Twelve-tone technique (also dodecaphony) is a method of musical composition devised by Arnold Schoenberg. Music using the technique is called twelve-tone music. 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 it is Schoenberg's method which is historically the most prevalent and considered easily the most important. Musical composition is: an original piece of music the structure of a musical piece the process of creating a new piece of music // A musical composition A piece of music exists in the form of a written composition in musical notation or as a single acoustic event (a live performance... Arnold Schoenberg, Los Angeles, 1948 For the American music critic and journalist, see Harold Charles Schonberg. ... Josef Mattias Hauer (March 19, 1883 – September 22, 1959) was an Austrian composer and music theorist. ... In music, a hexachord is a collection of six tones. ... // Linguistic usage A trope is a rhetorical figure of speech that consists of a play on words, i. ...

Schoenberg himself described the system as a "method of composing with 12 notes which are related only to one another".

The technique

The basis of twelve-tone technique is the tone row or set, an ordered arrangement of the twelve notes of the chromatic scale (the twelve equal tempered pitch classes), or, rather, an ordered arrangement of intervals which produce those notes. When the technique is strictly applied, an entire piece must be built up from statements of any transposition of this tone row in strict order or transformations of this row. Both melody and harmony may be created in this way. The set may be used in succession or simultaneously, the latter of which may be ordered up or down, or not. Given twelve pitch classes, there are 12! (12 factorial) possible tone rows, though invariance often reduces the number of distinct rows. In music, a tone row or note row is a permutation, an arrangement or ordering, of the twelve notes of the chromatic scale. ... Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as sets and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. ... The chromatic scale is any musical scale that contains more than one consecutive half-step (in other words two adjacent pairs of scale degrees or members which are separated by a semitone). ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... In music theory, an interval is the relationship between two notes or pitches, the lower and higher members of the interval. ... Transformation may refer to: In molecular biology: In genetics transformation involves the genetic alteration of a cell resulting from the introduction, uptake and expression of foreign DNA. In cell division, the transformation process converts normal cells into cells that will continue to divide without limit. ... Look up melody in Wiktionary, the free dictionary. ... Harmony is the use and study of pitch simultaneity and chords, actual or implied, in music. ... In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ...

The initial tone row, or set form, used is called the prime series (P), untransposed it is P₀. P can be used starting on any one of the twelve notes of the chromatic scale (P_χ) - so long as the intervals are the same, the rows are equivalent. P_χ = P₀ + χ. In music theory, an interval is the relationship between two notes or pitches, the lower and higher members of the interval. ...

Additionally, P can be transformed in three basic ways: it can be turned backwards to get the retrograde (R) or turned upsidedown to give the inversion (I) or the reverse contour direction. I(χ) = 12 - P_χ. These three transformative techniques can be combined to give the retrograde inversion (RI). As with the prime series, R, I and RI can be transposed to any note of the chromatic scale. This article is about retrograde motion. ... In music theory, the word inversion has several meanings. ... Counterpoint is a musical technique involving the simultaneous sounding of separate musical lines. ...

thus:

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. Multiplication is indicated by MX, X being the multiplier. As with the other compound operations multiplication is carried out and then transposition. P₀ = M1₀, I₀ = M11₀, M7₀=I(M5₀). Thus, for the untransposed form of all: Charles Wuorinen (born June 9, 1938 in New York City) is an American composer. ... In music and musical set theory, multiplication modulo 12 is a basic operation which may be performed on pitch or pitch class sets. ...

Even numbers remain unchanged under M7 and all odd numbers become transposed by a tritone. The chromatic scale may be mapped onto the circle of fourths with M5, and the circle of fifths with M7.

Suppose the prime series is as follows:

An example of a tone row. ...

Then the retrograde is the prime series in reverse order:

A tone row, the retrograde of Image:Example tone row. ...

The inversion is the prime series with the intervals inverted (so that a rising minor third becomes a falling minor third): In music theory, an interval is the relationship between two notes or pitches, the lower and higher members of the interval. ...

A tone row, the inversion of Image:Example tone row. ...

And the retrograde inversion is the inverted series in retrograde:

A tone row, the retrograde inversion of Image:Example tone row. ...

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). Categories: Wikipedia cleanup | Stub ... Invariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the...

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). In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... In music, dynamics refers to the volume or loudness of the sound or note, in particular to the range from soft (quiet) to loud. ...

