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In music, modernism is an aesthetic stance underlying the period of change and development in musical language that occurred around the turn of the 20th century, a period of diverse reactions in challenging and reinterpreting older categories of music, innovations that led to new ways of organizing and approaching harmonic, melodic, sonic, and rhythmic aspects of music, and changes in aesthetic worldviews in close relation to the larger identifiable period of modernism in the arts of the time. The operative word most associated with it is "innovation".^[1] Its leading feature is a "linguistic plurality", which is to say that no musical language, or modernist style, ever assumed a dominant position.^[2]

Inherent within musical modernism is the conviction that music is not a static phenomenon defined by timeless truths and classical principles, but rather something which is intrinsically historical and developmental. While belief in musical progress or in the principle of innovation is not new or unique to modernism, such values are particularly important within modernist aesthetic stances.

— Edward Campbell (2010, p. 37) [emphasis added]

Examples include the celebration of Arnold Schoenberg's rejection of tonality in chromatic post-tonal and twelve-tone works and Igor Stravinsky's move away from symmetrical rhythm.^[3]

Authorities typically regard musical modernism as an historical period or era extending from about 1890 to 1930, and apply the term "postmodernism" to the period or era after 1930.^[4][5] For the musicologist Carl Dahlhaus the purest form was over by 1910.^[6] However, there are other historians and critics who argue that modernism was revived after World War II. For example, Paul Griffiths notes that, while Modernism "seemed to be a spent force" by the late 1920s, after World War II, "a new generation of composers—Boulez, Barraqué, Babbitt, Nono, Stockhausen, Xenakis" revived modernism".^[7]

Carl Dahlhaus describes modernism as:

an obvious point of historical discontinuity ... The "breakthrough" of Mahler, Strauss, and Debussy implies a profound historical transformation ... If we were to search for a name to convey the breakaway mood of the 1890s (a mood symbolized musically by the opening bars of Strauss's Don Juan) but without imposing a fictitious unity of style on the age, we could do worse than revert to Hermann Bahr's term "modernism" and speak of a stylistically open-ended "modernist music" extending (with some latitude) from 1890 to the beginnings of our own twentieth-century modern music in 1910.^[8]

Eero Tarasti defines musical modernism directly in terms of "the dissolution of the traditional tonality and transformation of the very foundations of tonal language, searching for new models in atonalism, polytonalism or other forms of altered tonality", which took place around the turn of the century.^[9]

Daniel Albright proposes a definition of musical modernism as, "a testing of the limits of aesthetic construction" and presents the following modernist techniques or styles: Expressionism, the New Objectivity, Hyperrealism, Abstractionism, Neoclassicism, Neobarbarism, Futurism, and the Mythic Method.^[10]

Conductor and scholar Leon Botstein describes musical modernism as "...a consequence of the fundamental conviction among successive generations of composers since 1900 that the means of musical expression in the 20th century must be adequate to the unique and radical character of the age",^[11] which led to a reflection in the arts of the progress of science, technology and industry, mechanization, urbanization, mass culture and nationalism.

Similarly, Eric Pietro defines Modernism in his narrative Listening In: Music, Mind, and the Modernist as, “…a desire to find ‘ever more accurate representations of psychological states and processes’ by virtue of its links with the ‘historical crisis of the nineteenth century.’” From what we can understand with this information, there are two distinguishable concepts emphasizing Modernism: the first being music mirroring narrative depictions of the mind; and the second being music as a vocabulary that faces the possibility of describing psychological behaviors in language.^[12]

The term "modernism" (and the term "post-modern") has occasionally been applied to some genres of popular music, but not with any very clear definition.

