Aleatoric music
Encyclopedia
Aleatoric music is music
in which some element of the composition
is left to chance
, and/or some primary element of a composed work's realization is left to the determination of its performer(s). The term is most often associated with procedures in which the chance element involves a relatively limited number of possibilities.
The term became known to European composers through lectures by acoustician
Werner Meyer-Eppler
at the Darmstadt International Summer Courses for New Music
in the beginning of the 1950s. According to his definition, "a process is said to be aleatoric ... if its course is determined in general but depends on chance in detail" (Meyer-Eppler 1957, 55). Meyer-Eppler's German terms Aleatorik (noun) and aleatorisch (adjective), however, both mean "aleatory". By mistakenly rendering them, his translator inadvertently created a new English word, "aleatoric", which quickly became fashionable (Jacobs 1966).
. A later genre was the Musikalisches Würfelspiel
or musical dice
game, popular in the late 18th and early 19th century. (One such dice game is attributed to Wolfgang Amadeus Mozart
.) These games consisted of a sequence of musical measures, for which each measure had several possible versions, and a procedure for selecting the precise sequence based on the throwing of a number of dice (Boehmer 1967, 9–47).
The French artist Marcel Duchamp
composed two pieces between 1913 and 1915 based on chance operations. One of these, Erratum Musical written for three voices, was published in 1934. American composer John Cage
's Music of Changes
(1951) is often considered the first piece to be conceived largely through random procedures (Randel 2002, 17), though his indeterminacy
is of a different order from Meyer-Eppler's concept. Cage later asked Duchamp: "How is it that you used chance operations when I was just being born?" (Lotringer 1998).
in the early 20th century. Henry Cowell
adopted Ives’s ideas during the 1930s, in such works as the Mosaic Quartet (String Quartet No. 3, 1934), which allows the players to arrange the fragments of music in a number of different possible sequences. Cowell also used specially devised notations to introduce variability into the performance of a work, sometimes instructing the performers to improvise a short passage or play ad libitum (Griffiths 2001). Later American composers, such as Alan Hovhaness
(beginning with his Lousadzak of 1944) used procedures superficially similar to Cowell's, in which different short patterns with specified pitches and rhythm are assigned to several parts, with instructions that they be performed repeatedly at their own speed without coordination with the rest of the ensemble. Some scholars regard the resultant blur as "hardly aleatory, since exact pitches are carefully controlled and any two performances will be substantially the same" (Rosner and Wolverton 2001) although, according to another writer, this technique is essentially the same as that later used by Witold Lutosławski (Fisher 2010). Depending on the vehemence of the technique, Hovhaness’s published scores annotate these sections variously, for example as “Free tempo / humming effect” (Hovhaness: Lousadzak, p.3) and “Repeat and repeat ad lib, but not together” (Meditation on Orpheus, p.2).
In Europe, following the introduction of the expression "aleatory music" by Meyer-Eppler, the French composer Pierre Boulez
was largely responsible for popularizing the term (Boulez 1957), using it to describe works that give the performer certain liberties with regard to the sequencing and repetition of parts. The French composer and broadcaster Pierre Schaeffer
developed the term jeu (French for play) in reference to a technique of allowing random sounds to enter into a musical composition.
Other early European examples of aleatory music include Klavierstück XI
(1956) by Karlheinz Stockhausen
, which features 19 elements to be performed in a sequence to be determined in each case by the performer (Boehmer 1967, 72). A form of limited aleatory was used by Witold Lutosławski (beginning with Jeux Vénitiens in 1960–61) (Rae 2001), where extensive passages of pitches and rhythms are fully specified, but the rhythmic coordination of parts within the ensemble is subject to an element of chance. Lutosławski calls these sections 'ad libitum'. In some works by Krzysztof Penderecki
characteristic sequences are repeated quickly, producing a kind of oscillating sound.
There has been considerable confusion of the terms aleatory and indeterminate/chance music. One of Cage's pieces, HPSCHD, (see also his books of changes for more musical examples) itself composed using chance procedures, uses music from Mozart's Musikalisches Würfelspiel, referred to above, as well as original music. Still, both the aesthetic aims as well as the number of elements controlled by chance make the two methods clearly different.
