Tone row

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In music, a tone row or note row (German : Reihe or Tonreihe), also series or set, [1] 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.

Music form of art using sound

Music is an art form and cultural activity whose medium is sound organized in time. General definitions of music include common elements such as pitch, rhythm, dynamics, and the sonic qualities of timbre and texture. Different styles or types of music may emphasize, de-emphasize or omit some of these elements. Music is performed with a vast range of instruments and vocal techniques ranging from singing to rapping; there are solely instrumental pieces, solely vocal pieces and pieces that combine singing and instruments. The word derives from Greek μουσική . See glossary of musical terminology.

German language West Germanic language

German is a West Germanic language that is mainly spoken in Central Europe. It is the most widely spoken and official or co-official language in Germany, Austria, Switzerland, South Tyrol (Italy), the German-speaking Community of Belgium, and Liechtenstein. It is also one of the three official languages of Luxembourg and a co-official language in the Opole Voivodeship in Poland. The languages which are most similar to German are the other members of the West Germanic language branch: Afrikaans, Dutch, English, the Frisian languages, Low German/Low Saxon, Luxembourgish, and Yiddish. There are also strong similarities in vocabulary with Danish, Norwegian and Swedish, although those belong to the North Germanic group. German is the second most widely spoken Germanic language, after English.

Set theory (music) branch of music theory that categorizes musical objects and describes their relationships by using sets and permutations of pitches and pitch classes, rhythmic onsets, beat classes, etc.

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.

Contents

History and usage

Tone row of Karlheinz Stockhausen's Gruppen fur drei Orchester
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the registrally fixed pitches of which correspond with duration units and metronome marks. Stockhausen Gruppen fur drei Orchester series.png
Tone row of Karlheinz Stockhausen's Gruppen für drei Orchester Loudspeaker.svg   Play   the registrally fixed pitches of which correspond with duration units and metronome marks.

Tone rows are the basis of Arnold Schoenberg's twelve-tone technique and most types of serial music. Tone rows were widely used in 20th-century contemporary music, e.g. Dmitri Shostakovich's use of twelve-tone rows, "without dodecaphonic transformations", [3] [4] though one has been identified in the A minor prelude from book II of The Well-Tempered Clavier (1742) by J. S. Bach, [5] and by the late eighteenth century was a well-established technique, found in works such as Mozart's C major String Quartet, K. 157 (1772), String Quartet in E-flat major, K. 428, String Quintet in G minor, K. 516 (1790), and the Symphony in G minor, K. 550 (1788). [6] Beethoven also used the technique but, on the whole, "Mozart seems to have employed serial technique far more often than Beethoven". [7] Hans Keller claims that Schoenberg was aware of this serial practice in the classical period, and that "Schoenberg repressed his knowledge of classical serialism because it would have injured his narcissism." [8]

Arnold Schoenberg Austrian-American composer

Arnold Schoenberg or Schönberg was an Austrian, and later American, composer, music theorist, teacher, writer, and painter. He was associated with the expressionist movement in German poetry and art, and leader of the Second Viennese School. With the rise of the Nazi Party, Schoenberg's works were labeled degenerate music, because they were modernist and atonal. He immigrated to the United States in 1934.

Twelve-tone technique method of musical composition devised by Arnold Schönberg to ensure that all 12 notes of the chromatic scale are equally often, so that the music avoids being in a key

The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and 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 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, eventually adopted it in their music.

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.

Theory and compositional techniques

Principal forms of the tone row of Anton Webern's Variations for piano, op. 27. Each hexachord fills in a chromatic fourth, with B as the pivot (end of P1 and beginning of IR8), and thus linked by the prominent tritone in the center of the row
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. Webern - Piano Variations op. 27 tone row.png
Principal forms of the tone row of Anton Webern's Variations for piano, op. 27. Each hexachord fills in a chromatic fourth, with B as the pivot (end of P1 and beginning of IR8), and thus linked by the prominent tritone in the center of the row Loudspeaker.svg   Play  .