For example, the cultural studies professor Andrew Goodwin writes that "given the confusion of the terms, the identification of postmodern texts has ranged across an extraordinarily divergent, and incoherent profusion of textual instances ... Secondly, there are debates within popular music about pastiche and authenticity. 'Modernism' means something quite different within each of these two fields ... This confusion is obvious in an early formative attempt to understand rock music in postmodern terms".^[13] Goodwin argues that instances of modernism in popular music are generally not cited because "it undermines the postmodern thesis of cultural fusion, in its explicit effort to preserve a bourgeois notion of Art in opposition to mainstream, 'commercial' rock and pop".^[14]

Author Domenic Priore writes that: "the concept of Modernism was bound up in the very construction of the Greater Los Angeles area, at a time when the city was just beginning to come into its own as an international, cultural center",;^[15] it appears that the word is used here as an equivalent of the term "modern". Priore cites "River Deep – Mountain High" by Ike & Tina Turner (1966) and "Good Vibrations" by the Beach Boys (1966). Desiring "a taste of Modern, avant-garde R&B" for the latter's recording, group member and song co-writer Brian Wilson considered the music "advanced rhythm and blues", but received criticism from his bandmates, who derided the track for being "too Modern" during its making.^[16]

In the final decade leading up to the turn of the 20th century, the Romantic era in music had entered into its late period where great changes were occurring. Amongst the biggest changes were with the traditional tonal system, which was now being regularly stretched to its limits by composers such as Gustav Mahler who began incorporating progressive tonality^[17] into his pieces. The Impressionists such as Claude Debussy also began experimenting with ambiguous tonality and exotic scales. "The perception of Debussy’s compositional language as decidedly post-romantic/Impressionistic—nuanced, understated, and subtle—is firmly solidified among today’s musicians and well-informed audiences."^[18] Although this isn’t the first time composers began pushing the limits of tonality as can be seen in the works of Richard Wagner in Tristan und Isolde^[19] and in the works of Franz Liszt in Bagatelle sans tonalité,^[20] these practices became far more commonplace within the late romantic period. This break with tonality finally came to a critical point in 1908 when Arnold Schoenberg composed the second string quartet, Op. 10, with soprano. The last movement of this piece contains no key signature,^[21] marking a decisive transition point from Romanticism into Modernism.

Within this newly established Modernist era, several new parallel movements were founded as a reaction against late romanticism. The most prominent of these movements included Expressionism with Arnold Schoenberg and the Second Viennese School being its main promoters, Primitivism with Igor Stravinsky being its most influential composer, and Futurism with Luigi Russolo being one of its main proponents.

Musical expressionism is closely associated with the music of the Second Viennese School during their "free atonal" period from 1908 to 1921.^[22] One of the main goals of this movement was to avoid "traditional forms of beauty" to convey powerful feelings in their music. ^[23] In essence, Expressionist music often features a high level of dissonance, extreme contrasts of dynamics, constant changing of textures, "distorted" melodies and harmonies, and angular melodies with wide leaps.^[24]

Primitivism was a movement that aimed to rescue the most archaic folklore of certain regions with a modern or modernist language. Similar to nationalism in its eagerness to rescue the local traditions, primitivism also incorporated irregular metrics and accentuations, a greater use of percussion and other timbres, modal scales, and polytonal harmony. Important works of this style include The Firebird (1910), Petrushka (1911), The Rite of Spring (1913), and The Miraculous Mandarin (1926). Within this movement, the two giants of this movement were the Russian Igor Stravinsky and the Hungarian Béla Bartók, although the work of both far exceeds the name "primitivist".

Italian composers such as Francesco Balilla Pratella and Luigi Russolo aided in developing musical Futurism. This genre attempts to recreate everyday sounds and place them within a "Futurist" context. The "Machine Music" of George Antheil (starting with his Second Sonata, "The Airplane") and Alexander Mosolov (most notoriously his Iron Foundry) developed from this.

The process of extending musical vocabulary by exploring all available tones was pushed further by the use of Microtones. This can be seen in works of composers such as Charles Ives, Julián Carrillo, Alois Hába, John Foulds, Ivan Wyschnegradsky, Harry Partch and Mildred Couper. Microtones are intervals that are smaller than a semitone; human voices and unfretted strings can easily produce them by going in between the "normal" notes, however other musical instruments will have more difficulty in achieving the same result. The piano and organ have no way of producing them at all, aside from retuning or from major reconstruction.