Schoenberg's idea in developing the technique was for it to act as a replacement for tonal harmony as a basic grounding force for music. 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). Tonality is a system of writing music according to certain hierarchical pitch relationships around a center or tonic. ... Harmony is the use and study of pitch simultaneity and chords, actual or implied, in music. ... Atonality describes music that does not conform to the system of tonal hierarchies, which characterizes the sound of classical European music between the seventeenth and nineteenth centuries. ... The musical interval of a half step, semitone, or minor second is the relationship between the leading tone and the first note (the root or tonic) in a major scale. ... The chromatic scale is any musical scale that contains more than one consecutive half-step (in other words two adjacent pairs of scale degrees or members which are separated by a semitone). ... The tonic is the first note of a musical scale, and in the tonal method of music composition it is extremely important. ... In music, the dominant is the fifth degree of the scale. ...

History of the technique's use

Founded by Austrian composer Arnold Schoenberg around the late 1910s, 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. Arnold Schoenberg, Los Angeles, 1948 For the American music critic and journalist, see Harold Charles Schonberg. ... The Second Viennese School was a group of composers made up of Arnold Schoenberg and those who studied under him in early 20th century Vienna. ... Alban Maria Johannes Berg (February 9, 1885 – December 24, 1935) was an Austrian composer. ... Anton Webern (December 3, 1883 – September 15, 1945) was an Austrian composer. ... Hanns Eisler (July 6, 1898 - September 6, 1962) was a German and Austrian composer. ... Arnold Schoenberg, Los Angeles, 1948 For the American music critic and journalist, see Harold Charles Schonberg. ... Rudolph Réti (November 27, 1885 - February 7, 1957) was a musical analyst, composer and pianist. ... Luciano Berio (October 24, 1925 – May 27, 2003) was an Italian composer. ... Pierre Boulez Pierre Boulez (IPA: /pjɛʁ.buˈlÉ›z/) (born March 26, 1925) is a conductor and composer of classical music. ... Luigi Dallapiccola (February 3, 1904 – February 19, 1975) was an Italian composer known for his lyrical twelve-tone compositions. ... Igor Fyodorovitch Stravinsky (Russian: ) (June 17, 1882 – April 6, 1971) was a Russian-French-American composer of modern classical music. ... Serialism is a rigorous system of composing music in which various elements of the piece are ordered according to a pre-determined ordered set or sets, and variations on them. ...

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 is partitioning, the use of methods to create segments from sets, most often through registral difference. In music using the twelve tone technique a derived row is a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. ... In music, especially in musical set theory, a trichord is a collection of three pitch classes, often one of the four ordered trichords in a tone row or set form. ... Diminution, from Italian diminuimento, is a musical term used to mean different things in the context of melodies and intervals or chords. ... The tetrachord is a concept of music theory borrowed from ancient Greece. ... A major third is the larger of two commonly occuring musical intervals that span three diatonic scale degrees. ...

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. In music using the twelve tone technique 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, see George Perle. In music using the twelve tone technique invariance describes the portions of rows which have been so designed that they remain invariant under the allowable transformations (inversion, retrograde, retrograde-inversion, multiplication). ... Anton Webern (December 3, 1883 – September 15, 1945) was an Austrian composer. ... George Perle (born May 6, 1915 in Bayonne, New Jersey) is a composer and musicologist who has studied with Ernst Krenek. ...

Other

Also, some composers have used cyclic permutation, or rotation, where the row is taken in order but using a different starting note. A cyclic permutation is a permutation that shifts all elements of given ordered set by a fixed offset, with the elements shifted off the end inserted back at the beginning in the same order, i. ...

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. Arnold Schoenberg, Los Angeles, 1948 For the American music critic and journalist, see Harold Charles Schonberg. ... Leonard Bernstein in 1971 Leonard Bernstein (August 25, 1918 – October 14, 1990) was an American composer and orchestra conductor. ... Candide is a comic operetta by Leonard Bernstein, based on the novella of the same name by Voltaire. ...

Further reading

• Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern by George Perle, ISBN 0520074300
• Simple Composition by Charles Wuorinen, ISBN 0938856065.

Arnold Schoenberg, Los Angeles, 1948 For the American music critic and journalist, see Harold Charles Schonberg. ... Alban Maria Johannes Berg (February 9, 1885 – December 24, 1935) was an Austrian composer. ... Anton Webern (December 3, 1883 – September 15, 1945) was an Austrian composer. ... George Perle (born May 6, 1915 in Bayonne, New Jersey) is a composer and musicologist who has studied with Ernst Krenek. ... Charles Wuorinen (born June 9, 1938 in New York City) is an American composer. ...

Sources

• Chase, Gilbert (1992). America's Music: From the Pilgrims to the Present. University of Illinois Press, ISBN 0252062752.
• Reti, Rudolph (1958). Tonality, Atonality, Pantonality: A study of some trends in twentieth century music. Westport, Connecticut: Greenwood Press. ISBN 0313204780.

External links

Wikiquote has a collection of quotations related to:

Twelve-tone technique

• New Transformations: Beyond P, I, R, and RI by Larry Solomon
• Set Theory Glossary
• Twelve tone techniques