Note Row Music Definition Essay

Adaptations of the technique and aesthetic have become so integrated into concert music today that few listeners even notice anymore. Patrons at the season-opening gala of the Chamber Music Society of Lincoln Center at the Rose Theater last month might have been upset had they thought they were going to be confronted with some gnarly 12-tone piece. Yet the program included the premiere of Bruce Adolphe’s spiky “Crossing Broadway” for chamber ensemble, music that impishly blurs the distinctions between skittish 12-tone riffs and jazzy scat. The audience listened with pleasure and gave Mr. Adolphe a warm ovation.

The 12-tone movement was supposed to have engendered a revolution. In 1979 Mr. Wuorinen declared victory, at least in the realm of “serious” music. “While the tonal system, in an atrophied or vestigial form, is still used today in popular and commercial music, and even occasionally in the works of backwards-looking serious composers, it is no longer employed by serious composers of the mainstream,” he wrote. “It has been replaced or succeeded by the 12-tone system.”

Obviously this declaration was premature and ultimately wrong. Beginning in the mid-1960s a backlash emerged against 12-tone dogma. Minimalism, post-Minimalism and neo-Romanticism took root, along with various hard-to-label approaches to composition that found invigorating ways to combine tonal and atonal elements.

Before long audiences developed such animosity toward 12-tone music that its adherents, if not abandoning the technique, were disavowing the label. “If anyone writes program notes and says that I am a serial or a 12-tone composer, I am infuriated,” Donald Martino said in a 1997 interview. “I don’t want to prejudice people with that.”

By now the popular perception is that 12-tone music is passé if not dead. Yet the revolution may have surreptitiously succeeded, at least in part. Almost every composer of significance has had to come to grips with the method. You could say that once the 12-tone commando squad was defeated, the victors picked through the spoils, taking what they liked and ignoring the rest.

Now that decades of hostility are past, maybe it is time to reacknowledge the pervasive impact of this path-breaking development. It’s hard to ignore that at its worst the battle was debilitating. On one side were 12-tone composers who claimed the intellectual high ground, usually from secure posts in universities; on the other, composers who clung in various ways to tonal languages, cared about connecting with audiences and withstood the patronizing disdain of the tough-guy modernists in their midst. The critic Alex Ross tells the whole sorry story in his insightful, compulsively readable book, “The Rest Is Noise: Listening to the Twentieth Century,” coming out Oct. 23 from Farrar, Straus & Giroux.

Happily, the best young composers today feel entitled to borrow from anything and anyone, and more power to them. Talk to those young creators about the standoff between the Babbitts and the Coplands of 20th-century music, and they react as if you were trying to explain some archaic history, like the schism between the Brahms and Wagner wings of late 19th-century German music. Who cares?

From what I can tell the 12-tone technique is seldom adhered to strictly anymore. Even in Schoenberg’s day his followers, notably his devoted student Alban Berg, found deft ways to inject elements of haunting tonal harmony into astringent 12-tone scores, most amazingly in Berg’s unfinished opera “Lulu.”

Still, if the application of the technique is loose, the aesthetic of 12-tone music — its deliberately disorienting sound world, its burst-open harmonic palette, its leaping lines and every-which-way counterpoint, its gleeful avoidance of tonal centers — is very much alive among exciting composers of otherwise strikingly different styles. Think of Judith Weir, Stephen Hartke, Kaija Saariaho, Steven Stucky and Thomas Adès.

For those unversed in music theory it may be worth explaining with a little more specificity what 12-tone music is and how it came about.

Tonality is a means of organizing pitch in accordance with the physics of sound. A fundamental tone — say, C in a C major scale — is central; the other pitches relate to it in a hierarchy of importance based on natural overtone relationships. Whatever happens, the music keeps returning to that fundamental tonal mooring. Variety, expression and development result when a composer plays with expectations and introduces ambiguity, letting the music drift to remote pitches and chords that are not part of the basic major or minor scale.

As music developed in the late 19th century, Wagner, Mahler, Debussy, Strauss and other path breakers pushed at the boundaries of that mooring and weakened the pull of the tonal center. Ten years into the 20th century the whole business was in crisis, Schoenberg argued. .

So he started composing in a harmonic language unhinged from tonality: atonality, it has been called. His works in this style, Expressionistic pieces like “Erwartung,” sound as if they were conceived almost through harmonic free association.

Yet Schoenberg revered order, form and tradition. So he took a conceptual leap. If all 12 pitches in the octave are to be used more or less equally, why not devise a system that ensured a kind of equality?