In the United States, Charles Ives began to integrate American and European traditions as well as colloquial and church styles, while using innovative techniques in his harmony, rhythm, and form.^[25] His techniques included the use of polyrhythm, polytonality, tone clusters, quarter tones. and aleatoric elements. This new experimental style of composition influenced a number of American composers who came to be collectively known as the American Five.

In the early 1920s, Schoenberg developed the Twelve-tone technique, a method of musical composition which ensures that all 12 notes of the chromatic scale are sounded as often as one another in a composition while preventing the emphasis of any one note^[26] through the use of tone rows and the orderings of the 12 pitch classes. This new technique was quickly adopted by members of the Second Viennese School, namely Anton Webern who refined the system and became a massive influence to the development of Serialism.

After the end of World War I, Igor Stravinsky began to return to past Pre-Romantic compositional traditions for inspiration and wrote works that drew elements such as form, harmony, melody, structure from it. This style of music came to be known as Neoclassicism and it came to be the dominent style of composition during the Interwar period. Important works in this style includes; Pulcinella, Classical Symphony, Le Tombeau de Couperin, El retablo de maese Pedro, and Symphony: Mathis der Maler. A prominent group of mostly French composers known as Les Six were especially influenced by this compositional method.

A similar movement also took hold in Post-War Germany as a reaction against the sentimentality of late Romanticism and the emotional agitation of expressionism. Known as New Objectivity, this model of composition typically harkened back to baroque era models and made use of traditional forms as well as stable polyphonic structures, combined together with modern dissonance and jazz-inspired rhythms. Paul Hindemith was the most prominent composer of this style.

The 1930s proved to be a difficult time for the Modernist music scene in Europe after Adolf Hitler and the Nazi Party took power in Germany and the Austrofascists took power in Austria. As a result, most Modernist music which featured atonality, dissonance, and “disturbing rhythms” were deemed as degenerate music and banned. The music of Alban Berg, Hans Eisler, Paul Hindemith, Arnold Schoenberg, Anton von Webern, Kurt Weill, and other formerly prominent composers, as well as Jewish composers such as Felix Mendelssohn, Giacomo Meyerbeer, Jacques Offenbach and even George Gershwin and Irving Berlin, were no longer programmed or allowed to be performed.^[27] As a result of these new policies, many prominent Modernist composers such as Arnold Schoenberg and Igor Stravinsky were forced to flee to the United States while others such as Anton Webern were forced to compose their works in secret.

World War II was devastating for Europe and a new generation of composers had to pick up the pieces and reestablish the art music scene. Through the rediscovery and promotion of pre-war composers such as Anton Webern and Edgard Varèse, as well as the more recent developments initiated by the French composer Olivier Messiaen, Serialism came to be one of the dominant methods of composition within the art music establishment for the next few decades. Also influenced by other pioneering works of the Second Viennese School, starting in 1946, the Darmstädter Ferienkurse began an annual summer program in Darmstadt, Germany where Modernist forms of classical music were taught and promoted. Among the most important composers to emerge from these courses included Pierre Boulez, Bruno Maderna, Luigi Nono, and Karlheinz Stockhausen. Together, this group collectivley came to be known as the Darmstadt School. Among their primary goals was to reestablish and expand upon the serialist philosophies established by the likes of Arnold Schoenberg and Anton Webern. ^[28] Igor Stravinsky was also encouraged to explore serial music and the composers of the Second Viennese School, beginning Stravinsky's third and final distinct musical period, which lasted from 1954 until his death in 1971.^[29][30][31] However, some more traditionally based composers such as Dmitri Shostakovich and Benjamin Britten maintained a tonal style of composition despite the prominent serialist movement.