Tone rows are designated by letters and subscript numbers (e.g.: RI11, which may also appear as RI11 or RI-11). The numbers indicate the initial (P or I) or final (R or RI) pitch-class number of the given row form, most often with c=0. P indicates prime, a forward-directed right-side up form. I indicates inversion, a forward-directed upside-down form. R indicates retrograde, a backwards right-side up form. RI indicates retrograde-inversion, a backwards upside-down form. Transposition is indicated by a T number, for example P8 is a T(4) transposition of P4. [10]

A musical line which is the reverse of a previously or simultaneously stated line is said to be its retrograde or cancrizans. An exact retrograde includes both the pitches and rhythms in reverse. An even more exact retrograde reverses the physical contour of the notes themselves, though this is possible only in electronic music. Some composers choose to subject just the pitches of a musical line to retrograde, or just the rhythms. In twelve-tone music, reversal of the pitch classes alone—regardless of the melodic contour created by their registral placement—is regarded as a retrograde.

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 motive'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."

In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.

The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.

A twelve-tone composition will take one or more tone rows, called the "prime form", as its basis plus their transformations (inversion, retrograde, retrograde inversion, as well as transposition; see twelve-tone technique for details). These forms may be used to construct a melody in a straightforward manner as in Schoenberg's Op. 25 Minuet Trio, where P-0 is used to construct the opening melody and later varied through transposition, as P-6, and also in articulation and dynamics. It is then varied again through inversion, untransposed, taking form I-0. However, rows may be combined to produce melodies or harmonies in more complicated ways, such as taking successive or multiple pitches of a melody from two different row forms, as described at twelve-tone technique.

In music, a transformation consists of any operation or process that may apply to a musical variable, or rhythm in composition, performance, or analysis. Transformations include multiplication, rotation, permutation, prolation and combinations thereof.

Initially, Schoenberg required the avoidance of suggestions of tonality—such as the use of consecutive imperfect consonances (thirds or sixths)—when constructing tone rows, reserving such use for the time when the dissonance is completely emancipated. Alban Berg, however, sometimes incorporated tonal elements into his twelve-tone works, and the main tone row of his Violin Concerto hints at this tonality:

Tonality arranges pitches or chords to induce a hierarchy of perceived relations, stabilities, and attractions

Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is called the tonic. The root of the tonic chord forms the name given to the key; so in the key of C major, the note C is both the tonic of the scale and the root of the tonic chord. Simple folk music songs often start and end with the tonic note. The most common use of the term "is to designate the arrangement of musical phenomena around a referential tonic in European music from about 1600 to about 1910". Contemporary classical music from 1910 to the 2000s may practice or avoid any sort of tonality—but harmony in almost all Western popular music remains tonal. Harmony in jazz includes many but not all tonal characteristics of the European common practice period, sometimes known as "classical music".

Emancipation of the dissonance

The emancipation of the dissonance was a concept or goal put forth by composer Arnold Schoenberg and others, including his pupil Anton Webern. The phrase first appears in Schoenberg's 1926 essay "Opinion or Insight?". It may be described as a metanarrative to justify atonality. Jim Samson describes:

As the ear becomes acclimatized to a sonority within a particular context, the sonority will gradually become 'emancipated' from that context and seek a new one. The emancipation of the dominant-quality dissonances has followed this pattern, with the dominant seventh developing in status from a contrapuntal note in the sixteenth century to a quasi-consonant harmonic note in the early nineteenth. By the later nineteenth century the higher numbered dominant-quality dissonances had also achieved harmonic status, with resolution delayed or omitted completely. The greater autonomy of the dominant-quality dissonance contributed significantly to the weakening of traditional tonal function within a purely diatonic context.

Alban Berg Austrian composer

Alban Maria Johannes Berg was an Austrian composer of the Second Viennese School. His compositional style combined Romantic lyricism with twelve-tone technique.

