of the row and the horizontal columns (melodies) are not (and thus may contain non-adjacencies). For example, the layout of all possible 'even' cross partitions is as follows:Alegant 2010, 21. : One possible realization out of many for the ''order numbers'' of the 3⁴ cross partition, and one variation of that, are: 0 3 6 9 0 5 6 e 1 4 7 t 2 3 7 t 2 5 8 e 1 4 8 9 Thus if one's tone row was 0 e 7 4 2 9 3 8 t 1 5 6, one's cross partitions from above would be: 0 4 3 1 0 9 3 6 e 2 8 5 7 4 8 5 7 9 t 6 e 2 t 1 Cross partitions are used in Schoenberg's Op. 33a ''Klavierstück'' and also by

Berg Berg may refer to: People *Berg (surname), a surname (including a list of people with the name) * General Berg (disambiguation) * Berg Ng (born 1960), Hong Kong actor * Berg (footballer, born 1963), Ninimbergue dos Santos Guerra, Brazilian footba ...

but Dallapicolla used them more than any other composer.

Other

In practice, the "rules" of twelve-tone technique have been bent and broken many times, not least by Schoenberg himself. For instance, in some pieces two or more tone rows may be heard progressing at once, or there may be parts of a composition which are written freely, without recourse to the twelve-tone technique at all. Offshoots or variations may produce music in which: * the full chromatic is used and constantly circulates, but permutational devices are ignored * permutational devices are used but not on the full chromatic Also, some composers, including Stravinsky, have used

cyclic permutation In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k ...

, or rotation, where the row is taken in order but using a different starting note. Stravinsky also preferred the inverse-retrograde, rather than the retrograde-inverse, treating the former as the compositionally predominant, "untransposed" form. Although usually atonal, twelve tone music need not be—several pieces by Berg, for instance, have tonal elements. One of the best known twelve-note compositions is '' Variations for Orchestra'' by

. "Quiet", in

Leonard Bernstein Leonard Bernstein ( ; born Louis Bernstein; August 25, 1918 – October 14, 1990) was an American conductor, composer, pianist, music educator, author, and humanitarian. Considered to be one of the most important conductors of his time, he was th ...

's ''

Candide ( , ) is a French satire written by Voltaire, a philosopher of the Age of Enlightenment, first published in 1759. The novella has been widely translated, with English versions titled ''Candide: or, All for the Best'' (1759); ''Candide: or, The ...

'', satirizes the method by using it for a song about boredom, and

Benjamin Britten Edward Benjamin Britten, Baron Britten of Aldeburgh (22 November 1913 – 4 December 1976) was an English composer, conductor, and pianist. He was a central figure of 20th-century British music, with a range of works including opera, o ...

used a twelve-tone row—a "tema seriale con fuga"—in his ''Cantata Academica: Carmen Basiliense'' (1959) as an emblem of academicism.Brett 2007.

Schoenberg's mature practice

Ten features of Schoenberg's mature twelve-tone practice are characteristic, interdependent, and interactive:Haimo 1990, 41. #

Hexachord In music, a hexachord (also hexachordon) is a six- note series, as exhibited in a scale ( hexatonic or hexad) or tone row. The term was adopted in this sense during the Middle Ages and adapted in the 20th century in Milton Babbitt's serial t ...

al inversional combinatoriality # Aggregates # Linear

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

presentation # Partitioning # Isomorphic partitioning # Invariants # Hexachordal levels #

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 ...

, "consistent with and derived from the properties of the referential set" #

Metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...

, established through "pitch-relational characteristics" #

Multidimensional In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or Mathematical object, object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a ...

set presentations.

References

Notes

Sources

* Alegant, Brian. 2010. ''The Twelve-Tone Music of Luigi Dallapiccola''. Eastman Studies in Music 76. Rochester, New York: University of Rochester Press. . * Babbitt, Milton. 1960. "Twelve-Tone Invariants as Compositional Determinants". ''

The Musical Quarterly ''The Musical Quarterly'' is the oldest academic journal on music in America. Originally established in 1915 by Oscar Sonneck, the journal was edited by Sonneck until his death in 1928. Sonneck was succeeded by a number of editors, including C ...

'' 46, no. 2, Special Issue: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies (April): 246–259. . . * Babbitt, Milton. 1961. "Set Structure as a Compositional Determinant". ''

Journal of Music Theory The ''Journal of Music Theory'' is a peer-reviewed academic journal specializing in music theory and analysis. It was established by David Kraehenbuehl (Yale University) in 1957. According to its website, " e ''Journal of Music Theory'' fosters co ...

'' 5, no. 1 (Spring): 72–94. . * Benson, Dave. 2007
Music: A Mathematical Offering
'. Cambridge and New York: Cambridge University Press. . * Brett, Philip. "Britten, Benjamin." ''

Grove Music Online ''The New Grove Dictionary of Music and Musicians'' is an encyclopedic dictionary of music and musicians. Along with the German-language '' Die Musik in Geschichte und Gegenwart'', it is one of the largest reference works on the history and t ...

'' ed. L. Macy (Accessed 8 January 2007) * Chase, Gilbert. 1987. ''America's Music: From the Pilgrims to the Present'', revised third edition. Music in American Life. Urbana: University of Illinois Press. (cloth); (pbk). * * Haimo, Ethan. 1990. ''Schoenberg's Serial Odyssey: The Evolution of his Twelve-Tone Method, 1914–1928''. Oxford nglandClarendon Press; New York: Oxford University Press . * Hill, Richard S. 1936. "Schoenberg's Tone-Rows and the Tonal System of the Future". ''

'' 22, no. 1 (January): 14–37. . . * Lansky, Paul;

and Dave Headlam. 2001. "Twelve-note Composition". ''

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External links

Twelve tone square
to find all combinations of a 12 tone sequence

by Larry Solomon
Javascript twelve tone matrix calculator and tone row analyzer

by Ricci Adams
Twelve-Tone Technique, A Quick Reference
by Dan Román *
Dodecaphonic Knots and Topology of Words
by
Database on tone rows and tropes
{{Authority control Arnold Schoenberg 12 (number) Josef Matthias Hauer