The United States took a somewhat different direction to Modernism in comparison to their European counterparts in the early post-war era. American composers including John Cage, Morton Feldman, Earle Brown and Christian Wolff^[32] formed an informal circle musicians called the New York School. This group was far less concerned in working with serialism but rather focused on experimenting with chance. Their compositions influenced the music and events of the Fluxus group, and drew its name from Abstract Expressionist painters. However, composers such as Milton Babbitt, George Rochberg, and Roger Sessions fashioned their own extensions of the twelve-tone serialism of Schoenberg.

One of the most important and influential developments from the Modernist music scene in America was the concept of indeterminacy in music. Spearheaded by John Cage, this new composition approach left some aspects of a musical work open to chance or to the interpreter's free choice. This can be seen in Cage’s Music of Changes (1951), where the composer selects the duration, tempo, and dynamics by using the I Ching, an ancient Chinese book which prescribes methods for arriving at random numbers.^[33] Another example is Morton Feldman's "Intersection No. 2" (1951) for piano solo, written on coordinate paper. Time units are represented by the squares viewed horizontally, while relative pitch levels of high, middle, and low are indicated by three vertical squares in each row. The performer determines what particular pitches and rhythms to play.^[34]

In Europe, a similar method of composition developed. Coined as "aleatory music" by Meyer-Eppler and popularized by the French composer Pierre Boulez,^[35] this new compositional style did not completely give away its creation and performance to chance but rather the notated events are provided by the composer, but their arrangement is left to the determination of the performer. A prominent example of this style can be seen in Karlheinz Stockhausen's work Klavierstück XI (1956) where the nineteen events presented are composed and notated in a traditional way, but the arrangement of these events is determined by the performer spontaneously during the performance. Another example can be seen in Earle Brown's Available forms II (1962), where the conductor is asked to decide the order of the events at the very moment of the performance.^[36]

Major developments were also taking shape in Electronic music shortly after the end of World War II. In the late 1940s, acoustic engineer and radio scientist Pierre Schaeffer created a new style of composition called Musique concrète where recorded sounds are utilized as raw material.^[37] These recorded sounds are often modified through the application of audio signal processing and tape music techniques, and may be assembled into a form of sound collage.^{[nb 1]} Schaeffer’s pioneering works attracted and inspired a new generation of composers such as Olivier Messiaen, Karlheinz Stockhausen, Iannis Xenakis, Pierre Boulez, as well as others to try their hands into this new world and develop their own innovations.

Building upon aleatoric elements and electronic components, mathematics and scientific concepts were incorporated to produce Stochastic music. Pioneered by the works of Greek composer Iannis Xenakis, important examples of compositions drawing from concepts in physical science includes; the use of the statistical mechanics of gases in Pithoprakta, minimal constraints in Achorripsis, Markov chains in Analogiques, statistical distribution of points onto a plane in Diamorphoses, the use of normal distribution in ST/10 and Atrées, Brownian motion in N'Shima, game theory in Duel and Stratégie, the group theory in Nomos Alpha (for Siegfried Palm), and set theory in Herma and Eonta.^[39] Xenakis also frequently used computers to produce his compositions, such as the ST series including Morsima-Amorsima and Atrées. American composers such as Lejaren Hiller and Leonard Issacson also used generative grammars and Markov chains in their 1957 Illiac Suite.

Starting around 1944, Elliott Carter began to incorporate processes into his compositions such as in his Piano Sonata and First String Quartet. ^[40][41] Essentially notes through pitch and time were stretched into a long term change with limited transformations of musical events. This new compositional style came to be known as Process music and would become adopted by serialists during the 1960s. Minimalists would also come to embrase this approach in the coming decade.^[42] Other prominent examples of works that incorporate processes includes; Nr. 5, met zuivere tonen (1953), Kreuzspiel (1951),^[43] Plus-Minus (1963), Prozession (1967), It's Gonna Rain (1965), ^[44] Come Out (1966), ^[44] and Reed Phase (1966).