G, B, D, F#, A, C, E, G#, B, C#, E, F.
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G, B, D, F, A, C, E, G, B, C, E, F. Loudspeaker.svg   Play  

This tone row consists of alternating minor and major triads starting on the open strings of the violin, followed by a portion of an ascending whole tone scale. This whole tone scale reappears in the second movement when the chorale "Es ist genug" (It is enough) from Johann Sebastian Bach's cantata O Ewigkeit, du Donnerwort, BWV 60, which opens with consecutive whole tones based on Johann Rudolf Ahle's melody from 1662, is quoted literally in the woodwinds (mostly clarinet).

Chord (music) harmonic set of three or more notes

A chord, in music, is any harmonic set of pitches consisting of three or more notes that are heard as if sounding simultaneously.

In music, a whole tone scale is a scale in which each note is separated from its neighbours by the interval of a whole tone. In twelve-tone equal temperament, there are only two complementary whole tone scales, both six-note or hexatonic scales:

Chorale is the name of several related musical forms originating in the music genre of the Lutheran chorale:

"Mirror forms", P, R, I, and RI, of a tone row (from Arnold Schoenberg's Variations for Orchestra op. 31
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): "Called mirror forms because...they are identical.". Schoenberg - Variations for Orchestra op. 31 tone row mirror forms.png
"Mirror forms", P, R, I, and RI, of a tone row (from Arnold Schoenberg's Variations for Orchestra op. 31 Loudspeaker.svg   Play  ): "Called mirror forms because...they are identical.".

Some tone rows have a high degree of internal organisation. Here is the tone row from Anton Webern's Concerto for Nine Instruments Opus 24:

Tone row of Webern's Concerto Op. 24, composed of four trichords: P RI R I.
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Tone row of Webern's Concerto Op. 24, composed of four trichords: P RI R I. Loudspeaker.svg   Play  
B, B, D, E, G, F, G, E, F, C, C, A

If the first three notes are regarded as the "original" cell, then the next three are its retrograde inversion (backwards and upside down), the next three are retrograde (backwards), and the last three are its inversion (upside down). A row created in this manner, through variants of a trichord or tetrachord called the generator, is called a derived row. The tone rows of many of Webern's other late works are similarly intricate.

Webern's String Quartet Op. 28 tone row, composed of three tetrachords: P I RI, with P = the BACH motif.
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Webern's String Quartet Op. 28 tone row, composed of three tetrachords: P I RI, with P = the BACH motif. Loudspeaker.svg   Play  

The "set-complex" is the forty-eight forms of the set generated by stating each "aspect" or transformation on each pitch class. [1]

The all-interval twelve-tone row is a tone row arranged so that it contains one instance of each interval within the octave, 0 through 11.


The "total chromatic" (or "aggregate") [13] is the set of all twelve pitch classes. An "array" is a succession of aggregates. [13]

First array of four aggregates (numbered 1-4 at bottom) from Milton Babbitt's Composition for Four Instruments , each vertical line (four trichords labeled a-d) is an aggregate while each horizontal line (four trichords labeled a-d) is also an aggregate Array - Babbitt's Composition for Four Instruments.png
First array of four aggregates (numbered 1–4 at bottom) from Milton Babbitt's Composition for Four Instruments , each vertical line (four trichords labeled a–d) is an aggregate while each horizontal line (four trichords labeled a-d) is also an aggregate

An aggregate may be achieved through complementation or combinatoriality, such as with hexachords.

