Twelve-tone technique

Not to be confused withtwelve-tone equal temperament.

Arnold Schoenberg, inventor of the twelve-tone technique

Thetwelve-tone technique—also known asdodecaphony,twelve-tone serialism,and (in British usage)twelve-note composition—is a method ofmusical composition.The technique is a means of ensuring that all 12 notes of thechromatic scaleare sounded as often as one another in a piece of music while preventing the emphasis of any one notethrough the use oftone rows,orderings of the 12pitch classes.All 12 notes are thus given more or less equal importance, and the music avoids being in akey.

The technique was first devised by Austrian composerJosef 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 20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such asAaron CoplandandIgor 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 ofserialism.

Schoenberg's fellow countryman and contemporary Hauer also developed a similar system using unorderedhexachordsortropes—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

Though most sources will say it was invented by Austrian composerArnold Schoenbergin 1921 and first described privately to his associates in 1923, in factJosef Matthias Hauerpublished 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 theSecond Viennese School—Alban Berg,Anton Webern,and Schoenberg himself. Although, another important composer in this period wasElisabeth Lutyenswho wrote more than 50 pieces using the serial method.

The twelve tone technique was preceded by "freely"atonalpieces of 1908–1923 which, though "free", often have as an "integrative element... a minute intervalliccell"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 ofAlexander 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, theostinato".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."

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.In Hauer's breakthrough pieceNomos,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 Khunercontrasted 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 bytonalharmonies".As such, twelve-tone music is usuallyatonal,and treats each of the 12semitonesof thechromatic scalewith equal importance, as opposed to earlier classical music which had treated some notes as more important than others (particularly thetonicand thedominant note).

The technique became widely used by the fifties, taken up by composers such asMilton Babbitt,Luciano Berio,Pierre Boulez,Luigi Dallapiccola,Ernst Krenek,Riccardo Malipiero,and, after Schoenberg's death,Igor Stravinsky.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 producingserial music.Some even subjected all elements of music to the serial process.

Charles Wuorinensaid 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 composerScott Bradley,best known for his musical scores for works likeTom & JerryandDroopy Dog,utilized the 12-tone technique in his work. Bradley described his use thus:

The Twelve-Tone System provides the 'out-of-this-world' progressions so necessary to under-write the fantastic and incredible situations which present-day cartoons contain.

An example of Bradley's use of the technique to convey building tension occurs in theTom & Jerryshort "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 composedtone poemsthat were performed in concert in California.

Rock guitaristRon Jarzombekused a twelve-tone system for composingBlotted Science'sextended playThe 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

Main article:Tone row

The basis of the twelve-tone technique is thetone row,an ordered arrangement of the twelve notes of thechromatic scale(the twelveequal temperedpitch classes). There are fourpostulatesor preconditions to the technique which apply to the row (also called asetorseries), on which a work or section is based:

1. The row is a specific ordering of all twelve notes of the chromatic scale (without regard tooctaveplacement).
2. No note is repeated within the row.
3. The row may be subjected tointerval-preservingtransformations—that is, it may appear ininversion(denoted I),retrograde(R), orretrograde-inversion(RI), in addition to its "original" orprimeform (P).
4. The row in any of its four transformations may begin on any degree of the chromatic scale; in other words it may be freelytransposed.(Transposition being an interval-preserving transformation, this is technically covered already by 3.) Transpositions are indicated by anintegerbetween 0 and 11 denoting the number of semitones: thus, if the original form of the row is denoted P₀,then P₁denotes its transposition upward by one semitone (similarly I₁is an upward transposition of the inverted form, R₁of the retrograde form, and RI₁of 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 aset formorrow form.Every row thus has up to 48 different row forms. (Some rows have fewer due tosymmetry;see the sections onderived rowsandinvariancebelow.)

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 theintervalsinverted(so that a risingminor thirdbecomes a falling minor third, or equivalently, a risingmajor sixth):

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 thechromatic scale,meaning that 47permutationsof 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 asinvariance.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).

Prime, retrograde, inverted, and retrograde-inverted forms of the ascending chromatic scale. P and RI are the same (to within transposition), as are R and I.

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 toharmony.

Schoenberg's annotated opening of hisWind QuintetOp. 26 shows the distribution of the pitches of the row among the voices and the balance between the hexachords, 1–6 and 7–12, in the principal voice and accompaniment

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 (seeserialism).