Instead of the old tonal hierarchy, or his short-lived experiment in harmonic free-for-all, Schoenberg specified that the 12 pitches be put in an order, or row. Once a pitch was sounded, it was not to be repeated until the entire row had unfolded. There were countless ways around this dictum, however, because Schoenberg adapted his technique so that the row could be transposed, gone through backward or upside down, broken into smaller units that were mixed and matched, and so on. Wiggle room was built in from the start.

This description may make the technique sound like a rigid methodology, but Schoenberg found it liberating. “I find myself positively enabled to compose as freely and fantastically as one otherwise does only in one’s youth, and am nevertheless subject to a precisely definable aesthetic discipline,” he wrote to a colleague. Besides, tonal music relies on patterns too. Whole spans of pieces by Mozart and Beethoven are generated through default patterns of pitches: arpeggios, scale passages, chords and the like.

If the human ear is conditioned to find music with tonal moorings satisfying, if we are “embedded in a tonal universe,” as Leonard Bernstein once put it, then 12-tone music discombobulates those aural expectations and shakes up the universe. A Berlin critic of Schoenberg’s atonal works wrote indignantly that the music “kills tonal perception.” Exactly! And that’s what’s so exhilarating. You are invited to take a vacation from tonality, to experience music without a tonal safety net.

Other composers also found Schoenberg’s invention liberating. One branch took the systematizing principle radically further by placing rhythms and dynamics as well as pitches into predetermined series; hence the term serialism. In the 1960s and ’70s 12-tone music and serialism were treated like scientific disciplines by composers working within universities, where their research, to call it that, was of interest mainly to other composers.

Alas, there are many stories of gifted young composers who initially felt no choice but to adopt 12-tone technique. David Del Tredici, for one, eventually made the break and was in the vanguard of composers who rejected the dogma and reclaimed tonal languages. He emerged from that experience bitter.

William Bolcom went through this experience too, though with fewer psychic scars, from what he has said. He was drawn to the music of Mr. Boulez and Luciano Berio as a young man, but also to the music of Darius Milhaud, with whom he studied, as well as a wide range of American vernacular music. In time Mr. Bolcom too shed what he considered an academic approach to composing and developed a vibrantly eclectic language, rich with allusion to American song, jazz, ragtime and rock. Still, he and Mr. Del Tredici are stronger and more precise composers today for having been through the rigors of 12-tone composition.

Several giants explored 12-tone technique as well: not only Stravinsky, whose move into the enemy camp shook up the world of modern music, but also Messiaen and Copland. Whether their explorations were driven by a competitive desire to be in the front lines of modernism or by honest curiosity, each grew immensely from devising his own adaptations to the 12-tone system.

I don’t mean to romanticize 12-tone music, which has given us lots of terribly cerebral pieces. But prosaic, dull tonal works of every description continue to be written as well. I would much rather hear Mr. Babbitt’s scintillating 12-tone piano pieces than, say, the lushly tonal “Tempest Fantasy” by the Poulenc-infatuated Paul Moravec, a piece that beat out distinguished works by Peter Lieberson and Steve Reich for the 2004 Pulitzer Prize in music.

The refreshing lack of dogmatism among the new generation of composers seems to have spread to audiences as well. The brilliant pianist Pierre-Laurent Aimard segues from Beethoven to Boulez, from Liszt to Ligeti on the same program, and today’s audiences just follow along, open to everything. As Elliott Carter, the dean of modernist composers, approaches his 99th birthday, he keeps challenging us with complex and ingenious new scores and is cheered by young and old at every premiere.

I doubt that Mr. Wuorinen spends much time regretting dogmatic pronouncements he made in the heat of the battle, since he is far too busy enjoying the recent burst of enthusiasm for his music, thanks in part to champions like the conductor James Levine and the pianist Peter Serkin. Certainly Mr. Wuorinen’s Fourth Piano Concerto is evidence of a stunningly complex approach to writing music.

But in a brilliant performance by Mr. Serkin with Mr. Levine and the Boston Symphony Orchestra at Tanglewood in 2005, that concerto came across as audacious. You stopped thinking about tone rows and responded to the playfulness and ferocity of this formidable music.

In retrospect the major flaw in Schoenberg’s astute analysis of the crisis of tonality was the notion that pursuing an alternative had become a historical necessity. The development of 12-tone technique was no necessity. During the same period Bartok, Stravinsky and other giants were finding enthralling ways to adapt, transform and shake up tonality.

Instead Schoenberg’s great adventure was, as Mr. Ross puts it in his new book, “one man’s leap into the unknown.” But what a leap!

