![Arnold Schoenberg la 1948](https://upload.wikimedia.org/wikipedia/commons/a/ad/Arnold_Schoenberg_la_1948.jpg)
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 original piece or work of music, either vocal or instrumental, the structure of a musical piece or to the process of creating or writing a new piece of music. People who create new compositions are called ...
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, a method for composing with all 12 notes of th ...
, 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-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...
(1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "
Second Viennese School
The Second Viennese School (german: Zweite Wiener Schule, Neue Wiener Schule) 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. ...
" composers, who were the primary users of the technique in the first decades of its existence. 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 as often as one another in a piece of music while preventing the emphasis of any one note
[Perle 1977, 2.] through the use of
tone row
In music, a tone row or note row (german: Reihe 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 ar ...
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
Key or The Key may refer to:
Common meanings
* Key (cryptography), a piece of information that controls the operation of a cryptography algorithm
* Key (lock), device used to control access to places or facilities restricted by a lock
* Key (map ...
. Over time, the technique increased greatly in popularity and eventually became widely influential on 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, composition teacher, writer, and later a conductor of his own and other American music. Copland was referred to by his peers and critics as "the Dean of American Com ...
and
Igor Stravinsky
Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential composers of the ...
, 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 Musical composition, composition using series of pitches, rhythms, dynamics, timbres or other elements of music, musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, thou ...
.
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 theor ...
s or ''
tropes
Trope or tropes may refer to:
Arts, entertainment, and media
* Trope (cinema), a cinematic convention for conveying a concept
* Trope (literature), a figure of speech or common literary device
* Trope (music), any of a variety of different things ...
''—but with no connection to Schoenberg's twelve-tone technique. Other composers have created systematic use of the chromatic scale, but Schoenberg's method is considered to be historically and aesthetically most significant.
History of use
Though most sources will say it was invented by Austrian composer
Arnold Schoenberg
Arnold Schoenberg or Schönberg (, ; ; 13 September 187413 July 1951) was an Austrian-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...
in 1921 and first described privately to his associates in 1923, in fact
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, a method for composing with all 12 notes of th ...
published his "law of the twelve tones" in 1919, requiring that all twelve chromatic notes sound before any note is repeated. The method was used during the next twenty years almost exclusively by the composers of the
Second Viennese School
The Second Viennese School (german: Zweite Wiener Schule, Neue Wiener Schule) 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. ...
—
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 Friedrich Wilhelm von Webern (3 December 188315 September 1945), better known as Anton Webern (), was an Austrian composer and conductor whose music was among the most radical of its milieu in its sheer concision, even aphorism, and stea ...
, and Schoenberg himself.
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 a s ...
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
Cell may also refer to:
Locations
* Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
" 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 (; russian: Александр Николаевич Скрябин ; – ) was a Russian composer and virtuoso pianist. Before 1903, Scriabin was greatly influenced by the music of Frédéric Chopin and composed ...
,
Igor Stravinsky
Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential composers of the ...
,
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 H ...
,
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 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 include ...
".
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 ( srp, Рудолф Рети, translit=Rudolf Reti; 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, ...
, 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.
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 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, ...
".
As such, twelve-tone music is usually
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 a s ...
, and treats each of the 12
semitone
A semitone, also called a 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 two adjacent no ...
s 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 ...
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
In music, the dominant is the fifth scale degree () of the diatonic scale. It is called the ''dominant'' because it is second in importance to the first scale degree, the tonic. In the movable do solfège system, the dominant note is sung as "S ...
).
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 is particularly noted for his Serialism, serial and electronic music.
Biography
Babbitt was born in Philadelphia t ...
,
Luciano Berio
Luciano Berio (24 October 1925 – 27 May 2003) was an Italian composer noted for his experimental work (in particular his 1968 composition ''Sinfonia'' and his series of virtuosic solo pieces titled ''Sequenza''), and for his pioneering work ...
,
Pierre Boulez
Pierre Louis Joseph Boulez (; 26 March 1925 – 5 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 Western classical music.
Born in Mont ...
,
Luigi Dallapiccola
Luigi Dallapiccola (February 3, 1904 – February 19, 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, Croa ...
,
Ernst Krenek
Ernst Heinrich Krenek (, 23 August 1900 – 22 December 1991) was an Austrian, later American, composer of Czech origin. He explored atonality and other modern styles and wrote a number of books, including ''Music Here and Now'' (1939), a study ...
,
Riccardo Malipiero
Riccardo Malipiero Jr. (24 July 1914 – 27 November 2003) was an Italian composer, pianist, critic, and music educator. He was awarded the gold medal by the city of Milan in 1977 and by the city of Varese in 1984.John C. G. Waterhouse : , Cont ...
