Twelve-tone technique

Twelve-tone technique (also dodecaphony, or twelve-note technique) is a method of musical composition devised by Arnold Schoenberg. Music using the technique is called twelve-tone music. The basic idea is this: there are 12 notes in the chromatic scale. 12-tone music doesn't allow you to hit any one note more often than any another, so the music is always using all the notes as often as possible. It's never in a key. In fact, it actively avoids being in a key.

Josef Matthias Hauer also developed a similar system using unordered hexachords, or tropes, at the exact same time and country but with no connection to Schoenberg. Other composers have created systematic use of the chromatic scale, but Schoenberg's method is historically most significant.

Schoenberg himself described the system as a "Method of Composing with Twelve Tones Which are Related Only with One Another" (Schoenberg 1975, 218). However, the common usage at the present time is to describe this method as a form of serialism.

The technique

The basis of twelve-tone technique is the tone row, an ordered arrangement of the twelve notes of the chromatic scale (the twelve equal tempered pitch classes). 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! (12 factorial, i.e. 479,001,600) unique tone rows.

When twelve-tone technique is strictly applied, a piece consists of statements of certain permitted transformations of the prime series. These statements may appear serially, or may overlap, giving rise to harmony.

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 (I): I(χ) = 12 - P_χ.

The various transformations can be combined. The combination of the retrograde and inversion transformations is known as the retrograde 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 in its row and column headers:

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

More recently, composers such as Charles Wuorinen have also used multiplication of the row. However, there are only a few numbers which one may multiply a row by and still end up with twelve tones. (See Multiplication (music).)

Suppose the prime series is as follows:

Then the retrograde is the prime series in reverse order:

The inversion is the prime series with the intervals inverted (so that a rising minor third becomes a falling minor third):

And the retrograde inversion is the inverted series in retrograde:

P, R, I and RI can each be started on any of the twelve notes of the chromatic scale, meaning that 47 permutations 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 tranposed 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).

Note that 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 this internal consistency both positively and negatively.

When rigorously applied, the technique demands that one statement of the tone row must be heard in full (otherwise known as aggregate completion) before another can begin. Adjacent notes in the row can be sounded at the same time, and the notes can appear in any octave, but the order of the notes in the tone row must be maintained. Durations, dynamics and other aspects of music other than the pitch can be freely chosen by the composer, and there are also no rules about which tone rows should be used at which time (beyond them all being derived from the prime series, as already explained).

Schoenberg's idea in developing the technique was for it to "replace those structural differentiations provided formerly by tonal harmonies" (Schoenberg 1975, 218). As such, twelve-tone music is usually atonal, and treats each of the 12 semitones of the chromatic scale 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).

History of the technique's use

Founded by Austrian composer Arnold Schoenberg in 1921 and first described privately to his associates in 1923 (Schoenberg 1975, 213), the method was used during the next 20 years almost exclusively by the Second Viennese School (Alban Berg, Anton Webern, Hanns Eisler and Arnold Schoenberg himself). 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 (Reti, 1958). The technique became widely used by the fifties, taken up by composers such as Luciano Berio, Pierre Boulez, Luigi Dallapiccola 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 producing serial music. Some even subjected all elements of music to the serial process.

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

Charles Wuorinen claimed 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." (Chase 1992, p.587)

Derivation

Derivation is transforming segments of the full chromatic, less 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 (Perle 1977, 91–93).

Other

Also, some composers have used cyclic permutation, or rotation, where the row is taken in order but using a different starting note.

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 (Brett 2007).

See also

• List of twelve-tone pieces

Further reading

• Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern by George Perle, ISBN 0-520-07430-0
• Simple Composition by Charles Wuorinen, ISBN 0-938856-06-5.

Sources

• Brett, Philip. "Britten, Benjamin." Grove Music Online ed. L. Macy (Accessed 08 January 2007), http://www.grovemusic.com.
• Chase, Gilbert. 1992. America's Music: From the Pilgrims to the Present. University of Illinois Press, ISBN 0-252-06275-2.
• Perle, George. 1977. Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern. Fourth Edition. Berkeley, Los Angeles, and London: University of California Press. ISBN 0-520-03395-7
• Reti, Rudolph. 1958. Tonality, Atonality, Pantonality: A study of some trends in twentieth century music. Westport, Connecticut: Greenwood Press. ISBN 0-313-20478-0.
• Rufer, Josef. 1954. Composition with Twelve Notes Related Only to One Another (trans. Humphrey Searle; 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. ISBN 0-520-05294-3.
  • 207–208 "Twelve-Tone Composition (1923)"
  • 213–14 "'Schoenberg's Tone-Rows' (1936)"
  • 214–45 "Composition with Twelve Tones (1) (1941)"
  • 245–49 "Composition with Twelve Tones (2) (c.1948)"

External links

• New Transformations: Beyond P, I, R, and RI by Larry Solomon
• Mathematical puzzle featuring dodecaphony by Paul Niquette
• Javascript twelve tone matrix calculator and tone row analyzer
• Matrix generator from musictheory.net by Ricci Adams