The First Symphony of Alfred Schnittke
uses aleatory techniques as only one of a number of approaches to the 'chaos' of 20th century life (Schnittke also uses Ivesian
dissonance to similar effect).
s, where the order of movements
or section
s is indeterminate or left up to the performer
. Roman Haubenstock-Ramati
composed a series of influential "mobiles" such as Interpolation (1958).
However, "open form" in music is also used in the sense defined by the art historian Heinrich Wölfflin
(Renaissance und Barock, 1888) to mean a work which is fundamentally incomplete, represents an unfinished activity, or points outside of itself. In this sense, a "mobile form" can be either "open" or "closed". An example of a closed mobile musical composition is Stockhausen's Zyklus
(1959). Terry Riley
's In C
(1964) was composed of 53 short sequences; each member of the ensemble can repeat a given sequence as many times as desired before going on to the next, making the details of each performance of In C unique. However, because the overall course is fixed, it is a closed form.
' score for the film Images, but also notably in his more recent score for Steven Spielberg's Jurassic Park: The Lost World. Other film composers using this technique are Mark Snow
(X-Files: Fight the Future), Jerry Goldsmith
and others (Karlin and Wright 2004, 430–36).
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
in which some element of the composition
Aspect of music
An aspect of music is any characteristic, dimension, or element taken as a part or component of music.-European music:The traditional musicological or European-influenced aspects of music often listed are those elements given primacy in European-influenced classical music, so 7 basic elements of...
is left to chance
Randomness
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability of events....
, and/or some primary element of a composed work's realization is left to the determination of its performer(s). The term is most often associated with procedures in which the chance element involves a relatively limited number of possibilities.
The term became known to European composers through lectures by acoustician
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
Werner Meyer-Eppler
Werner Meyer-Eppler
Werner Meyer-Eppler , was a German physicist, experimental acoustician, phoneticist, and information theorist....
at the Darmstadt International Summer Courses for New Music
Darmstadt New Music Summer School
Initiated in 1946 by Wolfgang Steinecke, the Internationale Ferienkurse für Neue Musik, Darmstadt , held annually until 1970 and subsequently every two years, encompass both the teaching of composition and interpretation and include premières of new works...
in the beginning of the 1950s. According to his definition, "a process is said to be aleatoric ... if its course is determined in general but depends on chance in detail" (Meyer-Eppler 1957, 55). Meyer-Eppler's German terms Aleatorik (noun) and aleatorisch (adjective), however, both mean "aleatory". By mistakenly rendering them, his translator inadvertently created a new English word, "aleatoric", which quickly became fashionable (Jacobs 1966).
Early precedents
Compositions that could be considered a precedent for aleatory composition date back to at least the late 15th century, with the genre of the catholicon, exemplified by the Missa cuiusvis toni of Johannes OckeghemJohannes Ockeghem
Johannes Ockeghem was the most famous composer of the Franco-Flemish School in the last half of the 15th century, and is often considered the most...
. A later genre was the Musikalisches Würfelspiel
Musikalisches Würfelspiel
A Musikalisches Würfelspiel was a system for using dice to randomly 'generate' music. These games were quite popular throughout Western Europe in the 18th century...
or musical dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...
game, popular in the late 18th and early 19th century. (One such dice game is attributed to Wolfgang Amadeus Mozart
Wolfgang Amadeus Mozart
Wolfgang Amadeus Mozart , baptismal name Johannes Chrysostomus Wolfgangus Theophilus Mozart , was a prolific and influential composer of the Classical era. He composed over 600 works, many acknowledged as pinnacles of symphonic, concertante, chamber, piano, operatic, and choral music...
.) These games consisted of a sequence of musical measures, for which each measure had several possible versions, and a procedure for selecting the precise sequence based on the throwing of a number of dice (Boehmer 1967, 9–47).
The French artist Marcel Duchamp
Marcel Duchamp
Marcel Duchamp was a French artist whose work is most often associated with the Dadaist and Surrealist movements. Considered by some to be one of the most important artists of the 20th century, Duchamp's output influenced the development of post-World War I Western art...
composed two pieces between 1913 and 1915 based on chance operations. One of these, Erratum Musical written for three voices, was published in 1934. American composer John Cage
John Cage
John Milton Cage Jr. was an American composer, music theorist, writer, philosopher and artist. A pioneer of indeterminacy in music, electroacoustic music, and non-standard use of musical instruments, Cage was one of the leading figures of the post-war avant-garde...