A "secondary set" is a tone row which is derived from or, "results from the reversed coupling of hexachords", when a given row form is immediately repeated. [14] For example, the row form consisting of two hexachords:

0 1 2 3 4 5 / 6 7 8 9 t e

when repeated immediately results in the following succession of two aggregates, in the middle of which is a new and complete aggregate beginning with the second hexachord:

0 1 2 3 4 5 / 6 7 8 9 t e / 0 1 2 3 4 5 / 6 7 8 9 t e secondary set:  [6 7 8 9 t e / 0 1 2 3 4 5]

A "weighted aggregate" is an aggregate in which the twelfth pitch does not appear until at least one pitch has appeared at least twice, supplied by segments of different set forms. [15] It seems to have been first used in Milton Babbitt's String Quartet No. 4. An aggregate may be vertically or horizontally weighted. An "all-partition array" is created by combining a collection of hexachordally combinatorial arrays. [16]

Pierre Boulez's Second Piano Sonata series
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consists of three cells: A) an ascending perfect fifth followed by a tritone and a perfect fourth, B) a descending perfect fifth followed by an ascending major second and a descending augmented fifth, and B1) B inverted. Pierre Boulez - Second Piano Sonata series.png
Pierre Boulez's Second Piano Sonata series Loudspeaker.svg   Play   consists of three cells: A) an ascending perfect fifth followed by a tritone and a perfect fourth, B) a descending perfect fifth followed by an ascending major second and a descending augmented fifth, and B1) B inverted.

Nonstandard tone rows

Prime form of five-note tone row from Igor Stravinsky's In memoriam Dylan Thomas.
Play  (help*info) Stravinsky - In memoriam Dylan Thomas five-tone row.png
Prime form of five-note tone row from Igor Stravinsky's In memoriam Dylan Thomas. Loudspeaker.svg   Play  

Schoenberg specified many strict rules and desirable guidelines for the construction of tone rows such as number of notes and intervals to avoid. Tone rows that depart from these guidelines include the above tone row from Berg's Violin Concerto which contains triads and tonal emphasis, and the tone row below from Luciano Berio's Nones which contains a repeated note making it a 'thirteen-tone row':

Thirteen-note tone row from Luciano Berio's Nones , symmetrical about the central tone with one note (D) repeated.
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Thirteen-note tone row from Luciano Berio's Nones , symmetrical about the central tone with one note (D) repeated. Loudspeaker.svg   Play  

Igor Stravinsky used a five-tone row, chromatically filling out the space of a major third centered tonally on C (C-E), in one of his early serial compositions, In memoriam Dylan Thomas.

In his twelve-tone practice Stravinsky preferred the inverse-retrograde (IR) to the retrograde-inverse (RI), [20] [21] [22] as for example in his Requiem Canticles :

Basic row forms from Stravinsky's Requiem Canticles: P R I IR Stravinsky - Requiem Canticles basic row forms.png
Basic row forms from Stravinsky's Requiem Canticles: P R I IR
Unordered sets from the second of Stockhausen's Klavierstucke I-IV
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which, "retained only the rudiments of the 12-note series." Stockhausen - Klavierstucke I-IV 2 series.png
Unordered sets from the second of Stockhausen's Klavierstücke I–IV Loudspeaker.svg   Play   which, "retained only the rudiments of the 12-note series."
Unordered sets from the third of Stockhausen's Klavierstucke I-IV
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. Stockhausen - Klavierstucke I-IV 3 series.png
Unordered sets from the third of Stockhausen's Klavierstücke I–IV Loudspeaker.svg   Play  .

Ben Johnston uses a "just tone row" (see just intonation) in works including String Quartets Nos. 6 and 7. Each permutation contains a just chromatic scale, however, transformations (transposition and inversion) produce pitches outside of the primary row form, as already occurs in the inversion of P0. The pitches of each hexachord are drawn from different otonality or utonality on A+ utonality, C otonality and utonality, and E- otonality, outlining a diminished triad.

Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2
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and
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. Ben Johnston String Quartet No. 7, mov. 2 just tone row.png
Primary forms of the just tone row from Ben Johnston's String Quartet No. 7, mov. 2 Loudspeaker.svg   Play   and Loudspeaker.svg   Play hexachords  .

