Twelve-tone technique

Last updated

Arnold Schoenberg, inventor of twelve-tone technique Arnold Schoenberg la 1948.jpg
Arnold Schoenberg, inventor of 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 first devised by Austrian composer Josef Matthias Hauer,[ not verified in body ] 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. 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 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 fellow countryman and contemporary Hauer also developed a similar system using unordered hexachords or tropes —but with no connection to Schoenberg's twelve-tone technique.[ contradictory ] 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

Though most sources will say[ when? ] it was invented by Austrian composer Arnold Schoenberg in 1921 and first described privately to his associates in 1923, in fact Josef Matthias Hauer published his "law of the twelve tones" in 1919, requiring that all twelve chromatic notes sound before any note is repeated. [8] [ failed verification ] The method was used during the next twenty years almost exclusively by the composers of the Second Viennese SchoolAlban Berg, Anton Webern, and Schoenberg himself.

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". [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 tonal harmonies". [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 said 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 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. [16]

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. [17] Apart from his work in cartoon scores, Bradley also composed tone poems that were performed in concert in California. [18]

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." [19]

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: [20]

  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.)


Suppose the prime form of the row is as follows:

Example tone row.png

Then the retrograde is the prime form in reverse order:

Retrograde tone row.png

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):

Inversion tone row.png

And the retrograde inversion is the inverted row in retrograde:

Retrograde inversion tone row.png

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).

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. P-R-I-RI.png
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 to harmony.

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 Schoenberg - Wind Quintet opening.png
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

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

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 [22] (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). [23]

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

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:


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 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 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.


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. [25]

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", [26] 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

Aggregates spanning several local set forms in Schoenberg's Von heute auf morgen. Aggregate Von Heute auf Morgen.png
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). [28]

For example, the layout of all possible 'even' cross partitions is as follows: [29]


One possible realization out of many for the order numbers of the 34 cross partition, and one variation of that, are: [29]

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. [30]


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:

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. [31]

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. [32]

Schoenberg's mature practice

Ten features of Schoenberg's mature twelve-tone practice are characteristic, interdependent, and interactive: [33]

  1. Hexachordal inversional combinatoriality
  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

Related Research Articles

In music, a tone row or note row, also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.

Atonality Music that lacks a tonal center or key

Atonality in its broadest sense is music that lacks a tonal center, or key. Atonality, in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on a single, central triad is not used, and the notes of the chromatic scale function independently of one another. More narrowly, the term atonality describes music that does not conform to the system of tonal hierarchies that characterized European classical music between the seventeenth and nineteenth centuries. "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments".

In music, serialism is a method of composition using series of pitches, rhythms, dynamics, timbres or other musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though some of his contemporaries were also working to establish serialism as a form of post-tonal thinking. Twelve-tone technique orders the twelve notes of the chromatic scale, forming a row or series and providing a unifying basis for a composition's melody, harmony, structural progressions, and variations. Other types of serialism also work with sets, collections of objects, but not necessarily with fixed-order series, and extend the technique to other musical dimensions, such as duration, dynamics, and timbre.

In music, the mystic chord or Prometheus chord is a six-note synthetic chord and its associated scale, or pitch collection; which loosely serves as the harmonic and melodic basis for some of the later pieces by Russian composer Alexander Scriabin. Scriabin, however, did not use the chord directly but rather derived material from its transpositions.

In music using the twelve-tone technique, derivation is the construction of a row through segments. A derived row is a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton Webern often used derived rows in his pieces. A partition is a segment created from a set through partitioning.

In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates. Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."

Complement (music)

In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.

Permutation (music)

In music, a permutation (order) of a set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. Different permutations may be related by transformation, through the application of zero or more operations, such as transposition, inversion, retrogradation, circular permutation, or multiplicative operations. These may produce reorderings of the members of the set, or may simply map the set onto itself.

Josef Matthias Hauer Austrian composer and music theorist

Josef Matthias Hauer was an Austrian composer and music theorist. He is best known for developing, independent of and a year or two before Arnold Schoenberg, a method for composing with all 12 notes of the chromatic scale. Hauer was also an important early theorist of twelve-tone music and composition.

Retrograde inversion

Retrograde inversion is a musical term that literally means "backwards and upside down": "The inverse of the series is sounded in reverse order." Retrograde reverses the order of the motif's pitches: what was the first pitch becomes the last, and vice versa. This is a technique used in music, specifically in twelve-tone technique, where the inversion and retrograde techniques are performed on the same tone row successively, "[t]he inversion of the prime series in reverse order from last pitch to first."

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.

An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords, when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b). The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].

Trope (music) Concepts in music

A trope or tropus may refer to a variety of different concepts in medieval, 20th-, and 21st-century music.

<i>Composition for Four Instruments</i>

Composition for Four Instruments (1948) is an early serial music composition written by American composer Milton Babbitt. It is Babbitt's first published ensemble work, following shortly after his Three Compositions for Piano (1947). In both these pieces, Babbitt expands upon the methods of twelve-tone composition developed by Arnold Schoenberg. He is notably innovative for his application of serial techniques to rhythm. Composition for Four Instruments is considered one of the early examples of “totally serialized” music. It is remarkable for a strong sense of integration and concentration on its particular premises—qualities that caused Elliott Carter, upon first hearing it in 1951, to persuade New Music Edition to publish it.

