Aleatoric music
Aleatoric music (also aleatory music or chance music; from the Latin word alea, meaning "dice") is music in which some element of the composition is left to chance 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 Darmstadt International Summer Courses for New Music in the beginning of the 1950s. According to his definition, "a process is said to be aleatoric (from Lat. alea=dice) if its course is determined in general but depends on chance in detail" (Meyer-Eppler 1957, 55).

# Early precedents
An early genre of composition that could be considered a precedent for aleatoric compositions were the Musikalische Wurfelspiele or Musical Dice Games, 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.
Somewhat related to chance music is composer Alan Hovhaness's use of what he called the "spirit murmur" (beginning with his 1944 piano concerto Lousadzak), in which instruments repeat a melodic or rhythmic phrase for a certain amount of time in an uncoordinated fashion. Similar textures had been employed by Charles Ives, as early as The Unanswered Question (1908).
American composer John Cage's Music of Changes (1951) is the first piece to be conceived largely through random procedures (Randel 2002, 17), though for just this reason his indeterminacy is of a different order from Meyer-Eppler's concept.

# Modern usage
The French composer Pierre Boulez was largely responsible for popularizing the term, using it to describe works that give the performer certain liberties with regard to the sequencing and repetition of parts, an approach pioneered by avant-garde American composer-theorist Henry Cowell in his Mosaic Quartet (String Quartet No. 3, 1935). The term was intended by Boulez to distinguish his work from pieces composed through the application of chance operations by John Cage and Cage's aesthetic of indeterminate music or indeterminacy.
Early examples of aleatoric music include Klavierstuck XI (1956) by Karlheinz Stockhausen, which features 19 elements to be performed in changing sequences; certain orchestral works of  Witold Lutosawski (from after 1959) which contain passages where the musical content is not precisely dictated (Lutosawski calls this 'ad libitum'); and 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, itself composed using chance procedures, uses music from Mozart's Musikalisches Wurfelspiel, referred to above, as well as original music. He also generally used coin-tossing and other procedures depending on designs involving a pre-defined number of choices to be made. Still, both the aesthetic aims as well as the number of elements controlled by chance make the two methods clearly different.

# "Open form" chance music
Open form is a term sometimes used for mobile or polyvalent musical forms, where the order of movements or sections 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 Wolfflin (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 Momente (1962-64/69). 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 he or she chooses before going on to the next (similar to Hovhaness's "spirit murmer", only with a fixed pulsing rhythm), making the details of each performance of In C unique though, because the overall course is fixed, it is a closed form.