, and, after Schoenberg's death,
Igor Stravinsky
Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential composers of the ...
. 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 Musical composition, composition using series of pitches, rhythms, dynamics, timbres or other elements of music, musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, thou ...
. 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 performed his works and other 20th-century music as pianist and conductor.
He composed more than ...
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 work like ''
Tom & Jerry
''Tom and Jerry'' is an American animated media franchise and series of comedy short films created in 1940 by William Hanna and Joseph Barbera. Best known for its 161 theatrical short films by Metro-Goldwyn-Mayer, the series centers on the riva ...
'' and ''
Droopy Dog
Droopy is an animated character from the golden age of American animation. He is an anthropomorphic white Basset Hound with a droopy face; hence his name. He was created in 1943 by Tex Avery for theatrical cartoon shorts produced by the Metro-G ...
'', 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
This is a complete list of the 164 shorts in the ''Tom and Jerry'' series produced and released between 1940 and 2014. Of these, 162 are theatrical shorts, one is a made-for-TV short, and one is a 2-minute sketch shown as part of a telethon.
...
", from 1953. 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, which illustrates or evokes the content of a poem, short story, novel, painting, landscape, or other (non-musical) source. The German term ''T ...
s that were performed in concert in California.
Rock guitarist
Ron Jarzombek
Ron Jarzombek is an American guitarist best known for his work with WatchTower, Spastic Ink, and Blotted Science, featuring Alex Webster of Cannibal Corpse. His most recent project is Terrestrial Exiled. He released a brand new song, "Beyon ...
used a twelve-tone system for composing
Blotted Science
Blotted Science is an instrumental progressive metal supergroup headed by Ron Jarzombek (Watchtower, Spastic Ink), bassist Alex Webster ( Cannibal Corpse) and drummer Hannes Grossmann (ex- Obscura, ex-Necrophagist). They formed under the name Mac ...
's
extended play
An extended play record, usually referred to as an EP, is a musical recording that contains more tracks than a single but fewer than an album or LP record. ''
The Animation of Entomology
Blotted Science is an instrumental progressive metal supergroup headed by Ron Jarzombek (Watchtower, Spastic Ink), bassist Alex Webster (Cannibal Corpse) and drummer Hannes Grossmann (ex- Obscura, ex-Necrophagist). They formed under the name Machin ...
''. 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 ''
tone row
In music, a tone row or note row (german: Reihe 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 ar ...
'', an ordered arrangement of the twelve 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 ...
(the twelve
equal tempered
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, w ...
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). There are four
postulates
An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
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 ( 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 ...
placement).
# No note is repeated within the row.
# The row may be subjected to
interval-preserving
transformations—that is, it may appear in ''
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
'' (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
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
. (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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
between 0 and 11 denoting the number of semitones: thus, if the original form of the row is denoted P
0, then P
1 denotes its transposition upward by one semitone (similarly I
1 is an upward transposition of the inverted form, R
1 of the retrograde form, and RI
1 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 (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
; 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
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval est ...
inverted (so that a rising
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 ...
becomes a falling minor third, or equivalently, a rising
major sixth
In music from Western culture, 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 ...
):
:
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
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 ...
, meaning that 47
permutations
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
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 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 ...
.
Needless to say, 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
serialism
In music, serialism is a method of Musical composition, composition using series of pitches, rhythms, dynamics, timbres or other elements of music, musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, thou ...
).
Properties of transformations
The tone row chosen as the basis of the piece is called the ''prime series'' (P). Untransposed, it is notated as P
0. Given the twelve
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 of the chromatic scale, there are 12
factorial
In mathematics, the factorial of a non-negative denoted is the 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 (n-1) \times (n-2) \t ...
(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.
* reversal in time, giving the ''
retrograde'' (R)
* reversal in pitch, giving the ''
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
'' (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
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 becomes the last, and ...
'' (''RI'').
:
thus, each cell in the following table lists the result of the transformations, a
four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
, 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 (also known as the minor flatted fifth) is a triad consisting of two minor thirds above the root. It is a minor triad with a lowered ( flattened) fifth. When using chord symbols, it may be indicated by the s ...
. A derived set can also be generated from any
tetrachord
In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency propo ...
that excludes the interval class 4, a
major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
, between any two elements. The opposite, ''partitioning'', uses methods to create segments from sets, most often through
registral difference
A register is the "height" or range of a note, set of pitches or pitch classes, melody, part, instrument, or group of instruments. A higher register indicates higher pitch.
*Example 1: Violins are in a higher register than cellos.
In woodwind ...
.
Combinatoriality
Combinatoriality
In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates (all twelve tones). Whittall, Arnold ...