's Music of Changes
Music of Changes
Music of Changes is a piece for solo piano by John Cage. Composed in 1951 for pianist and friend David Tudor, it is Cage's earliest fully indeterminate instrumental work. The process of composition involved applying decisions made using the I Ching, a Chinese classic text that is commonly used as a...
(1951) is often considered the first piece to be conceived largely through random procedures (Randel 2002, 17), though his indeterminacy
Indeterminacy in music
Indeterminacy in music, which began early in the twentieth century in the music of Charles Ives, and was continued in the 1930s by Henry Cowell and carried on by his student, the experimental music composer John Cage beginning in 1951 , came to refer to the movement which grew up around Cage...
is of a different order from Meyer-Eppler's concept. Cage later asked Duchamp: "How is it that you used chance operations when I was just being born?" (Lotringer 1998).
Modern usage
The earliest significant use of aleatory features is found in many of the compositions of American Charles IvesCharles Ives
Charles Edward Ives was an American modernist composer. He is one of the first American composers of international renown, though Ives' music was largely ignored during his life, and many of his works went unperformed for many years. Over time, Ives came to be regarded as an "American Original"...
in the early 20th century. Henry Cowell
Henry Cowell
Henry Cowell was an American composer, music theorist, pianist, teacher, publisher, and impresario. His contribution to the world of music was summed up by Virgil Thomson, writing in the early 1950s:...
adopted Ives’s ideas during the 1930s, in such works as the Mosaic Quartet (String Quartet No. 3, 1934), which allows the players to arrange the fragments of music in a number of different possible sequences. Cowell also used specially devised notations to introduce variability into the performance of a work, sometimes instructing the performers to improvise a short passage or play ad libitum (Griffiths 2001). Later American composers, such as Alan Hovhaness
Alan Hovhaness
Alan Hovhaness was an Armenian-American composer.His music is accessible to the lay listener and often evokes a mood of mystery or contemplation...
(beginning with his Lousadzak of 1944) used procedures superficially similar to Cowell's, in which different short patterns with specified pitches and rhythm are assigned to several parts, with instructions that they be performed repeatedly at their own speed without coordination with the rest of the ensemble. Some scholars regard the resultant blur as "hardly aleatory, since exact pitches are carefully controlled and any two performances will be substantially the same" (Rosner and Wolverton 2001) although, according to another writer, this technique is essentially the same as that later used by Witold Lutosławski (Fisher 2010). Depending on the vehemence of the technique, Hovhaness’s published scores annotate these sections variously, for example as “Free tempo / humming effect” (Hovhaness: Lousadzak, p.3) and “Repeat and repeat ad lib, but not together” (Meditation on Orpheus, p.2).
In Europe, following the introduction of the expression "aleatory music" by Meyer-Eppler, the French composer Pierre Boulez
Pierre Boulez
Pierre Boulez is a French composer of contemporary classical music, a pianist, and a conductor.-Early years:Boulez was born in Montbrison, Loire, France. As a child he began piano lessons and demonstrated aptitude in both music and mathematics...
was largely responsible for popularizing the term (Boulez 1957), using it to describe works that give the performer certain liberties with regard to the sequencing and repetition of parts. The French composer and broadcaster Pierre Schaeffer
Pierre Schaeffer
Pierre Henri Marie Schaeffer was a French composer, writer, broadcaster, engineer, musicologist and acoustician of the 20th century. His innovative work in both the sciences —particularly communications and acoustics— and the various arts of music, literature and radio presentation after the end...
developed the term jeu (French for play) in reference to a technique of allowing random sounds to enter into a musical composition.
Other early European examples of aleatory music include Klavierstück XI
Klavierstücke (Stockhausen)
The Klavierstücke constitute a series of nineteen compositions by German composer Karlheinz Stockhausen.Stockhausen has said the Klavierstücke "are my drawings"...
(1956) by Karlheinz Stockhausen
Karlheinz Stockhausen
Karlheinz Stockhausen was a German composer, widely acknowledged by critics as one of the most important but also controversial composers of the 20th and early 21st centuries. Another critic calls him "one of the great visionaries of 20th-century music"...