In 1977, French composer Pierre Boulez founded the Institut de Recherche et Coordination Acoustique / Musique (IRCAM) whose aims included research into acoustics, instrumental design, and the use of computers in music.^[45] Spectralism, which originally arose in France during the early 1970s, had received much of its development and refinement through this institution. The composition of spectral music was often informed by sonographic representations and mathematical analysis of sound spectra, or by mathematically generated spectra. This new style also arose in part as a reaction against and an alternative to the primarily pitch focused aesthetics of the serialist and post-serialist compositions that were commonplace for the time.^[46] The two most prominent schools in spectral music were the French Ensemble l'Itinéraire headed by Gérard Grisey and Tristan Murail and the German Feedback group headed by Péter Eötvös and Claude Vivier. Likewise, spectral techniques would soon be adopted by a wider variety of composers such as Magnus Lindberg and Kaija Saariaho.^[47]

In the United Kingdom, a lose group of composers began writing scores in an increasingly complex musical notation that was often atonal, highly abstract, and dissonant in sound. Coined as New Complexity, earliest prominent mention being from Richard Toop’s article "Four Facets of the New Complexity",^[48] this new style gained traction in continental Europe, particularly through the Darmstädter Ferienkurse in the 1980s and 1990s. The most influential figures of this movement were Brian Ferneyhough and Michael Finnissy.

During the 1960s and 1970s, a backlash began to emerge against the strict serialism promoted by groups such as the Darmstadt School which had essentially taken over the academic musical establishment. In America, a new form of art music called Minimal music had emerged as a reaction against the perceived extreme and unsurpassable complexity of serialism.^[49] Instead minimal music focuses on the repetition of slowly changing common chords in steady rhythms, often overlaid with a lyrical melody in long, arching phrases.^[50]

Europe also experienced a similar backlash against strict serialism as can be seen in the emergence of the New Simplicity movement spearheaded by composers such as Wolfgang Rihm. In general, these composers strove for an immediacy between the creative impulse and the musical result, which contrasts with the elaborate precompositional planning characteristic of the High Modernists. Some writers argue that Darmstadt School representative Karlheinz Stockhausen, had anticipated this reaction through a radical simplification of his style accomplished between 1966 and 1975, which culminated in his Tierkreis melodies.^[51][52][53]

Impressionism was a movement among various composers in Western classical music from about 1890 to 1920, whose music focuses on mood and atmosphere.^[54] Just like Impressionism in painting and Impressionism in literature musical impressionism tries to represent impressions of moments. The most prominent feature of impressionist music is the timbre and instrumentation. Layerings of musical levels are typical but also includes: a profound but not intrusive bass, moving middle voices and a significant motif in the upper voices, and is not subject to the laws of the usual classical-romantic processing (diminution, secession, etc.) but is treated rather associatively. The most noteworthy composers of this movement includes Claude Debussy, Erik Satie, and Maurice Ravel.

Expressionism was a movement in music where composers sought a subjective immediacy of expression, drawn as directly as possible from the human soul. To achieve this, a break with tradition in regards to traditional aesthetics and the previous forms was desired. Stylistically, the changed function of dissonances is particularly striking; they appear on an equal footing with consonances and are no longer resolved – what was also called the "emancipation of dissonance". The tonal system is largely dissolved and expanded into atonality. Musical characteristics include: extreme pitches, extreme dynamic contrasts (from whispering to screaming, from pppp to ffff), jagged melody lines with wide leaps; metrically unbound, free rhythm and novel instrumentation. Form: asymmetrical period structure; rapid succession of contrasting moments; often very short "aphoristic" pieces. The main representatives of this movement are the composers of the Second Viennese School: Arnold Schoenberg, Anton Webern and Alban Berg.

The Second Viennese School were a group of composers consisting of Arnold Schoenberg and his pupils, most notably Alban Berg and Anton Webern, as well as close associates in early 20th-century Vienna. Their music was initially characterized by late-Romantic expanded tonality. However their compositional style would evolve to a totally chromatic expressionism without a firm tonal center, often referred to as atonality. Even later on beginning in the early 1920s, this group would adopt Schoenberg's serial twelve-tone technique. Greatly promoted by critics and musicologists such as Theodor Adorno, the music of the Second Viennese School would take over in intellectual circles and the art music establishment especially after the conclusion of WW2.