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
Anton Webern
Anton Friedrich Wilhelm von Webern (3 December 188315 September 1945), better known as Anton Webern (), was an Austrian composer and conductor whose music was among the most radical of its milieu in its sheer concision, even aphorism, and stea ...
and
Arnold Schoenberg
Arnold Schoenberg or Schönberg (, ; ; 13 September 187413 July 1951) was an Austrian-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...
.
''Invariance'' is defined as the "properties of a set that are preserved under
ny given
NY most commonly refers to:
* New York (state), a state in the Northeastern United States
* New York City, the most populous city in the United States, located in the state of New York
NY, Ny or ny may also refer to:
Places
* North Yorkshire, ...
operation, 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 theorist. As a composer, his music was largely atonal, using methods similar to the twelve-tone technique of the Second Viennese School. This serialist style, an ...
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 an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
and
derived
Derive may refer to:
* Derive (computer algebra system), a commercial system made by Texas Instruments
* ''Dérive'' (magazine), an Austrian science magazine on urbanism
*Dérive, a psychogeographical concept
See also
*
*Derivation (disambiguatio ...
.
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 3
4 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)
*Berg Ng (born 1960), Hong Kong actor
* Berg (footballer) (born 1989), Brazilian footballer
Former states
*Berg (state), county and duchy of the Holy ...
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 (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...
, 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
Arnold Schoenberg
Arnold Schoenberg or Schönberg (, ; ; 13 September 187413 July 1951) was an Austrian-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...
. "Quiet", in
Leonard Bernstein
Leonard 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 the first America ...
'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 (22 November 1913 – 4 December 1976, aged 63) was an English composer, conductor, and pianist. He was a central figure of 20th-century British music, with a range of works including opera, other ...
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 theor ...
al
inversional combinatoriality
In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates (all twelve tones). Whittall, Arnold ...
#
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
Level or levels may refer to:
Engineering
*Level (instrument), a device used to measure true horizontal or relative heights
*Spirit level, an instrument designed to indicate whether a surface is horizontal or vertical
*Canal pound or level
*Regr ...
#
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 ...
, "consistent with and derived from the properties of the referential set"
#
Metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
, established through "pitch-relational characteristics"
#
Multidimensional set presentations.
See also
*
List of dodecaphonic and serial compositions
This is an incomplete list of musical pieces composed in the twelve-tone technique and pieces that use serialism.
List of musical pieces composed in the twelve-tone technique
* Second Viennese School
** Alban Berg
*** '' Kammerkonzert'', for p ...
*
All-interval twelve-tone row
In music, an all-interval twelve-tone row, series, or chord, is a twelve-tone tone row arranged so that it contains one instance of each interval within the octave, 1 through 11 (an ordering of every interval, 0 through 11, that contains each ...
*
All-interval tetrachord
An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords (to within inversion), when expressed in prime form. In set theory (music), s ...
*
All-trichord hexachord
In music, the all-trichord hexachord is a unique hexachord that contains all twelve trichords, or from which all twelve possible trichords may be derived. The prime form of this set class is and its Forte number is 6-Z17. Its complement is 6-Z43 ...
*
Pitch interval
In musical set theory, a pitch interval (PI or ip) is the number of semitones that separates one pitch from another, upward or downward.Schuijer, Michiel (2008). ''Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts'', Eastman Studie ...
*
List of tone rows and series
This is a list of tone rows and series. For a list of unordered collections, see set (music), Forte number, list of set classes, and trope (music).
Twelve tone rows
Other lengths
Fewer than twelve
;Five
;Six
;Eight
;Nine
;Ten
;Elev ...
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 Car ...
'' 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 c ...
'' 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 theo ...
'' 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". ''
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 Car ...
'' 22, no. 1 (January): 14–37. . .
*
Lansky, Paul;
George Perle
George Perle (6 May 1915 – 23 January 2009) was an American composer and music theorist. As a composer, his music was largely atonal, using methods similar to the twelve-tone technique of the Second Viennese School. This serialist style, an ...
and Dave Headlam. 2001. "Twelve-note Composition". ''
The New Grove Dictionary of Music and Musicians
''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 theo ...
'', second edition, edited by
Stanley Sadie
Stanley John Sadie (; 30 October 1930 – 21 March 2005) was an influential and prolific British musicologist, music critic, and editor. He was editor of the sixth edition of the '' Grove Dictionary of Music and Musicians'' (1980), which was publ ...
and
John Tyrrell. London: Macmillan.
*
Leeuw, Ton de. 2005. ''Music of the Twentieth Century: A Study of Its Elements and Structure'', translated from the Dutch by Stephen Taylor. Amsterdam: Amsterdam University Press. . Translation of ''Muziek van de twintigste eeuw: een onderzoek naar haar elementen en structuur''. Utrecht: Oosthoek, 1964. Third impression, Utrecht: Bohn, Scheltema & Holkema, 1977. .