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 theprime series(P). Untransposed, it is notated as P₀.Given the twelvepitch classesof the chromatic scale, there are 12factorial(479,001,600) tone rows, although this is far higher than the number ofuniquetone 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:

• transpositionup or down, giving P_χ.
• reversing the order of the pitches, giving theretrograde(R)
• turning each interval direction to its opposite, giving theinversion(I).

The various transformations can be combined. These give rise to a set-complex of forty-eight forms of the set, 12 transpositions of thefourbasic forms: P, R, I, RI. The combination of the retrograde and inversion transformations is known as theretrograde inversion(RI).

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, afour-group,in its row and column headers:

P:	RI:	R:	I:
RI:	P	I	R
R:	I	P	RI
I:	R	RI	P

However, there are only a few numbers by which one maymultiplya row and still end up with twelve tones. (Multiplication is in any case not interval-preserving.)

Derivation

Main article:Derived row

Derivationis transforming segments of the full chromatic, fewer than 12 pitch classes, to yield a complete set, most commonly using trichords, tetrachords, and hexachords. Aderived setcan be generated by choosing appropriate transformations of anytrichordexcept 0,3,6, thediminished triad.A derived set can also be generated from anytetrachordthat excludes the interval class 4, amajor third,between any two elements. The opposite,partitioning,uses methods to create segments from sets, most often throughregistral difference.

Combinatoriality

Main article:Combinatoriality

Combinatorialityis 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

Invariantformations 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 byAnton WebernandArnold Schoenberg.

Invarianceis 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 ofmathematical invariance.George Perledescribes their use as "pivots" or non-tonal ways of emphasizing certainpitches.Invariant rows are alsocombinatorialandderived.

Cross partition

Aggregates spanning several local set forms in Schoenberg'sVon heute auf morgen.

See also:Derived row § Partition and mosaic

Across partitionis 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:

62	43	34	26
**	***	****	******
**	***	****	******
**	***	****
**	***
**
**

One possible realization out of many for theorder numbersof the 3⁴cross 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'sOp. 33aKlavierstückand also byBergbutDallapicollaused 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 usedcyclic permutation,or rotation, where the row is taken in order but using a different starting note. Stravinsky also preferred theinverse-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 isVariations for OrchestrabyArnold Schoenberg."Quiet", inLeonard Bernstein'sCandide,satirizes the method by using it for a song about boredom, andBenjamin Brittenused a twelve-tone row—a "tema seriale con fuga" —in hisCantata 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:

1. Hexachordalinversionalcombinatoriality
2. Aggregates
3. Linearsetpresentation
4. Partitioning
5. Isomorphic partitioning
6. Invariants
7. Hexachordallevels
8. Harmony,"consistent with and derived from the properties of the referential set"
9. Metre,established through "pitch-relational characteristics"
10. Multidimensionalset presentations.

See also

• List of dodecaphonic and serial compositions
• All-interval twelve-tone row
• All-interval tetrachord
• All-trichord hexachord
• Pitch interval
• List of tone rows and series