, which features 19 elements to be performed in a sequence to be determined in each case by the performer (Boehmer 1967, 72). A form of limited aleatory was used by Witold Lutosławski (beginning with Jeux Vénitiens in 1960–61) (Rae 2001), where extensive passages of pitches and rhythms are fully specified, but the rhythmic coordination of parts within the ensemble is subject to an element of chance. Lutosławski calls these sections 'ad libitum'. In some works by Krzysztof Penderecki
Krzysztof Penderecki
Krzysztof Penderecki , born November 23, 1933 in Dębica) is a Polish composer and conductor. His 1960 avant-garde Threnody to the Victims of Hiroshima for string orchestra brought him to international attention, and this success was followed by acclaim for his choral St. Luke Passion. Both these...
characteristic sequences are repeated quickly, producing a kind of oscillating sound.
There has been considerable confusion of the terms aleatory and indeterminate/chance music. One of Cage's pieces, HPSCHD, (see also his books of changes for more musical examples) itself composed using chance procedures, uses music from Mozart's Musikalisches Würfelspiel, referred to above, as well as original music. Still, both the aesthetic aims as well as the number of elements controlled by chance make the two methods clearly different.
The First Symphony of Alfred Schnittke
Alfred Schnittke
Alfred Schnittke ; November 24, 1934 – August 3, 1998) was a Russian and Soviet composer. Schnittke's early music shows the strong influence of Dmitri Shostakovich. He developed a polystylistic technique in works such as the epic First Symphony and First Concerto Grosso...
uses aleatory techniques as only one of a number of approaches to the 'chaos' of 20th century life (Schnittke also uses Ivesian
Charles Ives
Charles Edward Ives was an American modernist composer. He is one of the first American composers of international renown, though Ives' music was largely ignored during his life, and many of his works went unperformed for many years. Over time, Ives came to be regarded as an "American Original"...
dissonance to similar effect).
"Open form" chance music
Open form is a term sometimes used for mobile or polyvalent musical formMusical form
The term musical form refers to the overall structure or plan of a piece of music, and it describes the layout of a composition as divided into sections...
s, where the order of movements
Movement (music)
A movement is a self-contained part of a musical composition or musical form. While individual or selected movements from a composition are sometimes performed separately, a performance of the complete work requires all the movements to be performed in succession...
or section
Section (music)
In music, a section is "a complete, but not independent musical idea". Types of sections include the introduction or intro, exposition, recapitulation, verse, chorus or refrain, conclusion, coda or outro, fadeout, bridge or interlude...
s is indeterminate or left up to the performer
Musician
A musician is an artist who plays a musical instrument. It may or may not be the person's profession. Musicians can be classified by their roles in performing music and writing music.Also....* A person who makes music a profession....
. Roman Haubenstock-Ramati
Roman Haubenstock-Ramati
Roman Haubenstock-Ramati was a composer and music editor who worked in Kraków, Tel Aviv and Vienna.Haubenstock-Ramati studied composition, music theory, violin and philosophy in Kraków and Lemberg from 1937 to 1940. Among his teachers were Artur Malawski and Józef Koffler. From 1947 to 1950 he was...
composed a series of influential "mobiles" such as Interpolation (1958).
However, "open form" in music is also used in the sense defined by the art historian Heinrich Wölfflin
Heinrich Wölfflin
Heinrich Wölfflin was a famous Swiss art critic, whose objective classifying principles were influential in the development of formal analysis in the history of art during the 20th century. He taught at Basel, Berlin and Munich in the generation that raised German art history to pre-eminence...
(Renaissance und Barock, 1888) to mean a work which is fundamentally incomplete, represents an unfinished activity, or points outside of itself. In this sense, a "mobile form" can be either "open" or "closed". An example of a closed mobile musical composition is Stockhausen's Zyklus
Zyklus
Zyklus für einen Schlagzeuger is a composition by Karlheinz Stockhausen, assigned Number 9 in the composer's catalog of works. It was composed in 1959 at the request of Wolfgang Steinecke as a test piece for a percussion competition at the Darmstadt Summer Courses, where it was premièred on 25...
(1959). Terry Riley
Terry Riley
Terrence Mitchell Riley, is an American composer intrinsically associated with the minimalist school of Western classical music and was a pioneer of the movement...
's In C
In C
In C is a semi-aleatoric musical piece composed by Terry Riley in 1964 for any number of people, although he suggests "a group of about 35 is desired if possible but smaller or larger groups will work"...
(1964) was composed of 53 short sequences; each member of the ensemble can repeat a given sequence as many times as desired before going on to the next, making the details of each performance of In C unique. However, because the overall course is fixed, it is a closed form.
