Twelve-tone technique

thumb|upright|[[Arnold Schoenberg, inventor of the twelve-tone technique]]

The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded equally often in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. 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, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" 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 and Igor Stravinsky, 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". It is commonly considered a form of serialism.

Schoenberg's fellow countryman and contemporary Hauer also developed a similar system using unordered hexachords 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 Arnold Schoenberg. He recalls using it in 1921 and describing it to pupils two years later. Simultaneously, Josef Matthias Hauer 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 Second Viennese School—Alban Berg, Anton Webern, and Schoenberg himself. However, another important composer in this period, Elisabeth Lutyens, wrote over 50 pieces using the serial method.

The twelve tone technique was preceded by "freely" atonal pieces of 1908–1923 which, though "free", often have as an "integrative element ... a minute intervallic cell" 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, Igor Stravinsky, Béla Bartók, Carl Ruggles, and others. 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".</blockquote>

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."

Rudolph Reti, 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, providing a "positive premise" for atonality.

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 poems that were performed in concert in California.

Rock guitarist Ron Jarzombek used a twelve-tone system for composing Blotted Science's extended play 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 tone row, an ordered arrangement of the twelve notes of the chromatic scale (the twelve equal tempered pitch classes). There are four postulates 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 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 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.)]]

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).

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 pitch classes of the chromatic scale, there are 12 factorial (479,001,600 Without definite starting and ending pitches, there are 836,017 distinct 12 tone cycles.

Appearances of P can be transformed from the original in three basic ways:

transposition up or down, giving Pχ.
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 (RI).

:{| class="wikitable"

|RI is:

|RI of P,

|R of I,

|and I of R.

|R is:

|R of P,

|RI of I,

|and I of RI.

|I is:

|I of P,

|RI of R,

|and R of RI.

|P is:

|R of R,

|I of I,

|and RI of RI.

thus, each cell in the following table lists the result of the transformations, a four-group, in its row and column headers:

:{| class="wikitable"

|P:

|RI:

|R:

|I:

|RI:

|R:

|RI

|I:

|RI

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 except 0,3,6, the diminished triad. A derived set can also be generated from any tetrachord that excludes the interval class 4, a major third, 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 Anton Webern and Arnold Schoenberg.

Invariance is defined as the "properties of a set that are preserved under [any given] 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 describes their use as "pivots" or non-tonal ways of emphasizing certain pitches. Invariant rows are also combinatorial and derived.

===Cross partition===

thumb|Aggregates spanning several local set forms in Schoenberg's [[Von heute auf morgen.]]

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:

:{|

|style="width:30px;"|62

|style="width:30px;"|43

|style="width:30px;"|34

|style="width:30px;"|26

|**||***||****||******

|**||***||****

|**||***

|**

One possible realization out of many for the order numbers of the 34 cross partition, and one variation of that, are:

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, 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. "Quiet", in Leonard Bernstein's Candide, satirizes the method by using it for a song about boredom, and Benjamin Britten used a twelve-tone row—a "tema seriale con fuga"—in his Cantata Academica: Carmen Basiliense (1959) as an emblem of academicism.

Schoenberg's mature practice

Ten features of Schoenberg's mature twelve-tone practice are characteristic, interdependent, and interactive:

Hexachordal inversional combinatoriality
Aggregates
Linear set presentation
Partitioning
Isomorphic partitioning
Invariants
Hexachordal levels
Harmony, "consistent with and derived from the properties of the referential set"
Metre, established through "pitch-relational characteristics"
Multidimensional 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 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 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 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 [England] Clarendon Press; New York: Oxford University Press .
Hill, Richard S. 1936. "Schoenberg's Tone-Rows and the Tonal System of the Future". The Musical Quarterly 22, no. 1 (January): 14–37. . .
Lansky, Paul; George Perle and Dave Headlam. 2001. "Twelve-note Composition". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie 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, 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. New York: The Macmillan Company. (Original German ed., 1952)
Schoenberg, Arnold. 1975. Style and Idea, edited by Leonard Stein 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 11, no. 2 (Spring–Summer): 257–264. .
Spies, Claudio. 1965. "Notes on Stravinsky's Abraham and Isaac". Perspectives of New Music 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).

External links

Twelve tone square to find all combinations of a 12 tone sequence
New Transformations: Beyond P, I, R, and RI by Larry Solomon
Javascript twelve tone matrix calculator and tone row analyzer
Matrix generator from musictheory.net 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