See also

A literary parallel of the tone row is found in Georges Perec's poems which use each of a particular set of letters only once.[ citation needed ]

"Tone row" may also be used to describe other musical collections or scales, such as in Arabic music.[ citation needed ]

Sources

  1. 1 2 George Perle, 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, 1977): 3. ISBN   0-520-03395-7.
  2. Ton de Leeuw, Music of the Twentieth Century: A Study of Its Elements and Structure, translated from the Dutch by Stephen Taylor (Amsterdam: Amsterdam University Press, 2005), 174. 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.
  3. Andrew Kirkman and Alexander Ivashkin, Contemplating Shostakovich: Life, Music and Film: Life, Music and Film. (Farnham: Ashgate Publishing, 2013): [unpaginated]. ISBN   9781409472025.
  4. Stephen C. Brown, "Twelve-Tone Rows and Aggregate Melodies in the Music of Shostakovich," Journal of Music Theory, Vol. 59, No. 2 (Fall 2015): 191–234.
  5. Discovery Reveals Bach's Postmodern Side. Weekend Edition Sunday, NPR, 6 September 2009.
  6. Hans Keller, "Strict Serial Technique in Classical Music", Tempo, New Series, no. 37 (Autumn, 1955): 12–24; citations on 14–21.
  7. Keller 1955, 22–23.
  8. Keller 1955, 23.
  9. Leeuw 2005, 158.
  10. George Perle, Twelve-Tone Tonality, second edition, revised and expanded (Berkeley: University of California Press, 1996): 3. ISBN   0-520-20142-6.
  11. Leeuw 2005, 154. Italics original.
  12. Whittall 2008, 97.
  13. 1 2 3 Whittall 2008, 271.
  14. Perle 1977, 100; Perle 1996, 20.
  15. Haimo 1990, 183.
  16. Evan Allan Jones, Intimate Voices: The Twentieth-Century String Quartet. Volume2: Shostakovich to the Avant-garde. Dmitri Shostakovich: The String Quartets (Rochester: University of Rochester Press, 2009): 228. ISBN   9781580463225.
  17. Leeuw 2005, 166.
  18. Whittall 2008, 127.
  19. Whittall 2008, 195.
  20. Claudio Spies, "Notes on Stravinsky's Abraham and Isaac", Perspectives of New Music 3, no. 2 (Spring-Summer 1965): 104–26, citation on 118.
  21. Joseph N. Strauss, "Stravinsky's Serial 'Mistakes'", The Journal of Musicology 17, no. 2 (Spring 1999): 231–71, citation on 242.
  22. 1 2 Whittall 2008, 139.
  23. 1 2 Leeuw 2005, 176–77.
  24. John Fonville, "Ben Johnston's Extended Just Intonation: A Guide for Interpreters", Perspectives of New Music 29, no. 2 (Summer, 1991): 106-137, citation on 127.

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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.

The term "partition" is also French for the sheet music of a transcription.

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.

<i>Lyric Suite</i> (Berg)

The Lyric Suite is a six-movement work for string quartet written by Alban Berg between 1925 and 1926 using methods derived from Arnold Schoenberg's twelve-tone technique. Though publicly dedicated to Alexander von Zemlinsky, the work has been shown to possess a "secret dedication" and to outline a "secret programme" Perle 1977a,.

Otonality and Utonality

Otonality and utonality are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone (identity), respectively. For example: 1/1, 2/1, 3/1,... or 1/1, 1/2, 1/3,....

An Otonality is that set of pitches generated by the numerical factors (...identities)...over a numerical constant in the denominator. Conversely, a Utonality is the inversion of an Otonality, a set of pitches with a numerical constant in the numerator over the numerical factors...in the denominator.

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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.

In music theory, the word inversion has distinct, but related, meanings when applied to intervals, chords, voices, and melodies. The concept of inversion also plays an important role in musical set theory.

Fritz Heinrich Klein was an Austrian composer.

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Suite for Piano (Schoenberg) twelve tone piece for piano composed between 1921 and 1923

Arnold Schoenberg's Suite for Piano, Op. 25, is a twelve tone piece for piano composed between 1921 and 1923.

<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.