References

Sources

• Alegant, Brian. 2010.The Twelve-Tone Music of Luigi Dallapiccola.Eastman Studies in Music 76. Rochester, New York: University of Rochester Press.ISBN978-1-58046-325-6.
• Babbitt, Milton.1960. "Twelve-Tone Invariants as Compositional Determinants".The Musical Quarterly46, no. 2, Special Issue: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies (April): 246–259.doi:10.1093/mq/XLVI.2.246.JSTOR740374(subscription required).
• Babbitt, Milton. 1961. "Set Structure as a Compositional Determinant".Journal of Music Theory5, no. 1 (Spring): 72–94.JSTOR842871(subscription required).
• Benson, Dave. 2007Music: A Mathematical Offering.Cambridge and New York: Cambridge University Press.ISBN978-0-521-85387-3.
• Brett, Philip. "Britten, Benjamin."Grove Music Onlineed. 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.ISBN0-252-00454-X(cloth);ISBN0-252-06275-2(pbk).
• Crawford, Caroline andFelix Khuner.1996.Felix Khuner: A Violinist's Journey from Vienna's Kolisch Quartet to the San Francisco Symphony and Opera Orchestras,intro. by Tom Heimberg.Regional Oral History Office,The Bancroft Library.Berkeley: University of California.
• Haimo, Ethan. 1990.Schoenberg's Serial Odyssey: The Evolution of his Twelve-Tone Method, 1914–1928.Oxford [England] Clarendon Press; New York: Oxford University PressISBN0-19-315260-6.
• Hill, Richard S. 1936. "Schoenberg's Tone-Rows and the Tonal System of the Future".The Musical Quarterly22, no. 1 (January): 14–37.doi:10.1093/mq/XXII.1.14.JSTOR739013(subscription required).
• Lansky, Paul;George Perleand Dave Headlam. 2001. "Twelve-note Composition".The New Grove Dictionary of Music and Musicians,second edition, edited byStanley SadieandJohn 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.ISBN90-5356-765-8.Translation ofMuziek van de twintigste eeuw: een onderzoek naar haar elementen en structuur.Utrecht: Oosthoek, 1964. Third impression, Utrecht: Bohn, Scheltema & Holkema, 1977.ISBN90-313-0244-9.
• Loy, D. Gareth, 2007.Musimathics: The Mathematical Foundations of Music,Vol. 1. Cambridge, Massachusetts and London: MIT Press.ISBN978-0-262-12282-5.
• Neighbour, Oliver.1954. "The Evolution of Twelve-Note Music".Proceedings of the Royal Musical Association,volume 81, issue 1: 49–61.doi:10.1093/jrma/81.1.49
• 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.ISBN0-520-03395-7
• 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.ISBN978-0-520-07430-9.
• Reti, Rudolph.1958.Tonality, Atonality, Pantonality: A Study of Some Trends in Twentieth Century Music.Westport, Connecticut: Greenwood Press.ISBN0-313-20478-0.
• Rufer, Josef.1954.Composition with Twelve Notes Related Only to One Another,translated byHumphrey Searle.New York: The Macmillan Company. (Original German ed., 1952)
• Schoenberg, Arnold.1975.Style and Idea,edited byLeonard Steinwith translations by Leo Black. Berkeley & Los Angeles: University of California Press.ISBN0-520-05294-3.
  • 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 Music11, no. 2 (Spring–Summer): 257–264.JSTOR832323(subscription required).
• Spies, Claudio.1965. "Notes on Stravinsky'sAbraham and Isaac".Perspectives of New Music3, no. 2 (Spring–Summer): 104–126.JSTOR832508(subscription required).
• Whittall, Arnold.2008.The Cambridge Introduction to Serialism.Cambridge Introductions to Music. New York: Cambridge University Press.ISBN978-0-521-86341-4(cloth)ISBN978-0-521-68200-8(pbk).

Further reading

• Covach, John. 1992. "The Zwölftonspiel of Josef Matthias Hauer".Journal of Music Theory36, no. 1 (Spring): 149–84.JSTOR843913(subscription required).
• Covach, John. 2000. "Schoenberg's 'Poetics of Music', the Twelve-tone Method, and the Musical Idea". InSchoenberg and Words: The Modernist Years,edited by Russell A. Berman and Charlotte M. Cross, New York: Garland.ISBN0-8153-2830-3
• Covach, John. 2002, "Twelve-tone Theory". InThe Cambridge History of Western Music Theory,edited by Thomas Christensen, 603–627. Cambridge: Cambridge University Press.ISBN0-521-62371-5.
• Krenek, Ernst.1953. "Is the Twelve-Tone Technique on the Decline?"The Musical Quarterly39, 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.ISBN3-902796-03-0
• Sloan, Susan L. 1989. "Archival Exhibit: Schoenberg's Dodecaphonic Devices".Journal of the Arnold Schoenberg Institute12, no. 2 (November): 202–205.
• Starr, Daniel. 1978. "Sets, Invariance and Partitions".Journal of Music Theory22, no. 1 (Spring): 1–42.JSTOR843626(subscription required).
• Wuorinen, Charles.1979.Simple Composition.New York: Longman.ISBN0-582-28059-1.Reprinted 1991, New York: C. F. Peters.ISBN0-938856-06-5.

External links

• Twelve tone squareto find all combinations of a 12 tone sequence
• New Transformations: Beyond P, I, R, and RIby Larry Solomon
• Javascript twelve tone matrix calculator and tone row analyzer
• Matrix generator from musictheory.netby Ricci Adams
• Twelve-Tone Technique, A Quick Referenceby Dan Román
• Twelve Tones by mathemusician Vi HartonYouTube
• Dodecaphonic Knots and Topology of WordsbyFranck Jedrzejewski
• Database on tone rows and tropes