Continue reading the main story

Twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition devised by Austrian composer Arnold Schoenberg (1874–1951) and associated with the "Second Viennese School" 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 are sounded as often as one another in a piece of music while preventing the emphasis of any one note[3] 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. 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 even actively opposed the technique, such as Aaron Copland and Igor Stravinsky,[clarification needed] 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".[4] It is commonly considered a form of serialism.

Schoenberg's countryman and contemporary Josef Matthias Hauer also developed a similar system using unordered hexachords or tropes—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.[5]

History of use[edit]

Invented by Austrian composer Arnold Schoenberg in 1921 and first described privately to his associates in 1923,[8] the method was used during the next twenty years almost exclusively by the composers of the Second Viennese School—Alban Berg, Anton Webern, Hanns Eisler and Schoenberg himself.

The twelve tone technique was preceded by "freely" atonal pieces of 1908–23 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".[9] 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.[10] 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.[11] "Essentially, Schoenberg and Hauer systematized and defined for their own dodecaphonic purposes a pervasive technical feature of 'modern' musical practice, the ostinato".[10] 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.[12]

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."[13]

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,[14][page needed] providing a "positive premise" for atonality.[3] 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.[13]

Schoenberg's idea in developing the technique was for it to "replace those structural differentiations provided formerly by tonalharmonies".[4] 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).

The technique became widely used by the fifties, taken up by composers such as Milton 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 producing serial music. Some even subjected all elements of music to the serial process.

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."[15]

American composer Scott Bradley, best known for his musical scores for work like Tom & Jerry and Droopy Dog, utilized the 12-tone technique in his work. Bradley had learned the concept as a student of Schoenberg.[16] 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.[17]

Also a serious composer, Bradley composed tone poems that were performed in concert in California.[18] An example of Bradley's use of the technique to convey building tension occurs in the Tom & Jerry short Puttin' on the Dog, 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.[19][20]

Tone row[edit]

Main article: 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 temperedpitch 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:[21]

  1. The row is a specific ordering of all twelve notes of the chromatic scale (without regard to octave placement).
  2. No note is repeated within the row.
  3. 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).
  4. 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.[2])

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; see the sections on derived rows and invariance below.)

Example[edit]

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 inverted (so that a rising minor third becomes a falling minor third, or equivalently, a rising major 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 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 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[edit]

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.

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).

Properties of transformations[edit]

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![23]) (factorial, i.e. 479,001,600[13]) 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).[24]

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

  • transposition up or down, giving Pχ.
  • reversal in time, giving the retrograde (R)
  • reversal in pitch, 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).

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:

P:RI:R:I:
RI:PIR
R:IPRI
I:RRIP

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[edit]

Main article: Derived row

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[edit]

Main article: 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[edit]

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.[26]

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",[27] 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[edit]

See also: Derived row § Partition and mosaic

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).[29]

For example, the layout of all possible 'even' cross partitions is as follows:[30]

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

One possible realization out of many for the order numbers of the 34 cross partition, and one variation of that, are:[30]

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 but Dallapicolla used them more than any other composer.[31]

Other[edit]

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.[32]

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.[33]

Schoenberg's mature practice[edit]

Ten features of Schoenberg's mature twelve-tone practice are characteristic, interdependent, and interactive:[34]

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

See also[edit]

References[edit]

Notes[edit]

  1. ^Whittall 2008, 26.
  2. ^ abPerle 1991, 145.
  3. ^ abPerle 1977, 2.
  4. ^ abSchoenberg 1975, 218.
  5. ^Whittall 2008, 25.
  6. ^Leeuw 2005, 149.
  7. ^Leeuw 2005, 155–57.
  8. ^Schoenberg 1975, 213.
  9. ^Perle 1977, 9–10.
  10. ^ abPerle 1977, 37.
  11. ^Neighbour 1955, 53.
  12. ^John Covach quoted in Whittall 2008, 24.
  13. ^ abcWhittall 2008, 24.
  14. ^Reti 1958
  15. ^Chase 1987, 587.
  16. ^AllMusic Biography: Scott Bradley
  17. ^Cartoon Composer Scott Bradley
  18. ^IMDB Biography, Scott Bradley
  19. ^Tom and Jerry Episode 16 Puttin' on the Dog Part 2
  20. ^"bradley+repeats+the+scale+immediately"+piccolo+bassoon Tunes for ’Toons: Music and the Hollywood Cartoon
  21. ^Perle 1977, 3.
  22. ^Whittall 2008, 52.
  23. ^Loy 2007, 310.
  24. ^Benson 2007, 348.
  25. ^Haimo 1990, 27.
  26. ^Perle 1977, 91–93.
  27. ^Babbitt 1960, 249–50.
  28. ^Haimo 1990, 13.
  29. ^Alegant 2010, 20.
  30. ^ abAlegant 2010, 21.
  31. ^Alegant 2010, 22 and 24.
  32. ^Spies 1965, 118.
  33. ^Brett 2007.
  34. ^Haimo 1990, 41.