Film music
In contemporary film music, often aleatoric composition techniques are used to increase the sense of drama and urgency. The composer usually indicates in the score what level of freedom the performer has and gives indications as to the required pitch and rhythm of the aleatoric passage. Examples of extensive aleatoric writing can be found in John WilliamsJohn Williams
John Towner Williams is an American composer, conductor, and pianist. In a career spanning almost six decades, he has composed some of the most recognizable film scores in the history of motion pictures, including the Star Wars saga, Jaws, Superman, the Indiana Jones films, E.T...
' score for the film Images, but also notably in his more recent score for Steven Spielberg's Jurassic Park: The Lost World. Other film composers using this technique are Mark Snow
Mark Snow
Mark Snow is an American composer for film and television.Born in New York, he grew up in Brooklyn, graduating from the High School of Music and Art and, afterwards, the Juilliard School of Music...
(X-Files: Fight the Future), Jerry Goldsmith
Jerry Goldsmith
Jerrald King Goldsmith was an American composer and conductor most known for his work in film and television scoring....
and others (Karlin and Wright 2004, 430–36).
See also
- Aleatoricism
- Algorithmic music
- Generative musicGenerative musicGenerative music is a term popularized by Brian Eno to describe music that is ever-different and changing, and that is created by a system.- Theory :There are four primary perspectives on "Generative Music" Generative music is a term popularized by Brian Eno to describe music that is ever-different...
- Stochastic music
- Indeterminacy in musicIndeterminacy in musicIndeterminacy in music, which began early in the twentieth century in the music of Charles Ives, and was continued in the 1930s by Henry Cowell and carried on by his student, the experimental music composer John Cage beginning in 1951 , came to refer to the movement which grew up around Cage...
Sources
- Boehmer, KonradKonrad BoehmerKonrad Boehmer is a Dutch composer and writer of German birth.Boehmer was born in Berlin. His music reflects his Marxist political agenda, which is made explicit in many of his writings from the late 1960s and 1970s...
. 1967. Zur Theorie der offenen Form in der neuen Musik. Darmstadt: Edition Tonos. (Second printing 1988.) - Boulez, PierrePierre BoulezPierre Boulez is a French composer of contemporary classical music, a pianist, and a conductor.-Early years:Boulez was born in Montbrison, Loire, France. As a child he began piano lessons and demonstrated aptitude in both music and mathematics...
. 1957. "Aléa". Nouvelle Revue française, no. 59 (1 November). Reprinted in Pierre Boulez, Relevés d’apprenti, collected and presented by Paule Thévenin, 41–45. Paris: Éditions du Seuil, 1966. ISBN 2020019302. English as "Alea" in Pierre Boulez, Notes of an Apprenticeship, collected and presented by Paule Thévenin, translated from the French by Herbert Weinstock, 35–51. New York: Alfred A. Knopf, 1968. New English translation, as "Alea", in Pierre Boulez, Stocktakings from an Apprenticeship, collected and presented by Paule Thévenin, translated from the French by Stephen Walsh, with an introduction by Robert Piencikowski, 26–38. Oxford: Clarendon Press; New York: Oxford University Press, 1991. ISBN 0193112108. - Fisher, Lynn. 2010. "An Unlikely Musical Pioneer? Early Aleatory Counterpoint". AleaCounterpoint blog site (Accessed 22 June 2010).
- Griffiths, Paul. 2001. "Aleatory". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley SadieStanley SadieStanley Sadie CBE was a leading British musicologist, music critic, and editor. He was editor of the sixth edition of the Grove Dictionary of Music and Musicians , which was published as the first edition of the New Grove Dictionary of Music and Musicians.Sadie was educated at St Paul's School,...
and John TyrrellJohn Tyrrell (professor of music)John Tyrrell was born in Salisbury, Southern Rhodesia in 1942. He studied at the universities of Cape Town, Oxford and Brno. In 2000 he was appointed Research Professor at Cardiff University....
. London: Macmillan Publishers. - Hovhaness, Alan. 1944. Lousadzak Op.48, 3. New York: Peer International Corporation; 1958. Meditation on Orpheus Op.155, 2. New York: C. F. Peters Corporation.
- Jacobs, Arthur. 1966. "Admonitoric Note". The Musical Times 107, no. 1479 (May): 414.