Primitivism was a movement that aimed to incorporate the most archaic and often pagan folklore of certain regions in Europe into modernist musical compositions. Similar to nationalism in its eagerness to rescue the local traditions, primitivism also incorporated irregular metrics and accentuations, a greater use of percussion and other timbres, modal scales, and polytonal harmony. Within this movement, the most prominent composers were the Russian Igor Stravinsky and the Hungarian Béla Bartók, although the work of both far exceeds the name "primitivist".

Futurism was a movement originating in Italy which rejected tradition and introduced experimental sounds inspired by machinery. Much of this new genre’s origins can be traced to painter and composer Luigi Russolo, who in 1913 published his groundbreaking manifesto, The Art of Noises calling for the incorporation of noises of every kind into music.^[55] This inspired fellow Italian composers Francesco Balilla Pratella and Franco Casavola to follow in his footsteps. This new aesthetic also became quickly embraced by the Russian avant-garde creating a parallel movement of Russian Futurists. Among the most prominent Russian composers from this tradition includes Mikhail Matyushin and Nikolai Roslavets.

The American Five were a group of American experimental composers who often implemented polyrhythm, polytonality, tone clusters, quarter tones. and aleatoric elements within their music. Spearheaded by Charles Ives, they were noted for their unusual and often dissonant pieces which broke away from European compositional techniques to create a uniquely American style.^[56] The primary members of this group were Charles Ives, John J. Becker, Wallingford Riegger, Henry Cowell, and Carl Ruggles.^[57][58]

Neoclassicism was a movement, especially prevalent during the interwar period, in which composers sought to return to aesthetic precepts associated with the broadly defined concept of "classicism", namely order, balance, clarity, economy, and emotional restraint. As such, neoclassicism was a reaction against the unrestrained emotionalism and perceived formlessness of late Romanticism, as well as a "call to order" after the experimental ferment of the first two decades of the twentieth century. The neoclassical impulse found its expression in such features as the use of pared-down performing forces, an emphasis on rhythm and on contrapuntal texture, an updated or expanded tonal harmony, and a concentration on absolute music as opposed to Romantic program music. The main representatives of this movement are Igor Stravinsky and Sergei Prokofiev.

Les Six were a group of six composers, five of them French and one Swiss, who primarily worked in the Montparnasse region of Paris, France. Composing in the neoclassical style of Igor Stravinsky, their music was often seen as a reaction against both the late German Romanticism of Gustav Mahler and the Impressionistic chromaticism of Claude Debussy. They were also heavily inspired by the music of Erik Satie and the poetry of Jean Cocteau. The primary members of this group were Georges Auric, Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc, and Germaine Tailleferre.

The Darmstadt School refers to a group of composers who were associated with the Darmstädter Ferienkurse from the 1950s and 1960s centered in Darmstadt, Germany. Greatly influenced by Arnold Schoenberg's twelve-tone technique, they developed it further to implement Integral Serialism as the foundation to their compositions. They also often applied electroacoustic and aleatoric techniques into their works. Other key influences of the School included the works of Anton Webern, Edgard Varèse,^[59] and Olivier Messiaen's "Mode de valeurs et d'intensités" (from the Quatre études de rythme). The most prominent composers include Pierre Boulez, Bruno Maderna, Luigi Nono, and Karlheinz Stockhausen.

The New York School was an informal circle of experimental musicians and composers active in the 1950s and 1960s originating from New York City. They often drew inspiration from the Dada and contemporary avant-garde art movements. Their music often displayed indeterminacy, electroacoustic properties, and non-standard use of musical instruments. They were in particular greatly influenced by the pioneering experimental works of Charles Ives, Henry Cowell, and Edgard Varèse. The most prominent composers of this compositional school include John Cage, Earle Brown, Christian Wolff, Morton Feldman, and David Tudor.