* Loy, D. Gareth, 2007. ''Musimathics: The Mathematical Foundations of Music'', Vol. 1. Cambridge, Massachusetts and London: MIT Press. .
*
Neighbour, Oliver. 1954. "The Evolution of Twelve-Note Music". ''
Proceedings of the Royal Musical Association
In academia and librarianship, conference proceedings is a collection of academic papers published in the context of an academic conference or workshop. Conference proceedings typically contain the contributions made by researchers at the conferen ...
'', volume 81, issue 1: 49–61.
*
Perle, George. 1977. ''Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern'', fourth edition, revised. Berkeley, Los Angeles, and London: University of California Press.
* Perle, George. 1991. ''Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern'', sixth edition, revised. Berkeley: University of California Press. .
*
Reti, Rudolph. 1958. ''Tonality, Atonality, Pantonality: A Study of Some Trends in Twentieth Century Music''. Westport, Connecticut: Greenwood Press. .
*
Rufer, Josef. 1954. ''Composition with Twelve Notes Related Only to One Another'', translated by
Humphrey Searle
Humphrey Searle (26 August 1915 – 12 May 1982) was an English composer and writer on music. His music combines aspects of late Romanticism and modernist serialism, particularly reminiscent of his primary influences, Franz Liszt, Arnold Schoen ...
. New York: The Macmillan Company. (Original German ed., 1952)
*
Schoenberg, Arnold. 1975. ''Style and Idea'', edited by
Leonard Stein
Leonard David Stein (December 1, 1916 – June 24, 2004) was a musicologist, pianist, conductor, university teacher, and influential in promoting contemporary music on the American West Coast. He was for years Arnold Schoenberg's assistant, mus ...
with translations by Leo Black. Berkeley & Los Angeles: University of California Press. .
** 207–208 "Twelve-Tone Composition (1923)"
** 214–245 "Composition with Twelve Tones (1) (1941)"
** 245–249 "Composition with Twelve Tones (2) (c. 1948)"
* Solomon, Larry. 1973. "New Symmetric Transformations". ''
Perspectives of New Music
''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief).
''Perspectives'' was first ...
'' 11, no. 2 (Spring–Summer): 257–264. .
*
Spies, Claudio. 1965. "Notes on Stravinsky's ''Abraham and Isaac''". ''
Perspectives of New Music
''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief).
''Perspectives'' was first ...
'' 3, no. 2 (Spring–Summer): 104–126. .
*
Whittall, Arnold. 2008. ''The Cambridge Introduction to Serialism''. Cambridge Introductions to Music. New York: Cambridge University Press. (cloth) (pbk).
Further reading
* Covach, John. 1992. "The Zwölftonspiel of Josef Matthias Hauer". ''
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 c ...
'' 36, no. 1 (Spring): 149–84. .
* Covach, John. 2000. "Schoenberg's 'Poetics of Music', the Twelve-tone Method, and the Musical Idea". In ''Schoenberg and Words: The Modernist Years'', edited by Russell A. Berman and Charlotte M. Cross, New York: Garland.
* Covach, John. 2002, "Twelve-tone Theory". In ''The Cambridge History of Western Music Theory'', edited by Thomas Christensen, 603–627. Cambridge: Cambridge University Press. .
*
Krenek, Ernst. 1953. "Is the Twelve-Tone Technique on the Decline?" ''
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 Car ...
'' 39, no 4 (October): 513–527.
* Šedivý, Dominik. 2011. ''Serial Composition and Tonality. An Introduction to the Music of Hauer and Steinbauer'', edited by Günther Friesinger, Helmut Neumann and Dominik Šedivý. Vienna: edition mono.
* Sloan, Susan L. 1989.
Archival Exhibit: Schoenberg's Dodecaphonic Devices. ''Journal of the Arnold Schoenberg Institute'' 12, no. 2 (November): 202–205.
* Starr, Daniel. 1978. "Sets, Invariance and Partitions". ''
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 c ...
'' 22, no. 1 (Spring): 1–42. .
*
Wuorinen, Charles. 1979. ''Simple Composition''. New York: Longman. . Reprinted 1991, New York: C. F. Peters. .
External links
Twelve tone squareto find all combinations of a 12 tone sequence
by Larry Solomon
Javascript twelve tone matrix calculator and tone row analyzerby Ricci Adams
Twelve-Tone Technique, A Quick Referenceby Dan Román
*
Dodecaphonic Knots and Topology of Wordsby
Database on tone rows and tropes
{{Authority control
Arnold Schoenberg
12 (number)