Sources[edit]

  • Alegant, Brian. 2010. The Twelve-Tone Music of Luigi Dallapiccola. Eastman Studies in Music 76. Rochester, NY: University of Rochester Press. ISBN 978-1-58046-325-6.
  • Babbitt, Milton. 1960. "Twelve-Tone Invariants as Compositional Determinants". Musical Quarterly 46, no. 2, Special Issue: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies (April): 246–59. doi:10.1093/mq/XLVI.2.246. JSTOR 740374(subscription required).
  • Babbitt, Milton. 1961. "Set Structure as a Compositional Determinant". Journal of Music Theory 5, no. 1 (Spring): 72–94. JSTOR 842871(subscription required).
  • Benson, Dave. 2007 Music: A Mathematical Offering. Cambridge and New York: Cambridge University Press. ISBN 978-0-521-85387-3.
  • Brett, Philip. "Britten, Benjamin." Grove Music Online ed. L. Macy (Accessed 8 January 2007), http://www.grovemusic.com.
  • Chase, Gilbert. 1987. America's Music: From the Pilgrims to the Present, revised third edition. Music in American Life. Urbana: University of Illinois Press. ISBN 0-252-00454-X (cloth); ISBN 0-252-06275-2 (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 ISBN 0-19-315260-6.
  • Hill, Richard S. 1936. "Schoenberg's Tone-Rows and the Tonal System of the Future". Musical Quarterly 22, no. 1 (January): 14–37. doi:10.1093/mq/XXII.1.14. JSTOR 739013(subscription required).
  • 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 Publishers.
  • 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. ISBN 90-5356-765-8. Translation of Muziek van de twintigste eeuw: een onderzoek naar haar elementen en structuur. Utrecht: Oosthoek, 1964. Third impression, Utrecht: Bohn, Scheltema & Holkema, 1977. ISBN 90-313-0244-9.
  • Loy, D. Gareth, 2007. Musimathics: The Mathematical Foundations of Music, Vol. 1. Cambridge, Mass. and London: MIT Press. ISBN 978-0-262-12282-5.
  • 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. ISBN 0-520-03395-7
  • 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. ISBN 978-0-520-07430-9.
  • 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, 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. ISBN 0-520-05294-3.
    • 207–208 "Twelve-Tone Composition (1923)"
    • 214–45 "Composition with Twelve Tones (1) (1941)"
    • 245–49 "Composition with Twelve Tones (2) (c.1948)"
  • Solomon, Larry. 1973. "New Symmetric Transformations". Perspectives of New Music 11, no. 2 (Spring-Summer): 257–64. JSTOR 832323(subscription required).
  • Spies, Claudio. 1965. "Notes on Stravinsky's Abraham and Isaac". Perspectives of New Music 3, no. 2 (Spring–Summer): 104–26. JSTOR 832508(subscription required).
  • Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music. New York: Cambridge University Press. ISBN 978-0-521-86341-4 (cloth) ISBN 978-0-521-68200-8 (pbk).

Further reading[edit]

  • Covach, John. 1992. "The Zwölftonspiel of Josef Matthias Hauer". Journal of Music Theory 36, no. 1 (Spring): 149–84. JSTOR 843913(subscription required).
  • 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. ISBN 0-8153-2830-3
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  • 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. ISBN 3-902796-03-0
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External links[edit]

Schoenberg, inventor of twelve-tone technique
Josef Matthias Hauer's "athematic" dodecaphony in Nomos Op. 19[1]( Play (help·info))
The principal forms, P1 and I6, of Schoenberg's Piano Piece, op. 33a, tone row  Play (help·info) feature hexachordal combinatoriality and contains three perfect fifths each, which is the relation between P1 and I6 and a source of contrast between, "accumulations of 5ths," and, "generally more complex simultaneity".[7] For example group A consists of B♭-C-F-B♮ while the, "more blended," group B consists of A-C♯-D♯-F♯.
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.
Schoenberg's annotated opening of his Wind Quintet Op. 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[22]
Hexachord invariance in Schoenberg's Concerto for Violin.[25] Play (help·info) The last hexachord of P0 (C–C♯–G–A♭–D–F) contains the same pitches as the first hexachord of I5 (D–C♯–A♭–C–G–F).

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