Fritz Heinrich Klein was an Austrian composer.

Post-tonal music theory is the set of theories put forward to describe music written outside of, or 'after', the tonal system of the common practice period. It revolves around the idea of 'emancipating dissonance', that is, freeing the structure of music from the familiar harmonic patterns that are derived from natural overtones. As music becomes more complex, dissonance becomes indistinguishable from consonance.

Suite for Piano (Schoenberg)

Arnold Schoenberg's Suite for Piano, Op. 25, is a 12-tone piece for piano composed between 1921 and 1923. The work is the earliest in which Schoenberg employs a row of "12 tones related only to one another" in every movement: the earlier 5 Stücke, Op. 23 (1920–23) employs a 12-tone row only in the final waltz movement, and the Serenade, Op. 24, uses a single row in its central Sonnet. The basic tone row of the suite consists of the following pitches: E–F–G–D–G–E–A–D–B–C–A–B.

<i>A Sermon, a Narrative and a Prayer</i>

A Sermon, a Narrative and a Prayer is a cantata for alto and tenor singers, a narrator, chorus, and orchestra by Igor Stravinsky, composed in 1960–61. It belongs to the composer’s serial period, and lasts a little over a quarter of an hour in performance.

The Tone Clock, and its related compositional theory Tone-Clock Theory, is a post-tonal music composition technique, developed by composers Peter Schat and Jenny McLeod. Music written using tone-clock theory features a high economy of musical intervals within a generally chromatic musical language. This is because tone-clock theory encourages the composer to generate all their harmonic and melodic material from a limited number of intervallic configurations. Tone-clock theory is also concerned with the way that the three-note pitch-class sets can be shown to underlie larger sets, and considers these triads as a fundamental unit in the harmonic world of any piece. Because there are twelve possible triadic prime forms, Schat called them the 'hours', and imagined them arrayed in a clock face, with the smallest hour in the 1 o'clock position, and the largest hour in the 12 o'clock position. A notable feature of Tone-Clock Theory is 'tone-clock steering': transposing and/or inverting hours so that each note of the chromatic aggregate is generated once and once only.

The String Quartet No.1 is a piece for two violins, viola and cello, composed by Robert Gerhard between 1951 and 1955, premiered at Dartington in 1956. This work marks a turning point in Gerhard's style and composition processes, because in one hand, he recovers some old techniques such as the sonata form in the first movement, along with others not as old like the 12-tone technique. Gerhard brilliantly develops, combines and transforms these resources along with new systematic processes created by himself, so that it leads to a new and broad theoretical framework that will be essential to his music thereafter.



  1. Whittall 2008, 26.
  2. 1 2 Perle 1991, 145.
  3. 1 2 Perle 1977, 2.
  4. 1 2 Schoenberg 1975, 218.
  5. Whittall 2008, 25.
  6. Leeuw 2005, 149.
  7. Leeuw 2005, 155–157.
  8. Schoenberg 1975, 213.
  9. Perle 1977, 9–10.
  10. 1 2 Perle 1977, 37.
  11. Neighbour 1955, 53.
  12. John Covach quoted in Whittall 2008, 24.
  13. 1 2 3 Whittall 2008, 24.
  14. Reti 1958
  15. Chase 1987, 587.
  16. Yowp (7 January 2017). "Tralfaz: Cartoon Composer Scott Bradley".
  17. Goldmark, Daniel (2007). Tunes for 'Toons: Music and the Hollywood Cartoon. Univ of California Press. p. 71. ISBN   978-0-520-25311-7.
  18. Scott Bradley at IMDb
  19. Mustein, Dave (2 November 2011). "Blotted Science's Ron Jarzombek: The Twelve-tone Metalsucks Interview". MetalSucks. Retrieved 19 January 2021.
  20. Perle 1977, 3.
  21. Whittall 2008, 52.
  22. Loy 2007, 310.
  23. Benson 2007, 348.
  24. Haimo 1990, 27.
  25. Perle 1977, 91–93.
  26. Babbitt 1960, 249–250.
  27. Haimo 1990, 13.
  28. Alegant 2010, 20.
  29. 1 2 Alegant 2010, 21.
  30. Alegant 2010, 22, 24.
  31. Spies 1965, 118.
  32. Brett 2007.
  33. Haimo 1990, 41.


Further reading

  • 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
  • Covach, John. 2002, "Twelve-tone Theory". In The Cambridge History of Western Music Theory, edited by Thomas Christensen, 603–627. Cambridge: Cambridge University Press. ISBN   0-521-62371-5.
  • Krenek, Ernst. 1953. "Is the Twelve-Tone Technique on the Decline?" The Musical Quarterly 39, 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. ISBN   3-902796-03-0
  • Sloan, Susan L. 1989. "Archival Exhibit: Schoenberg's Dodecaphonic Devices". Journal of the Arnold Schoenberg Institute 12, no. 2 (November): 202–205.
  • Starr, Daniel. 1978. "Sets, Invariance and Partitions". Journal of Music Theory 22, no. 1 (Spring): 1–42. JSTOR   843626 (subscription required).
  • Wuorinen, Charles. 1979. Simple Composition. New York: Longman. ISBN   0-582-28059-1. Reprinted 1991, New York: C. F. Peters. ISBN   0-938856-06-5.