- Karlin, Fred, and Rayburn Wright. 2004. On the Track: A Guide to Contemporary Film Scoring, second edition. New York: Routledge. ISBN 0415941350 (cloth); ISBN 0415941369 (pbk).
- Lieberman, David. 2006. "Game Enhanced Music Manuscript." In GRAPHITE '06: Proceedings of the 4th International Conference on Computer Graphics and Interactive Techniques in Australasia and South East Asia, Kuala Lumpur (Malaysia), November 29–December 2, 2006, edited by Y. T. Lee, Siti Mariyam Shamsuddin, Diego Gutierrez, and Norhaida Mohd Suaib, 245–50. New York: ACM Press. ISBN 1-59593-564-9.
- Lotringer, Sylvère. 1998. "Duchamp Werden". In Crossings: Kunst zum Hören und Sehen [exhibition catalogue]), 55-61. Vienna: Kunsthalle
- Meyer-Eppler, WernerWerner Meyer-EpplerWerner Meyer-Eppler , was a German physicist, experimental acoustician, phoneticist, and information theorist....
. 1957. "Statistic and Psychologic Problems of Sound", translated by Alexander Goehr. Die ReiheDie ReiheDie Reihe was a German-language music journal, edited by Herbert Eimert and Karlheinz Stockhausen and published by Universal Edition between 1955 and 1962 . An English edition was published, under the original German title, between 1957 and 1968 by the Theodore Presser Company , in association...
1 ("Electronic Music"): 55–61. Original German edition, 1955, as "Statistische und psychologische Klangprobleme", Die Reihe 1 ("Elektronische Musik"): 22–28. - Pimmer, Hans. 1997. Würfelkomposition: zeitgenössische Recherche: mit Betrachtungen über die Musik 1799. Munich: Akademischer Verlag. ISBN 3929115905.
- Prendergast, Mark J. 2000. The Ambient Century: from Mahler to Trance: The Evolution of Sound in the Electronic Age. London: Bloomsbury. ISBN 0747542139.
- Rae, Charles Bodman. 2001. "Lutosławski, Witold (Roman)". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
- Randel, Don Michael. 2002. The Harvard Concise Dictionary of Music and Musicians. ISBN 0-674-00978-9.
- Rosner, Arnold, and Vance Wolverton. 2001. "Hovhaness [Hovaness], Alan [Chakmakjian, Alan Hovhaness]". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley SadieStanley SadieStanley Sadie CBE was a leading British musicologist, music critic, and editor. He was editor of the sixth edition of the Grove Dictionary of Music and Musicians , which was published as the first edition of the New Grove Dictionary of Music and Musicians.Sadie was educated at St Paul's School,...
and John TyrrellJohn Tyrrell (professor of music)John Tyrrell was born in Salisbury, Southern Rhodesia in 1942. He studied at the universities of Cape Town, Oxford and Brno. In 2000 he was appointed Research Professor at Cardiff University....
. London: Macmillan Publishers. - Stone, Susan. 2005. "The Barrons: Forgotten Pioneers of Electronic Music", NPR Music (7 February). (Accessed 23 September 2008)
- Wölfflin, HeinrichHeinrich WölfflinHeinrich Wölfflin was a famous Swiss art critic, whose objective classifying principles were influential in the development of formal analysis in the history of art during the 20th century. He taught at Basel, Berlin and Munich in the generation that raised German art history to pre-eminence...
. 1888. Renaissance und Barock: Eine Untersuchung über Wesen und Entstehung der Barockstils in Italien. Munich: T. Ackermann. English edition: Renaissance and Baroque. Translated by Kathrin Simon, with an introduction by Peter Murray. London: Collins, 1964; Ithaca: Cornell University Press, 1967.
External links
- Art of the States: aleatoric aleatoric musical works by American composers
- Mozart's Musikalisches Würfelspiel - online version of Mozart's dice game
- Encyclopaedia Britannica Aleatoric music
- http://www.sciencenews.org/20010901/mathtrek.asp
- http://www.anigraphical.com/
- An exploration of the stochastic randomness in modern electronic music
- John Cage's Indeterminacy
- A Visual Interpretation of Indeterminacy: Found Photos Paired with One-minute Short Stories
- BIT BYTE BEAT Make aleatoric music with your heartbeat
- Video of an Aleatoric Musical Instrument