The twelve-tone technique is a method of musical composition developed by Austrian composer Arnold Schoenberg where 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^[26] 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. Schoenberg’s technique would first be adopted by other members of the Second Viennese School, most notably Alban Berg and Anton Webern. However its usage would greatly expand after WW2 through its promotion by the Darmstadt School, American composers such as Milton Babbitt, and its adoption by Igor Stravinsky after phasing out of his Neoclassical period in the early 1950s.

Serialism is a method of composition in which a fixed series of notes, usually the twelve notes of the chromatic scale, are used to generate the harmonic and melodic basis of a piece and are subject to change only in specific ways. 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. Serialism of the pre-WW2 Second Viennese School was composed in which a recurring series of ordered elements (normally a set—or row—of pitches or pitch classes) is used in order or manipulated in particular ways to give a piece unity. In post-WW2 Europe, Integral serialism which was developed mainly by the Darmstadt School, incorporated use of series for aspects such as duration, dynamics, and register as well as pitch.^[60] The most prominent composers of this compositional technique include Arnold Schoenberg, Anton Webern, Alban Berg, Karlheinz Stockhausen, Pierre Boulez, Luigi Nono, Milton Babbitt, Igor Stravinsky, Henri Pousseur, Charles Wuorinen and Jean Barraqué.

Musique concrète (French; "concrete music"), is a form of electroacoustic music that utilises recorded sounds as a compositional resource. The compositional material is commonly modified through the application of audio signal processing and tape music techniques, and can be assembled into a sound collage structure.^[38] The theoretical basis of this compositional practice was developed by French composer Pierre Schaeffer beginning in the early 1940s. Other prominent composers who used or were influenced by this compositional technique include Pierre Henry, Luc Ferrari, Pierre Boulez, Karlheinz Stockhausen, Edgard Varèse, and Iannis Xenakis.

Indeterminacy in music is a compositional method in which some or all aspects of a musical work are left either to chance or to the performer’s free choice. Its first significant adoption can be attributed to the works of American composer Charles Ives written in the early 20th century. Ives’s ideas were further developed in the 1930s by Henry Cowell in such works as the Mosaic Quartet (String Quartet No. 3, 1934), which players are allowed to arrange the music fragments in a number of different possible sequences. During the 1950s, development of this technique reached its apex in the works of John Cage and the New York School where chance becomes adopted by a wide range of composers.

Aleatoric music is a compositional style in which some element of a composed work is left to indeterminacy, or in other words left to the determination of the performer(s). The term became known to European composers through the lectures of acoustician Werner Meyer-Eppler at the Darmstädter Ferienkurse in the early 1950s. Unlike their American counterparts however, many European composers did not completely leave the performance of their works to chance. Instead, they would compose and notate several separate paths within their music in which the performer is given the freedom to choose the arrangement. ^[36] The most prominent composers of this style include Pierre Boulez, Karlheinz Stockhausen, and Iannis Xenakis.

Stochastic music is a compositional style pioneered by Greek composer Iannis Xenakis in which mathematical processes often found within statistics, probability, and physics are used to generate scores. Stochastic processes can also be used in compositions to create a fixed notation in the piece or alternatively being produced in real time during a performance. Computers were also frequently used to produce this type of music. The most prominent composers of this style includes Iannis Xenakis, Gottfried Michael Koenig, Jean-Claude Risset, and Lejaren Hiller.

Process music is a compositional style in which a score is generated using a process that’s either audible to the audience or deliberately concealed. These processes can involve specific systems of picking and organizing notes through pitch and time, often involving a long term change with a limited amount of musical material, or transformations of musical events that are already relatively complex in an of themselves. Originating in serial compositions,^[61] this style also came to be widely adopted later on by the minimalists.^[42] Prominent composers of this style includes Karlheinz Stockhausen, Elliott Carter, Karel Goeyvaerts, and Steve Reich.

• List of modernist composers

• Written Tape Guides – The Contemporary Era, lectures by John Ronsheim (1927–1997), Antioch College